Citation

Material Information

Title:
Coordinatized Hjelmslev planes
Hjelmslev planes
Creator:
Bacon, Phyrne Youens, 1936-
Publication Date:
1974
Language:
English
Physical Description:
xxxiii, 267 leaves. : ; 28 cm.

Subjects

Subjects / Keywords:
Automorphisms ( jstor )
Functors ( jstor )
Geometric lines ( jstor )
Geometric planes ( jstor )
Homomorphisms ( jstor )
Isomorphism ( jstor )
Morphisms ( jstor )
Parallel lines ( jstor )
Property lines ( jstor )
Dissertations, Academic -- Mathematics -- UF
Geometry, Projective ( lcsh )
Mathematics thesis Ph. D
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 263-266.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
022774601 ( AlephBibNum )
14074879 ( OCLC )

Full Text

COO2DIINATIZED HJSLZ'SLEV LANES

By

PHYRNE YOUENS BACON

A DISSLRTATIO;N -RLSE:' D TC :EE GRADULI.TE COUNCIL, OF
THE UNIVERSITY CF- ELCiz']. A
IN PARTIAL FULFILlMENT OF THE RE-UI1RE: ::TS FOR THE
DEGREE OF DOCTOR OF t'HILOSOrHY

UNIVERSITY OF FLORIDA

1974

Copyright 1974 by Phyrne Youens Bacon

To my husband, Philip Bacon, to my-mother, Cynthia Tanner

Youens, and to the memories of my father, Willis George Youens,

Sr., M.D., and my maternal grandmother, Phyrne Claiborne Tanner.

The white people must think paper has some mysterious

power to help them on in the world. The Indian needs no

writings; words that are true sink deep into his heart where

they remain; he never forgets them.

Four Guns, Oglala Sioux (1891)

quoted in "I Have Spoken" compiled

by Virginia Irving Armstrong (1971),

The Swallow Press Inc., Chicago,

pages 130-131.

ACKNO'WLEL DGEMENTS

I would like to express my thanks and appreciation to

my advisor, David A. Drake, for his superior example, for his

contagious enthusiasm, and for his excellent suggestions.

I would like to thank the other members of my committee,

Ernest E. Shult, Kermit N. Sigmon, Mark L. Teply, Mark P. Hale,

Jr., and Billy Thomas, for their comments and suggestions.' I

would also like to thank George E. Strecker for his comments.

I would like to thank Benno Artmann, N. D. Lane, and

Willian J. LeVeque for having indirectly provided copies of

papers referenced in my bibliography, [Cyganova (1967)],

[Lorimer (1971)] and CSkornjakov (1964)] respectively, and I

would like to thank Wladimiro Scheffer for his translation of

[Cyganova (1967)].

I also wish to thank the many professors and fellow

students who have helped make my graduate study an interesting

and challenging experience. I would especially like to thank

Richard D. Present, William M. Bugg, W. Edward Deeds, E. G.

Harris, Harold C. Schweinler, Don D. Miller, David R. Hayes,

John G. Moore, Dennison R. Brown, George E. Strecker, W.

Edwin Clark, and Charles I. Babst.

ACKNOWLEDGEM!E NTS V

LIST OF DEFINED TERM-S viii

LIST OF FUNCTORS xvii

LIST OF SPECIAL NAPS xx

KEY TO CATEGORIES xxii

KEY TO SYMBOLS xxv

ABSTRACT xxxi

Sections

1. INTRODUCTION 1

2. HJELMSLEV PLANES 7

3. BITERNARY RINGS 44

4. SEMITRANSLATIONS AND GEOMETRY 69

5. SEMITRANSLATIONS AND ALGEBRA 86

6. PREQUASIRINGS AND QUASIRIT:GS 106

7. KERNELS OF QUASIRINGS 116

8. OTHER CENTRAL AXIAL AUTOMORPHIS:S 123

9. AH-RINGS 139

10. HJELMSLEV STRUCTURES 150

11. DESARGUECIAN PH-PLA.ES 199

12. PAPPIAN CONFIGURATIONS 208

Appendices

A. RESTRICTED BITERNARY RINGS 222

Appendices continued

B. QUASICONGRUENCES 243

BIBLIOGRAPHY 263

BIOGRAPHICAL SKETCH 267

LIST OF DEFINED TERMS

Term Subsection

natural isomorphism, functor 2.1

function, graph, compose, identity function,

underlying set, concrete morphism,

underlying set function, natural composition,

natural identity morphism, identity,

concrete category, homomorphisms, map, map 2.2

generated by, generated by, of, with 2.3

surjective, injective, bijective 2.4

equivalence, isomorphism 2.5

reciprocal equivalences 2.6

preserve, reflect 2.7

incidence structure, points, lines,

incidence relation, is incident with, lies on,

is a point of, is on, goes through,

is a line through, join, collinear, copunctal 2.8

incidence structure homomorphism 2.9

the category of incidence structures 2.10

the incidence structure induced from S by -,

the induced incidence structure 2.12

projective plane, ordinary projective plane 2.14

parallel, affine plane, ordinary affine plane 2.15

viii

LIST OF DEFINED TERS.S continued

Term Subsection

neighbor, projectively neighbor,

projectively neighbor, projective Hjelmslev plane,

PH-plane 2.17

projective Hjelmslev plane homomorphism,

the category of projective Hjelmslev planes,

the category of projective planes 2.19

parallel, affinely neighbor, affinely neighbor,

affine Hjelmslev plane, AH-plane 2.21

affine Hjelmslev plane homomorphism,

the category of affine Hjelmslev planes,

the category of affine planes 2.23

the gross structure, the neighbor map,

the gross structure, the neighbor map 2.25

is neighbor to, is not neighbor to,

is not parallel to 2.26

non-degenerate, degenerate 2.30

direction 2.35

quasiparallel, quasiparallel, quasiparallel 2.37

lined incidence structure, base line, affine points,

affine line, lined incidence structure homomorphism,

the category of lined PH-planes,

the category of lined projective planes,

the category of lined incidence structures 2.39

the generalized incidence structure,

the lined generalized incidence structure, 2.40

LIST OF DEFINED TERMS continued

Term Subsection

generalized point, generalized line,

generalized incidence structure, line, point,

incidence structure 2.40

neighbor, generalized neighbor relation,

neighbor relation, neighbor, neighbor relation,

generalized neighbor relation 2.43

derived from, derived from,

derived from H by use of, derived, extended to,

extended to, extensions of 2.51

embedding, incidence structure embedding of S

into S' 2.52

generalized AH-plane,

generalized AH-plane homomorphism,

the category of generalized AH-planes 2.58

AH-plane embedding 2.62

ternary field, zero, one, symbols,

ternary field homomorphism,

the category of ternary fields 3.1

neighbor, not neighbor, biternary ring,

right zero divisor 3.2

dual 3.3

symbols, zero, one, biternary field -3.4

biternary ring homomorphism 3.9

the category of biternary rings,

the category of biternary fields 3.10

x

LIST OF DEFINED TERMS continued

Term Subsection

coordinatized AH-plane, coordinatization,

symbols, y-coordinate, x-coordinate,

representation, representation, representations,

x-axis, v-axis, origin, unit point 3.11

xy-duals 3.12

coordinatized AH-plane homomorphism,

coordinatization homomorphism, neighbor map 3.14

the category of coordinatized affine Hjelmslev

planes, the category of coordinatized affine plnes 3.15

dual 3.20

the biternary field associated with 3.29

generated by,

the AH-plane generated by a biternary ring 3.31

dilatation, trace, semitranslation,

semitranslation with direction P, translation 4.1

(P,g)-endomorphism, central axial endomorphism,

center, axis, (P,g)-endomorphism,

central axial endomorphism, center, axis 4.3

neighbor, neighbor endomorphisms 4.6

j-angle, vertices, sides, triangle, f-related,

(r,g,)-related, (P,g.)-j-Desarguesian,

(rP,g)-H-Desarguesian 4.10

the canonical expansion of a (j-l)-angle to

* j-angle 4.12

(r',)-mimetic 4.13

LIST OF DEFINED TERMS continued

Term Subsection

(P,g)-transitive 4.17

order, infinite order 4.20

T'-multiplication, linear, linear 5.1

(k)-regular for s, (k)-regular,

(k)'-regular for s, (k)'-regular,

axially regular, regular 5.10

C-regular, regular, axially regular,

regular in the direction r 5.13

the category of axially regular biternary rings,

the category of axially regular coordinatized

AH-planes 5.21

the category of coordinatized translation AH-planes,

the category of regular biternary rings 5.26

quasifield, zero, one 6.2

second multiplication, prequasiring,

right zero divisor, zero, one 6.3

quasiring 6.5

dual 6.6

skew quasiring 6.7

prequasiring homomorphism 6.11

the category of prequasirings,

the category of quasirings 6.12

generated by, 6.19

LIST OF DEFINED TER5S continued

Term Subsection

the AH-plane generated by a prequasiring 6.19

biquasifield, the category of biquasifields,

the category of coordinatized translation affine

planes 6.20

kernel, the kernel of a quasiring 7.1

local ring 7.2

trace preserving, a trace preserving endomorphism

of the translation group 7.5

left modular for s, left modular,

strongly left modular, T'-weakly left modular for s 8.1

strongly (P,g.)-transitive 8.6

P-related, (P,g.)-related, (P,g,)-H-Desarguesian,

strongly (P,g,)-H-Desarguesian 8.8

((0),[0,0]')-normal for s, ((0),[0,0]')-normal,

T'-weakly ((0),[0,01)-normal for s,

T-weakly ((0),10,03')-normal for s 8.10

((0)',[0,01')-normal for s, ((0)',[0,01')-normal 3.15

affine Hjelmslev ring, AH-ring, Hjelmslev ring,

H-ring 9.1

AH-ring homomorphism, the category of AH-rings 9.2

kernel quasiring, the category of kernel quasirings 9.3

Desarguesian, the category of coordinatized

Desarguesian AH-planes 9.6

the AH-plane generated by an AH-ring,

generated by 9.8

xiii

LIST OF DEFINED TERMS continued

Term Subsection

Hjelmslev Desarguesian,

strongly Hjelmslev Desarguesian 9.9

the category of division rings 9.12

the category of coordinatized Desarguesian

affine planes 9.13

the Hjelmslev structure of an AH-ring S 10.3

near, near, near, near 10.6

the lined affine Hjelmslev structure of S 10.9

extended to, extended to (H(S),rgQ) through ,

extended to, extended to H(S) through 10.10

projectively Desarguesian 10.16

Desarguesian 10.17

(P,g)-automorphism, (P,g)-transitive 10.19

full j-vertex 10.26

Hjelmslev structure, near,

Hjelmslev structure homomorphism, gross structure,

neighbor map, the category of lined Hjelmslev

structures 10.36

full, a full lined Hjelmslev structure homo-

morphism 10.39

basis triple 10.46

Klingenberg coordinatization .10.47

Klingenberg coordinatization 10.49

induces % through k and X',

is induced through X and X' by a, extension, 10.52

xiv

LIST OF DEFINED TERI;S continued

Term Subsection

extension of v through X and X' 10.52

the canonical basis triple,

canonical coordinatization, basis triple 10.59

the category of Desarguesian AH-planes with

non-degenerate AH-plane hcmonorphisms 10.63

the category of lined Hjelmslev structures with

full lined Hjelmslev structure homomorphisms 10.66

the category of projectively Desarguesian

AH-planes with non-degenerate AH-plane hononorph-

isms, the category of lined Desarguesian rH-planes

with full lined Desarguesian PH-plane homomorphisms 10.67

triangle, vertex, side, full 11.3

Hjelmslov Desarguesian 11.5

(L, 1g,g',Pl',P)-Pappian configuration,

Pappian configuration, (A,l1g,g',p?',P)-Pappian,

Pappian, (g,g',P ',P",g")-Pappian, Pappian,

Pappian for the full triangle J 12.1

restricted biternary ring, right zero divisor,

symbols, zero, one A.1

restricted biternary ring homomorphism,

restricted biternary field, the category of

restricted biternary rings, the category of

restricted biternary fields A.6

the category of quasifields A.30

components, precongruence B.1

LIST OF DEFINED TERKS continued

Term Subsection

congruence B.2

semicongruence, projection map, quasicongruence B.3

quasicongruence homcmorphism B.4

pointed AH-plane, base point,

pointed AH-plane homomorphism, the category of

pointed translation AH-planes, the category of

pointed translation affine planes B.8

parallel B.9

quasicongruence cocrdinatization,

the canonical base point, the canonical quasi-

congruence coordinatization B.19

the category of translation AH-planes with

non-degenerate AH-plane homomorphisms B.20

LIST OF FUNCTCRS

Functor

G:A S

G :A H S
-g 9
H* :A -

-9 9

A:H A

:A -- A

A :A -- A

2:C -2 B2
B:r -T r

B' :* -4 B'

C:B2 --
v,: --. F'

C_* :F' --
-F

22
:C 2 2
B:0 -* V
-2
E':V ->. 8

V -

Subsection

2.40

2.40

2.42

2.48

2.47

2.59

2.59

3.16

3.18

3.28

3.28

3.29

3.29

5.21

5.21

5.26

5.26

6.1

6.14

6.16

6.16

xvii

LIST OF FUNCTORS continued

Functor

Q:C --Q
T

G':Q T

R*: C-* *
QY: Dn -- T
T T
2 s
K2 :(C) L

D,:(T s) -L

X:Q -* R
K
X':R --

R:C R

R':R -- D

D
R.:C -- R,

R*':R' C
D
H:A -4

Y:(B)n -48

D:((A ) )f (D)n

Sg
J :K A

J:K T
AP
K:Tp -* K

K :(T) -.

Z:B .Z

~ Z

CZ :Z -*Z C

^:z

Subsection

6.18

6.18

6.20

6.20

7.6

7.6

9.4

9.4

9.11

9.11

9.16

9.16

10.2

10.63

10.66

10.66

B.9

B.9

B.14

B.20

A.10

A.8

A.13

A.17

xviii

LIST OF FUNCTORS continued

Functor

Z .:F* -+ Z.

P *:Z* ?*
~Z

Subsection

A.7

A.7

LIST OF SPECIAL MAPS

Symbol In use Subsection

f w = (A,A',fQ) 2.2

K K:A A* 2.25

K K:H -- H* 2.25

TT TT(g) 2.35

g g(g ) 2.40

P P(TT) 2.40

S S(H,g) 2.47

T T(x,m,b) 3.2

T' T'(y,u,v) 3.2

; :OE -- M 3.11

+ a + b 5.1

x ab = ab 5.1

* a b 5.1

a.b 5.1

Z Z(x,m,a) 5.4

Z' Z'(y,u,b) 5.4

i rx. 10.2
1

i yis 10.2

G G(H(S),h) 10.9

G G 10.9

Kern Kern Q 7.1

xx

LIST OF SPECIAL MAPS continued

Symbol

t

z

ij

6, #

T"

P

r

In use

Y:S -- S/N

v:H(S) -- H(S/r;)

(v,0,0)t

o-z

cZ, tZ

Z.i
13
(a- H(L))t, (tr aj(C)) #
za
GA

3s
T"(y,u,v)

P
A
r(x,x',x"), rx.

(y,y',y")s, yis

-b, a b

J7(gD)

(OP3)

(OA4)

(PH3), ('2)

(AH4), (LF3)

(Ell)

(QF4)

( 11)

(R5), (RH6)

(R16)

J0

(OP )

(OA)

(PH ), (1 )

(AH ), ( )

(B )

(F )

(VJ )

(R ), (RH )

(R )

Subsection

10.12

10.12

10.21

10.21

10.43

10.44

10.44

10.53

10.59

10.62

A.1

B.19

10.2

10.2

5.3

B.9

2.14

2.15

2.17

2.21

3.2

6.2

6.3

9.1

A.1

KEY TO CATEGORIES

Equivalent categories are joined by [

Category Subsection

S incidence structures 2.10

S lined incidence structures 2.39
g
-* affine planes 2.23

lined projective planes 2.39
g
H* projective planes 2.19

AAH-planes 2.23

generalized AH-planes 2.58
9
H PH-planes 2.19

H lined PH-planes 2.39
g
coordinatized AH-planes 3.15

biternary rings 3.10

Restricted biternary rings A.6

-* coordinatized affine planes 3.15

biternary fields 3.10

restricted biternary fields A.6

ternary fields 3.1

xxii

KEY TO CATEGORIES

Cateory

"2

2

-V

-B

-.C

-rT*

-6.
r

-Q F*

K

-D^

-K*

-R*

H
S
A n
(T)

S- continued

Subsection

axially regular cocrdinatized

AH-planes

axially regular biternary rings

prequasirings

coordinatized translation

AH-planes

regular biternary rings

quasirings

coordinatized translation affine

planes

regular biternary fields

biquasifields

quasifields

coordinatized Desarguesian

AH-planes

kernel quasirings

AH-rings

coordinatized Desarguesian affine

planes

kernel biquasifields

division rings

Hjelmslev structures

translation AH-planes with non-

degenerate homomorphisms

left modules

xxiii

5.21

5.21

6.12

5.26

5.26

6.12

6.20

5.30

6.20

A.29

9.6

9.3

9.2

9.13

9.14

9.12

10.36

B.20

7.5

KEY TO CATEGCRIES

Category

,(S)n

^Sgc

[T
Tp

pT
P

- continued

Subsection

Desarguesian AH-planes with non-

degenerate homomorphisms 10.63

lined Hjelmslev structures with

full lined Hjelmslev structure 10.66

homomorphisms

quasicongruences B.6

pointed translation AH-planes B.8

congruences B.6

pointed translation affine planes B.8

Sxxiv

KEY TO SYMBOLS

Symbol Subsection

neighbor 2.17

2.21

3.2

6.3

9.1

A.1

not neighbor (see above)

parallel 2.15

2.21

Snot parallel (see above)

I quasiparallel 2.37

H not quasiparallel (see above)

near 10.6

4 not near (see above)

c~, r, TT automorphism 2.1

& a trace preserving endomorphism

of the translation group 7.4

p* the neighbor class of directions

which contains f 2.43

2.45

TT, A,, r direction 2.35

XXV

KEY TO SYMBCLS continued

Symbol

the direction containing

the line g

semitranslation

translation

W, w', any small

Greek letter

0

1

(CP,o,I)

A, A', (S,I\)

homomorphism

zero

one

the set of lines

the set of points

incidence structure

AH-plane

xxvi

TV(g)

0 .

r, T 1 (a, b)

Subsection

2.35

4.1

4.1

7.4

B.14

2.2

3.1

3.2

3.11

6.2

6.3

9.1

3.1

3.2

3.11

6.2

6.3

9.1

2.8

2.8

2.8

2.21

KEY TO SYMBOLS continued

Symbol

A* the gross structure of A

A" affine plane

(A,K), C, C' coordinatized AH-plane

AB the AH-plane generated by a

biternary ring B

the AH-plane generated by a

prequasiring V

the AH-plane generated by an

AH-ring S

B, B', (M,T,T')

C, C', (A,K)

D

Dr

E

E (A), E

biternary ring

coordinatized AH-plane

set of semitranslations

set of semitranslations with

direction V

unit Doint

the ring of trace preserving

endomorphisms of the translation

group of A

xxvii

Subsection

2.25

2.15

3.11

3.18

3.31

6.19

3.31

3.18

9.8

6.19

3.31

3.18

3.2

3.11

4.22

4.22

3.11

7.4

7.6

KEY TO SYMBOLS

Symbol

g, h, k

g9

g*, h', k*

gx

gy

H, H'

H*

H*

(H,g)

(H*,g')

I

K

K, (O,K)

K*, (o*,K')

(m)

[m,dl

mef
M, M*

- continued

Subsection

line 2.8

the class of lines neighbor to g 2.17

2.21

2.43

line of an affine or projective

plane

set of directions

x-axis

y-axis

PH-plane

the gross structure of H

projective plane

lined PH-plane

lined projective plane

incidence relation

coordinatization

quasicongruence

congruence

direction containing [m,01

a line not quasiparallel to gy

side of a j-angle

the set of symbols

xxviii

2.14

2.15

2.35

2.40

2.43

3.11

3.11

2.17

2.25

2.14

2.39

2.39

2.8

3.11

B.4

B.2

5.9

3.11

4.10

3.1

KEY TO SYMBOLS continued

Symbol

M, KY the set oA

(M,T,T')

(M/~,T")

(M*,T*), F'

(M,+,X,-)

(M,+,X, .)

(N,T,T"), R

N, NB

NV, N

NS, N

0

P, Q, R, S

P.

P", Q"

P.

(Pi; mef)

Q, P, R, S

Q*

Su

f symbols continued

biternary ring

induced ternary field

ternary field

prequasiring

quasiring

restricted biternary ring.

right zero divisors

right zero divisors

right zero divisors

origin

point

the class of points neighbor to

point of an affine or projective

plane

vertex of a j-angle

j-angle

point

the class of points neighbor to Q

xxix

absection

3.2

6.2

6.3

9.1

3.2

3.2

3.1

6.3

6.5

A.1

3.2

6.3

9.1

3.11

2.8

P 2.17

2.21

2.43

2.14

2.15

4.10

4.10

2.8

2.17

KEY TO SYMBOLS continued

Symbol

Q" the class of points neighbor

to Q continued

Q', P*

Q, (M,+,x,.)

R, (R,+,X)

R, (M,T,T")

(S,g)

S, R, (S,+,X)

S, (9,S)

T, T'

T"

T, (A,P)

(u) '

[u,v],

V, (M,+,X,-)

W

(w,o)

x, a, c

y, b, d

Subsection

point of an affine or projective

plane

quasiring

H-ring, AH-ring

restricted biternary ring

lined incidence structure

AH-ring

semicongruence

ternary operation

partial ternary operation

pointed translation AH-plane

direction containing [u,03'

line not quasiparallel to gx

prequasiring

the set of translations

the group of translations.

the x-coordinate of a point

the y-coordinate of a point

XXX

2.21

2.43

2.14

2.15

6.5

9.1

A.1

2.39

9.1

B.3

3.2

A.1

B.8

5.9

3.11

6.3

4.22

4.22

5.24

7.4

3.11

3.11

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in partial Fulfillment of the
Requirements for the Degree of Doctor of philosophyy

COORDINATIZED HJEL!SLEV PLANES

By

Phyrne Youens Bacon

June, 1974

Chairman: David A. Drake

r'ajor Department: Iathematics

A coordinatization may be thought of as an isomorphism

between a geometric structure and a geometric structure which has

been constructed from an algebraic structure. Affine Hjelmslev

planes (AH-planes) are coordinatized by using biternary rings;

translation AH-planes by using quasirings; Desarguesian AH-planes

by using AH-rings; and Desarguesian projective Hjelmslev planes

(Desarguesian PH-planes) by using H-rings.

An affine plane homomorphism <:A A' is an incidence

structure homomorphism which preserves the parallel relation.

If- is non-degenerate (that is, it does not map all the points

of A into points of a single line of A') then S is injective and

reflects the incidence and parallel relations. An AH-plane

homomorphism w:A -A A' is an incidence structure homomorphism

which preserves the parallel and neighbor relations. If w is

non-degenerate (that is, it does not map all the points of A

into points of A' which are all neighbor to points on some line

k' of A'), then w reflects the quasiparallel and the neighbor

relations.
Sxxxi

If attention is restricted to AH-plane homomorphisms

which take x-axis to x-axis, y-axis to y-axis and unit point to

unit point, then the following pairs of categories are equivalent:

biternary rings and coordinatized AH-planes, quasirings and

coordinatized translation AH-planes, AH-rings and coordinatized

Desarguesian AH-planes.

The category of quasicongruences is equivalent to the

category of pointed translation AH-planes, and the category of

Desarguesian AH-planes with non-degenerate AH-plane homomorph-

isms is equivalent to the category of lined Hjelmslev structures

with full lined Hjelsmlev structure homomorphisms. Desarguesian

PH-planes are Hjelmslev structures in which every two lines meet.

The directions of the x-axis and the y-axis are denoted by

(0) and (0)'. The translations of a coordinatized AH-plane C are

((0),g.)- and ((0)',g )-transitive if and only if the biternary

ring (M,T,T') of C satisfies the following conditions:

1) T and T' are linear.

2) The T- and T'-additions are equivalent: a + b = a b for

all a,b in M; that is, T(a,l,b) = T'(a,l,b) for all a,b in M.

3) (M,+) is a group.

4) xm + sm = (x + s)m and x-m + s.m = (x + s)-m for all

x,s,m in M where the two multiplications are defined by ab =

T(a,b,0) and a-b = T'(a,b,0) for all a,b in M.

A coordinatized AH-plane whose biternary ring satisfies the

conditions listed above is a translation AH-plane if and only if

Translation AH-planes, Desarguesian AH-planes, Pappian

xxxii

translation AH-planes, Desarguesian :H-planes and Pappian

Desarguesian PH-planes are each characterized geometrically,

in terms of algebraic properties of their coordinatizations, and

(for all except the sappian planes) in terms of properties of

their endomorphisms.

Algebraic characterizations are given of those coordina-

tized AH-planes which have a semitranslation (in an affine plane

a semitranslation is a translation) with direction (k) which

moves the origin to (s,sk), and of those coordinatized AH-planes

which have a ((0,0),g,)-endomorphism which moves (1,1) to (s,s).

There are similar results concerning ((0),[0,01')- and

((0)',[0,0]')-automorphisms where [0,0]' is the y-axis and (0)'

is the direction of the y-axis.

If H is a PH-plane; if s s2, s3 are the sides of a tri-

angle whose image in the gross structure of H is non-decenerate,

and if each of the three AH-planes A,, A2, A3 derived from H by

use of one of the sides sl, s2, s3 is Desarguesian, then H is

Desarguesian. There exists a Desarguesian 'AH-plane which cannot

be derived from any Desarguesian PH-plane.

xxiii

1. INTRODUCTION

A coordinatizaticn may be thought of as an isomorphism

between a geometric structure and a geometric structure which

has been constructed from an algebraic structure. Klingenberg

[(1955)] began the solution of the coordinatization problem for

Hjelmslev planes by constructing a projective Hjelmslev plane

from an H-ring, and he showed that this constructed FH-plane has

a number of properties. Lineburg [(1962)3 defined an algebraic

structure (which is here called a quasicongruence) and showed

that any translation affine Hjelmslev plane (translation AH-plane)

can be coordinatized (in the sense mentioned above) by using a

quasicongruence. Lcrimer [(1971)3 continued work on the coordi-

natization problem by constructing an affine Hjelmslev plane

from an AH-ring and then giving various theorems relating these

constructed planes to the class of Desarguesian AH-planes.

Cyganova [(1967)3 also did considerable work on the coordi-

natization problem: she undertook to define an algebraic

structure which would have essentially the same relation to

affine Hjelmslev planes that ternary fields have to affine

planes. Unfortunately, her arguments contain a number of

serious omissions, not all of which I have been able to repair.

Her algebraic system has a ternary operation and a partial

ternary operation. In Definition A.1, a similar (but different)

algebraic system is defined which is called a restricted

biternary ring.

In Definition 3.2, an algebraic system with two (complete)

ternary operations is defined: this system is called a biternary

ring. An AH-plane homomorphism which takes x-axis to x-axis,

y-axis to y-axis and unit point to unit point is called a

coordinatization hcmomorphism (Definitions 3.11 and 3.14). In

Theorem 3.27, it is shown that the category of biternary rings is

equivalent to the category of coordinatized affine Hjelmslev

planes (with coordinatization homomorphisms). This result is

used (along with others) to show that the category of quasirings

(these have two multiplications) is equivalent to the category

of coordinatized translation AH-planes (Corollary 6.18), and

that the category of AH-rings is equivalent to the category of

coordinatized Desarguesian AH-planes (Proposition 9.11). The

quasiring equivalence is used to prove the existence of a

module isomorphism (in fact a natural transformation between

functors) which relates the kernel of a quasiring to the ring

of trace preserving endomorphisms of the translation group of

the associated AH-plane (Theorem 7.7).

Even without considering possible algebraic connections,

there is considerable interaction between geometric properties

of a Hjelmslev plane and the existence of certain types of

endomorphisms of the plane. In Theorem 4.21, a geometric

characterization of those AH-planes whose automorphisms are

(P,g.)-transitive for any given direction V is given. In

Proposition 8.9, a geometric characterization of those AH-planes

whose automorphisms are (F,g.)-transitive for any given point P

is given. These results are used (together with some algebraic

ones) to give geometric characterizations of translation AH-planes

(Theorem 5.25), of Desarguesian AH-planes (Froposition 9.10) and

of Desarguesian PH-planes (Theorem 11.6).

An AH-plane homomorphism w:A -- A' is required to preserve

the incidence, parallel and neighbor relations. An affine plane

homomorphism c:A -- A' (c is an AH-plane homomorphism between

affine planes) is said to be non-degenerate if it does not map

all the points of A onto a single line of A'. A non-degenerate

affine plane homomorphism is an incidence structure embedding

and reflects both the incidence and parallel relations (ag
implies g II h) (Theorem 2.63). If Q:A -- A' is a non-degenerate

AH-plane homomorphism (that is, e does not map all the points of

A into points of A' which are all neighbor to points on some

line k' of A'), then c reflects the quasiparallel and the

neighbor relations (Corollary 2.65).

In Theorem 5.11, an algebraic characterization is given of

those coordinatized AH-planes which have a semitranslation with

direction (k) which moves the origin to (s,sk) (in an affine

plane a semitranslation is a translation); in Proposition 8.2,

of those coordinatized AH-planes which have a ((0j0),g.)-endo-

morphism which moves the unit point to the point (s,s).

Propositions 8.11 and 8.16 give similar results for ((0),[0,0]')-

and ((0)',[0,03')-automorphisms respectively: here [0,01' is the

y-axis and (0) and (0)' are the directions of the x-axis and the

y-axis respectively.

4
Theorem 5.29 shows that the translations of a coordinatized

AH-plane C are ((0),g )- and ((0)',g )-transitive if and only if

the biternary ring (N,T,T') of C satisfies the following

conditions:

1) T and T' are linear.

2) The T- and T'-additions are equivalent: a + b = a b for

all a,b in M; that is, T(a,l,b) T'(a,l,b) for all a,b in R.

3) (M,+) is a group.

4) xm + sm (x + s)m and x.m + s.m = (x + s).n for all

x,s,m in M where the two multiplications are defined by

ab T(a,b,0) and a-b T'(a,b,0) for all a,b in M.

Theorem 5.25 shows that a coordinatized AH-plane whose

biternary ring satisfies the conditions listed above is a

translation AH-plane if and only if the addition + is abelian.

Klingenberg [(1955)1 attempted to characterize in terms of

their automorphisms those PH-planes which are isomorphic to some

PH-plane constructed from an H-ring. His argument fails however.

In Theorem 11.6, such a characterization is given. This theorem

also indicates a geometric characterization of these PH-planes

(they are called Desarguesian PH-planes).

Lorimer [(1971)3 generalizes part of what Artin C(1957)]

calls "the fundamental theorem of projective geometry" by proving

some results relating the automorphisms of a Desarguesian AH-plane

which fix the origin to a set of semilinear transformations.

Theorem 10.63 shows that there is a functor Y from the category

of Desarguesian AH-planes with non-degenerate AH-plane homo-

morphisms to the category of AH-rings; if t:A -- A' is a morphism

in the first category, and if (S,8) and(S',9') are canonical

coordinatizations of A and A', then can be defined by ')P =

(Y(p)6P)Z' for some non-singular matrix Z' with first column

(1,0,0) ; if there is an AH-ring homomorphism 4 and a non-singular

matrix Z such that p can be defined by O'lP = (?GP)Z, then 2 = eZ'

and Y is defined by 4(s) = e(Y( )s)e- for some e 6 S'.

In Appendix B, a number of theorems of Lineburg's are

used (along with other arguments) to show that the category of

pointed translation AH-planes is equivalent to the category of

quasicongruences (Theorem B.16). In Proposition B.20, it is

shown that there is a functor K from the category of translation

AH-planes with non-degenerate AH-plane homomorphisms to the

category of quasicongruences which has the following property:

if :T -T T' is a non-degenerate AH-plane homomorphism between

translation AH-planes, and if (K,9) and (K', ') are the

canonical quasicongruence coordinatizations of T and T', then

p is defined (in terms of the algebras associated with the

coordinatizations) by yQ = KT(t)Q + O0.

In Proposition 11.4, it is shown that if H is a PH-plane;

if sl, s2, s3 are the sides of a triangle whose image in the

gross structure of H is non-degenerate, and if each of the three

AH-planes Al, A2, A3 derived from H by use- of one of the sides

sl, s2, 53 is Desarguesian, then H is Desarguesian.

In Proposition A.23, it is shown that the category of

restricted biternary rings is isomorphic to the category of

biternary rings; that is, there is an equivalence between them

which produces a one-to-one correspondence between the objects.

The development here depends only on well-known results

from the theories of categories, sets, algebra and affine and

projective planes. Paragraphs marked with asterisks are not

part of this development and are intended to relate results here

to other portions of the literature.

Since some of the treatment of Desarguesian AH-planes

given here resembles some of the recent work of J. U. Lorimer

and N. D. Lane C(1973)l and of J. W. Lorimer [(1973)a and (1973)b],

there are discussions of their recent papers and a brief history

of this research in Appendix A: A.24 through A.2S.

If it is desired, Appendix A (restricted biternary rings)

can be read immediately following Section 3 (biternary rings) and

Appendix B (quasicongruences) can be read immediately following

Section 5 (semitranslations and algebra). Nothing in any of the

sections depends on anything in the appendices.

2. HJEL:MSLEV PLANES

In this section we give a number of definitions (some of

which differ slightly from the usual ones and some of which are

entirely new), prove a few basic propositions, and discuss the

relationship between affine and projective HIjelmslev planes.

Proposition 2.31, Theorem 2.63, and Corollaries 2.64 and 2.65

concern non-degenerate homomorphisms. Proposition 2.56 concerns

isomorphisms.

2.1 Definitions. We will assume that the reader is familiar with

the elementary definitions of category theory. Those of the

necessary category theory definitions which are not given here

are stated by Mitchell [(1965), pages 1-5, 49, 52, 591. We will

commit a common notational inconsistency by usually denoting a

category with objects A and morphisms I by simply A; our names

for the various categories will reflect this notation. We will

use the term 'natural isomorphism' in plane of Mitchell's term

'natural equivalence', and the term 'functor' in place of

Mitchell's term 'covariant functor'.

2.2 Definitions. A function f from a set A to a set B is an

ordered triple (A,B,G ) where Gf is a subset of AxB such that,

if a-is an element of A, there is a unique element b of B such

that (a,b) 6 Gf G is called the graph of f. If (a,b) E Gf,

we write f(a) = b. Two functions f = (A,B,Gf) and g = (C,D,G )

compose to give gf = (A,D,G ) if and only if the domain C of

g is equal to the codomain B of f, and, if C = B, then gf is

defined by gf(a) = g(f(a)) for every a in A. We call the

function (A,A,G1 ) defined by 1A(a) = a for every a in A the
A
identity function on A.

Let A be a class such that for each A 6 A, there is a set

U(A), called the underlying set of A. An ordered triple ~ =

(A,B,f ) is said to be a concrete morphism (on A), and is

written w:A B, if A,B E A and f is a function from U(A) to

U(B); that is, f = (U(A),U(B),G ). If fj(a) = b for some a in

U(A), we write wa = b and w(a) = b. The function fL is said to be

the underlying set function of o, and we define U(w) to be f .

If < = (A,B,f) and p = (B,C,f ) are concrete morphisms on A,

we say that (A,C,f f ), denoted p(, is the natural composition

of p by c. 'e denote (A,A,lU(A)) by 1A and call 1A the natural

identity morphism on A or simply the identity on A.

A concrete catecorv A is a category whose morphisms are a

subclass M of the class of all concrete morphisms on A, whose

identity morphisms are the natural identity morphisms, and whose

composition is the natural composition. All of the categories

which we construct in what follows are concrete categories. Once

we have given a class A and a subclass M of concrete morphisms

called 'A homomorphisms', we will assume that the reader will be

able to identify the underlying sets of the objects in A and to

verify that the given subclass of concrete morphisms is closed

under natural composition and contains the natural identity

morphisms. Once this has been verified, it is immediate that A

is a concrete category since [A,B]A will always be a set; the

morphisms of A will always satisfy the requirement that (O) =

(p)C( where defined, and 1AC = W, plA = p where defined.

Instead of saying 'the concrete morphism t = (A,B,f )', we

will usually say 'the map w:A -- B'. In an abuse of terminology,

we say 'the map F:A B' when A,B are classes and F is a

subclass of AEB such that for every A in A there is a unique

B in B, written F(A), such that (A,B) E F.

2.3 Definitions. If there is a map F:A -* B which 'forgets'

structure, or which is the functor we construct from A to B,

we frequently say that a concrete morphism w = (A,A',f,) on

is a B homomorphism if (F(A),&(A'),f ) is a B homomorphism. For

example, if a = (A,A',f ) is a concrete morphism on AH-planes,

and if A = (S,U1), A' = (S',11) then we say 0 is an incidence

structure homomorphism if (S,S',f ) is an incidence structure

homomorphism. We abuse our terminology in other similar respects;

for example, we speak of the 'lines' of an AH-plane A = (S,I\)

when we mean the lines of S.

Once we have constructed a functor G:C -- D and shown that

G is an equivalence, we will call G(C) where C is.an object in C

the D-object generated by C. If F:D -- E is an obvious functor

which 'forgets' structure, we will call FG(C) the E-object

generated by C. For example, in Section 3 we construct an

equivalence C:B -- C, and if B is a biternary ring, we call C(B)

the coordinatized AH-plane generated by B, and we call the

10

AH-plane of C(B) the AH-plane generated by B. We will frequently

say 'of' in place of 'generated by', and 'with' in place of

'which generates the'.

2.4 Definitions. Let <:A -- B be a concrete morphism. If the

underlying set function of N is onto, a is said to be suriective.

If the underlying set function of c is one-to-one, m is said to

be injective. If a is both injective and surjective, it is said

to be bijective.

2.5 Definitions. A functor F:C -* D is said to be an equivalence

if and only if there is a functor G:D C together with natural

isomorphisms ::1- -- FG and %:GF 1-. If an equivalence

F: -" d.produces a correspondence between the objects of C and

D, then it is said to be an isomorphism if it is one-to-one, onto.

2.6 Definition. If F:C -* 5 and G:D -- are functors, and if

q:l -- FG and I:GF -- 1- are natural isomorphisms, we say that

F and G are reciprocal ecuivalences.

2.7 Definitions. If A is a class of objects such that a binary

relation PA is defined on the underlying set of each object A of

A, then a concrete morphism w:A -- B is said to preserve p if

whenever (a,b) PA for some a,b in U(A); it is also true that

((a),(b)) C< We say that 4 reflects p if whenever

(w(a),w(b))e p. for some a,b in U(A); it is also true that

(a,b) .A

2.8 Definitions. Let (CP,o,I) be a triple of sets. Then (CP,,I)

is said to be an incidence structure if % and i have no elements

in common and I r 1 X a. The elements of 'Y are called points,

the elements of o are called lines, and if P 6 ', g E 0 we write

P I g whenever (P,g) 6 I. The set I is called the incidence

relation. If P I g, we say P is incident with g, P lies on g,

P is a point of g, or P is on g. '.e also say g goes through P,

or g is a line through P. We use other similar geometric

language to designate incidence or non-incidence. For example,

g is said to join P,Q if [,Q I g; that is, if P I g and Q I g.

The capital letters P, 0, G, K will be used to designate points

unless otherwise indicated; similarly, the small letters g, h, k

will be used to designate lines unless otherwise indicated.

Points which all lie on a common line are said to be collinear,

and lines which all go through a common point are said to be

copunctal. We say Ig A hi = n if the cardinality of the set of

points incident with both g and h is n. We let g n h denote the

point or set of points incident with both g and h. We define

Ig q h h 1 kl and g h h 1 k similarly. We write PQ to denote the

line or set of lines joining P and 0. Occasionally we write

P v Q instead of PQ.

2.9 Definition. Let S = (r,j,I) and S' = '(C',oj',I') be

incidence structures. An incidence structure homomorphism

w:S -- S' is a concrete morphism which satisfies the following

conditions.

(1) fw is a function from 2 U O to 2' U o'.

(2) w(I) C qP'.

(3) (o) cj'.

(4) W preserves incidence; that is, if F I g, then wP I wg.

2.10 Definition. The class of incidence structures, the class

of incidence structure homomorphisms and the natural composition

of incidence structure homomorphisms form a category S which we

call the cateoryv of incidence structures.

Remark. We usually denote the incidence relation of any in-

cidence structure by I.

2.11 Proposition. In the category of incidence structures, a

morphism % is an isomorphism if and only if is a bijective

homomorphism which reflects incidence. //

2.12 Definitions. Let ~ be an equivalence relation on the points

and lines of an incidence structure S = ( ,O,I) such that no

point is equivalent to any line. Let 'f, 9' be the sets of

equivalence classes of points, lines. Let P*, g* be the equiva-

lence classes containing 2, g respectively. Let P* I* g* if and

only if there exist Q, h equivalent to P, g respectively such

that Q I h. Then S' = (f',*,I*) is called the incidence

structure induced from S by -, or the induced incidence structure.

2.13 Notation. Let S be a set and let ~ be an equivalence

relation on the elements of S. The ~-equivalence class contain-

ing an element s will frequently be written s*; and an arbitrary

*-equivalence class will frequently be denoted by a theretofore

unused letter with an asterisk; for example, t'. Once t' has

been used, however, t will denote an element of t'. Asterisks

used as superscripts do not necessarily denote equivalence

classes; it should be clear from the context what is meant in

each case.

2.14 Definition. Let H* be an incidence structure with

incidence relation I. One calls H* an (ordinary) protective plane

if the following three conditions hold.

(OP1) Whenever P" and Q' are distinct points of H*, there

is exactly one line g* such that P* I g* and Q* I g'.

(OP2) Whenever g* and h* are distinct lines of H*, there is

exactly one point P* such that P* I g* and P* I h*.

(OP3) There exist four points, no three of which are col-

linear; that is, no three of the four points lie on a common

line.

2.15 Definitions. Let S* be an incidence structure with

incidence relation I. Let i be an equivalence relation defined

on the lines of S*. If g* \ h', we say g* is parallel to h'.

One calls A* = (S*,1) an (ordinary) affine'olane if the following

four conditions hold.

(OAl) Whenever P* and Q* are distinct points of S*, there is

exactly one line g' such that P* I g* and Q* I g*.

(OA2) Whenever P* is a point and g* is a line such that P*

is not incident with g*, there is exactly one line h* such that

F* I h* and h* and g" have no point in cormon.

(OA3) S* has three points which are not collinear; that is,

S* has three points not all on the same line.

(OA4) Two lines g* and h* are parallel, g' 1 h*, if and only

if.g* and h* have no point in corm.on or g* = h*.

*Remark. It is well-known that if S* is an incidence structure

satisfying conditions (OAl), (CA2) and (OA3), then there is a

unique equivalence relation II such that (S*,l) is an affine plane

by our definition [Pickert (1955), pages 7-10).

The following result is well known.

2.16 Prooosition. If H is an affine plane projectivee plane),

then each line of H goes through at least two (three) distinct

points, and each point of H is incident wih at least three

distinct lines.

Proof. This is essentially shown in Pickert [(1955), pages

7, 9-111. //

2.17 Definitions. Let H = ('~,I,I) be an incidence structure.

Points P and Q are said to be projectivol. neighbor and one

writes P ~ Q whenever there are distinct lines g and h such that

P,Q I g and P,Q I h. Lines g and h are said to be protectively

neighbor and one writes g h whenever there are distinct points

P and Q such that P I g,h and Q I g,h. One calls H a

projective Hjel.slev clano (abbreviated PH-plane) whenever the

following three conditions are satisfied.

(PHI) If P and Q are points of H, there is at least one line g

such that P,Q I g.

(PH2) If g and h are lines of H, there is at least one point P

such that P I g,h.

(PH3) There is a surjective incidence structure homomorphism

f:H -- H' from H to a projective plane H' such that the following

two conditions are satisfied.

( l) If P,Q < '; then PP = 4Q 0 P ~ Q.

(92) If g,h a oa; then 4g = ?h t> g h.

If P is a point of K and if g is a line of H, one denotes the

~-equivalence classes containing P and g by P* and g' respective-

ly.

2.18 Proposition. Any projective plane is a projective

Hjelmslev plane. //

2.19 Definitions.' Let H,H' be PH-planes. A projective Hjelmslev

plane homomorohism w:H -- H' is an incidence structure homo-

morphism which preserves the relation 'projectively neighbor'.

We denote the category of projective Hielnslev lanes by H,

and we denote the category of projective planes with PH-plane

homomorphisms by H'.

2.20 Proposition. If H is a PH-plane, and if q:H -- H' is a map

satisfying conditions (PH3) of the definition of PH-plane, then

4 is a PH-plane homomorphism. //

2.21 Definitions. Let S = (,o,I) be an incidence structure,

and let 1 be an equivalence relation called the parallel

relation which is defined on the lines of S. Points P and Q are

said to be affinelv neighbor and one writes P ~ Q whenever

there are distinct lines g,h such that P,Q I g,h. Lines

g and h are said to be affinely neighbor and one writes g h

whenever to each point of each there corresponds a point of

the other which is affinely neighbor to it. One calls A = (S,t)

an affine Hjelmslcv plane (abbreviated AH-plane) whenever the

following four conditions are satisfied.

(AHl) If P and Q are points of S, there exists at least one

line g such that P,Q I g.

(AH2) Let P 1 g,h. Then P is the only point on both g and h

if and only if g is not affinely neighbor to h.

(AH3) If P is a point and if g is a line, then there is exactly

one line h such that P I h and g \ h.

(AH4) There is a nap 4:A -- A' from A to an affine plane A'

which is a surjective incidence structure honomorphism such that

the following three conditions are satisfied.

(1l) If P,Q e E-; then qP = Q40 P ~ Q.

(f2) If g,h a 03; thenqh = 4g <> g h.

(93) If Ig .h\ = 0 in A, then 4g ,Wh in A'.

If P is a point and if g is a line of A, one denotes the

--equivalence classes containing ? and g by P" and g"

respectively.

We usually denote the parallel relation of an AH-plane

by U.

2.22 Proposition. Any affine plane is an AH-plane.

Proof. Let A' be an affine plane. By Proposition 2.16, any line

g" of A* goes through at least two distinct points. In A* the

relation 'affinely neighbor' reduces to the relation 'is equal

to'. Using these results, one can easily show that A* is an

AH-plane. //

2.23 Definitions. Let A and A' be AH-planes. An affine Hjelmslev

plane homomorphism ":A -- A' is an incidence structure homomorph-

ism from S to S' which preserves the relations 'parallel' and

'affinely neighbor'.

We denote the category of affine Hjelmslev planes by A, and

we denote the category of affine planes with AH-plane homomorph-

isms by A'.

*Remark. We have shown [Bacon (1971), page 21, Corollary 3.12]

that there exist two non-isomorphic AH-planes which have

isomorphic incidence structures.

2.24 Propositon. If A is an AH-plane, and if J:A -> A' is a

map satisfying condition (AH4) of the definition of AH-plane,

then T is an AH-plane homomorphism. //

2.25 Definitions. If A = (S,U) is an AH-plane, we denote the

incidence structure induced by the equivalence relation

'affinely neighbor' by S*, and we denote the induced parallel

relation by II*: we say g* I* h* if and only if there are lines

k,m such that k ~ g; m h and k 11 m. We call A* = (S',I\*)

the aross structure of A. We call the concrete morchism

K:A -- A* defined by KP = P*; Kg = g" the neighbor man of A.

If H is a PH-plane, we denote the incidence structure

induced by the equivalence relation 'projectively neighbor' by

H', and we call H* the cross structure of H. We call the

concrete morphism K:H -- H* defined by KP = P*; Kg = g* the

neighbor map of H.

2.26 Definitions. Once the gross structures have been defined,

we extend the relations 'projectively neighbor' and 'affinely

neighbor' in the following way. Let H be a PH-plane or an

AH-plane. Let P be a point and g a line of H. We say P ~ g

and g P whenever the image of P is incident with the image

of g in the gross structure of H. Hereafter, the symbol ~ is

to be read 'is neighbor to' except where otherwise specified.

Its negation is written 9 and is read 'is not neighbor to.'

The negation of the symbol U is written V, and is read

'is not parallel to.'

2.27 Proposition. If w:A -* A' is an AH-plane homomorphism,

then P I wg 4> 3 h such that P I h and wh = ig.

Proof. Let h be a line through P, h I g; then .h = wg. //

2.28 Proposition. If w:H -- H' is a surjective PH-plane homo-

morphism, then wP I wg <> 3 h such that P I h and -h = wg.

Proof. Assume oP I wg. Let '':H' -- H" be a map satisfying

condition (PH3)- There is a line wk in H' such that 4'(GP) is

not on P'(uk). Since wk 4 tg, k 4 g. Since (wk C\ tg) ()P),

(k R g) 4 P. Thus, if we let h = (k n g) v P, then wh = wg. //

2.29 Proposition. If H is an AH-plane (PH-plane), and if

t:H -- H' is a map satisfying condition (AH4) (condition (PH3)),

then the gross structure H* of H is an affine plane projectivee

plane), and is isomorphic to H'; also, the neighbor map

K:H -- H* is a surjective AH-plane (PH-plane) homomorphism

satisfying condition (AH4) (condition (PH3)).

Proof. Define Q:H* -- H' by G9P =
well-defined bijection. If P* I g', then there are Q,h, Q ~ P,

h g, such that Q I h; hence GP* I Gg'. Since I is a surjection,

6 reflects incidence by Proposition 2.27 or 2.28: if OP' I Bg',

then PP I Sg and hence there is an h, h ~ g, such that P I h;

thus, P* I g'. Thus, 9 is an incidence structure isomorphism

and P* Q* < GP ~ 6Q; g* ~ h'* t 8g* Oh'. Hence S pre-

serves and reflects the neighbor relation on points (lines).

If H is an AH-plane, and if g* I* h*, then there are g' in

g' and h' in h* such that g' I h'. Then, since qg' = Gg' and

Th' = Gh*, we have that Gg* i Bh*, and hence 9 preserves the

parallel relation.

Assume that H is an AH-plane and that tg tI h. Let Lk be

a line not parallel to th. Since k -ph, \k \ h\ 0. Let

P E k A h. Let h' be a line such that h' U g; P I h'. Then

since h' II g, 4h' I g. Since qP I fh',Jh and Ih,4h' IL qg,

we have that qh' = (h. Thus h' h and h' I1 g; so that,

h* U* g*. Thus, 6 reflects the parallel relation.

Consequently, H* is an affine plane projectivee plane)

isomorphic to H'.

Since K = 6-19, K is a surjective AH-plane (PH-plane)

homomorphism satisfying condition (AH4) (condition (PH3)). //

2.30 Definition. Let t:A --* A' be an AH-plane (PH-plane) homo-

morphism. Let K' be the neighbor map of A'. If there are

three points R, R', R" of A whose images K'pR, K',R', K'RR" in

the gross structure of A' are not collinear, then we say that

Sis non-decenerate; otherwise we say that p is degenerate.

2.31 Proposition. If f:H -* H' is a non-degenerate AH-plane

(PH-plane) homomorphism, then pP Irg 4: 3 Q such that Q I g

and P = rQ; andpP I g 3h such that P I h andrg = h.

Proof. Assume H is a PH-plane and rP Ig. Let K' be the

neighbor map of H'. Since r is non-degenerate, there is a point

S such that A'S is not on K'Ig. Then P 4 S and IPS gl = 1,

since pP/S i g implies PS g. Let Q = PS n g; then /Q = pP.

Since is non-degenerate, there are three lines in H whose

images in (H')" are not copunctal. Thus, there is a'line pm in

H' such that (PP). is not on ( m)*. Let M = m R g. Since (pP)*

is not on (pm)*, P j t.M, and hence P L M. Let h = MP; then,

rh = r P = g and P I h.

21

Assume H is an AH-plane and pP I g. Let K, K' be the re-

spective neighbor maps of H,H'. By Proposition 2.27, there is a

line h such that P I h and -g = ph. Let m be a line such that

K'rm R K'lg. Let m' and g' be lines such that m' R\ m, g' I g and

P I m',g'. If \m' n g\ = 0, Km' = Kg'; so that g' m'. Hence

K'pm II K'pg, a contradiction. Thus, \m' 0 g\ / 0. If m' g,

then K'pm 11 K'g, a contradiction. Hence rm' 0 g\ = 1. Let Q =

m' A g. Then, if pm' g, K'm' 11 K'pg, and hence K'pm \\ k'Ig,

a contradiction. Hence im' 4 g and Q = //

2.32 Proposition. If ,:H H' is an AH-plane (PH-plane)

homomorphism which induces an isomorphism between the gross

structures, then r is non-degenerate. //

2.33 Proposition. If p:H -* H' is a surjective AH-plane (PH-

plane) homomorphism, then r is non-degenerate. Hence, if H is

an AH-plane (PH-plane), and if P is a point and g, a line of H,

then P ~ g if and only if there is a point Q on g such that

P Q. //

*Remark. Lineburg [(1962); pages 263, 264, 265; Satze 2.1, 2.3,

2.43 states the first two sentences of the- following proposition

for AH-planes. Klingenberg [(1955), page 101, S 5] states the

first sentence of the following proposition for PH-planes.

2.34 Proposition. Let H be an AH-plane (PH-plane); then, there

are at least three pairwise non-neighbor lines through each

point of H, and at least two (three pairwise) non-neighbor

points on each line of H. Hence each line is uniquely determined

by the set of points on it. Also, each point is uniquely

determined by the set of lines through it.

Proof. The first two (all three) sentences of the proposition

follow easily from Propositions 2.33 and 2.31.

Let H be an AH-plane and let (g\ P I gl = \gi Q I gl.

Let g,h be lines such that P,Q I g,h and g + h. Then by (AH2),

Ig R h\ = 1. Thus, P = Q. //

2.35 Definition. Let A be an AH-plane. A \I-equivalence class

of lines is called a direction. We denote the set of directions

by g If g is a line, the direction containing g will be

denoted by T(g). Arbitrary directions will be denoted by T, Z,

r or some other capital Greek letter.

2.36 Definition. Let A be an AH-plane. If P is a point and g

a line of A, we denote the unique line of A through P parallel to

g by L(P,g). If P is a point and r is a direction, we denote the

unique line of F through P by L(P,r).

2.37 Definition. If g,h are lines of an AH-plane A, and their

images under the neighbor map of A are parallel, we say g and h

are quasiparallel, and write g \\ h. If h is a line and r is a

direction, we write h II F and P r h and say 'h is quasiparallel

to r' and ' is cuasiparallel to h' if there is a line g in

23

r such that h \k g. Similarly, two directions f,- of A are said

to be quasiparallel, 7 1\ 2, if they map into the same parallel

class under the neighbor map of A. The negation of the symbol

\ is written 4.

2.38 Proposition. Let A be an AH-plane. Two lines h,k of A

have exactly one point in common if and only if they are not

quasiparallel. Also, \ is an equivalence relation.

Proof. Assume h k. Since the images of the two lines are not

parallel, the lines are not neighbor. If Ih n k\ = 0, then their

images would be parallel by (AH4). Hence by (AH2), lh 0 k\ = 1,

and h and k have exactly one point in common.

Assume Ih A k) = 1. Then h 4- k. Hence h* / k*. But

Ih* 0 k*'1 0; hence h* is not parallel to k*. Thus, h is not

quasiparallel to k. //

2.39 Definitions. If S is an incidence structure, and if g is a

line of S, we say (S,g) is a lined incidence structure with base

line g. If (S,g) is a lined incidence structure, the points of

S which are not neighbor to g are called the affine points of

(S,g). Any line of S which goes through an affine point is

called an affine line of (S,g). We say that u:(S,g) -- (S',g')

is a lined incidence structure homomorphism if w is an incidence

structure homomorphism such that i(g) = g', and such that w maps

the affine points of (S,g) into the set of affine points of

(S',g'). If S or w is also some special type of incidence

24

structure or some special type of incidence structure homomorph-

ism, we modify our terminology accordingly. We denote the

category of lined PH-planes by H the category of lined

projective planes by H* and the category of lined incidence

structures by S .

2.40 Construction of G:A -- S and G :A -- S Let A be an
------ ---~g g
AH-plane. Let g_ be the set of parallel classes of A. For

every parallel class TTin g let P(T) be a new point, and adjoin

P(T) to each line in RT. Let the P(T)'s be different for differ-

ent 9's. Let g(g ) be a new line incident with each of the new

points. Choose the P(T)'s and g(g ) in such a way that the new

point set '- U P( )I TT E g_. and the new line set g U g(g ) are

disjoint. Let G(A) be the incidence structure obtained by

adjoining the new points P(C), the new line g(g.) and the.new

incidences to the points, lines and incidences of A. Define

G (A) to be (G(A),g(g,)). G(A) is called the generalized
~g
incidence structure of A. G (A) is called the lined generalized
~g
incidence structure of A. A point of G(A) is called a

generalized point, and a line of G(A) is called a generalized

line. The incidence relation of G(A) is called the generalized

incidence of A. We call the original points, lines and

incidence structure of A, the affine points, affine lines and

affine incidence structure of A. Unless otherwise specified,

line (point; incidence structure) will mean affine line (affine

point; affine incidence structure) in an AH-plane.

If M:A -* A' is an AH-plane homomorphism, then a can be

extended in an obvious natural way to an incidence structure

homomorphism G(wi:G(A) -- G(A'), and to a lined incidence

structure homomorphism G (,3):G (A) -+ G (A'l.
~g -g gd

Remark. The definition of an affine point (affine line) of A

agrees with the definition of an affine point (affine line) of

G (A).
-g

2.41 Proposition. The map G:A -- S constructed above is a

functor from the category of AH-planes to the category of

incidence structures, and G :A -- S is a functor from A to the
~g g
category of lined incidence structures. //

2.42 Proposition. The map H- :A' -- H" defined by H* (A*) =

G (A*), H* (G) = G (w) is a functor from the category of affine
~g g ~g
planes to the category of lined projective planes.

Proof. Pickert [(1955), page 11, Satz 73 shows that if A" is

an affine plane, then G(A*) is a projective plane. Hence H* (A*)
-g
is a lined projective plane.

Since the neighbor relation in a projective plane is

trivial, if w:A" -- B" is a morphism in A', then H* (w) is a
g

lined projective plane homomorphism. Thus, H* is a functor. //
~ g

2.43 Definitions. Let A be an AH-plane. Let K:A --- A be the

neighbor map. Let R be a point or a line in G (A), and let S be

a point or a line in G (A). We say R is neighbor to S, and
-g

write R ~ S, whenever G (K)R = G (K)S, G (G)R I G (K)S or
-g -g ~g ~g
G (K)S I G (K)R in the lined projective plane H* (A*) = G (A*).
~g -g -g ~g
We call the relation ~ thus defined the generalized neighbor

relation of G (A), or the neighbor relation of G (A). One
'g ~g
can show that restricted to the affine points and affine lines

of G (A), the generalized neighbor relation agrees with the

neighbor relation induced from A. Once this has been shown

(see Proposition 2.44), extend the neighbor relation of A in

the obvious way: we say 'R is neighbor to S' in A (where R,S

can be a point, a line, a direction or g ) whenever R S in

G (A). If R is neighbor to S we write R ~ S; otherwise we
-g
write R + S. We call the (generalized) neighbor relation of A.

2.44 Proposition. Let A be an AH-plane. The restriction of the

generalized neighbor relation of G (A) to the affine points and
-g
affine lines of A is the relation 'neighbor' of A.

Proof. Observe that P ~ Q <4 KP = KQ; g ~ h c= Kg = Kh;

P g 4< KP I Kg; g ~ P 4 K P I Kg. //

Remark. Hereafter we will frequently not distinguish between

g(TT) and T; g(g_) and g_; G (A), G(A) and A; G (w), G(-) and 1.

2.45 Proposition. Let A be an AH-plane.

(1) Let h be a line and let r be a direction of A. The

following are equivalent.

a) h II P.

b) h ~ r.

c) "If P I h, there is a line g c P such that g 1I h and

P e g h.

(2) Let f,r be directions in A. The following are equivalent.

i) Zn' F.

ii) P.

iii) If h re, then h I\ F.

Proof. Part (1). Let h be a line, and let r be a direction of A.

Assume h \\ F. Then there is a line g in r such that g \ h.

Hence Kg \I Kh in A', and IT(g) I wh in H* (A*). Thus, h P.

Assume h ~ 7. Then K(r) I K(h). Let P I h, and let g =

L(P,r). Then since Kg = KP v (Vr) = h, we have that g U h.

Assume that if P I h, there is a line g 6 r such that

g I h and P I g. Let P I h. Then P I g, g \ h; hence h U r.

Part (2). Let f,r be directions in A.

Assume 2 I P. Then K(2) = K(F), and hence i r.

Assume I- Let h C& Let P I h. Let g = L(P,F).

Then since K(T) = C(r), we have that Kh = Kg; hence h I g. Thus,

h u r.

Assume that for every h C h I r. Let he Then

there is a g e such that h I g. Thus, K

2.46 Proposition. If is an AH-plane homomorphism, < preserves

the generalized neighbor relation defined above as well as the

quasiparallel relation.

Proof. Since < is an AH-plane homomorphism, t preserves the

'affinely neighbor' relation. Hence by Proposition 2.34, a pre-

serves the relations P ~ g and g ~ P. If 7 ~ then there are

lines g f h f such that g ~ h. Hence g ~ < h, and rP N <4.

If.. g, where g i g then there is a line h 6 1 such that h

is neighbor to g. Hence Kh ~ og, and &<~

Assume g \ h. Let P I g, and let h' = L(P,h). Then

h' g. Hence h' og and .h' 1 ah. Thus, .h 1 sg. //

2.47 Construction of S ) and A:H -- A. Let H be a PH-plane
(H,g) -- ~ g
and let g be a line of H, Remove all the points and lines neigh-

bor to g from the incidence structure of H along with all the

related incidences. Call the resulting incidence structure

S(H,g). Define a parallel relation on the lines of S(H,g) in the

following manner: h 1 k in S(H,g) if and only if h, k and g are

copunctal in H. Define A(H,g) to be (S(Hg)', If

J:(Hg) -* (H',g') is a lined PH-plane homomorphism, then w in-

duces a map from A(H,g) to A(H',g'). We call this induced map

A(w).

2.48 Proposition. The map A*:H A* defined by A*(H*,g')

A(H*,g'), A*(") = A(w) is a functor from the category of lined

projective planes to the category of affine planes, and the

functors H' and A* are reciprocal equivalences.

Proof. Assume that (H,g is a lined protective plane. Pickert
Proof. Assume that (H',g) is a lined projective plane. Pickert

[(1955), pages 9-10] shows that S(H, g*) is the incidence

structure of an affine plane. Two lines of A*(H',g') are

parallel if and only if they fail to meet in S (H g*) (and hence

meet at a point on g'.) Thus, A'(H*,g') = (S(H,g*),R) is an

affine plane.

If w:(H*,g-) -- (H',g*) is a morphism in H'g, then

w(P* \ P" I g' C {*' \ P I g' and w maps the affine points

(lines) of H* into the affine points (lines) of H'. Hence A'(w)

is an affine plane homomcrphism. It is easily seen that A* is a

functor.

Let A* and A' be affine planes. Observe that A'H* (A*) =

A*. If w:A* -- A" is an affine plane homomorphism, then

A*H* (a) = w. Define 'A* to be the identity map on A*. Then

.:A*H* --H 1- is a natural isomorphism.
-~g -A'
If (H',g') is a lined projective plane, define a map

I1(H g):(H',g') -- H* A*(H*,g*) by letting it be the identity
(H',g') ~ g~
on the affine points and the affine lines of (H',g*); by letting

it take a point P* on g* to P(TT) where TT is the set of affine

lines through P', and by letting it take g* to g(g,). It is

easily seen that 1.( ) is an isomorphism. If
(H ,g')
w:(H*,g') -- (H*,g*) is a morphism in H* then, since a lined

projective plane homomorphism is completely determined by its

action on the affine points and the affine lines, we have that

,g = (H* A *())(H*,g*) Hence :^H. H' -A is a

natural isomorphism. Thus, H' and A' are reciprocal

equivalences. //

30

2.49 Proposition. If (G,g_) is the lined generalized incidence

structure of an AH-plane A, then A can be obtained from (G,g.)

by a construction identical to that used to obtain the affine

plane A(H',g') from a lined projective plane (H',g'). //

2.50 ProDosition. The map A:H -- A constructed in Construction

2.47 is a functor from the category of lined projective Hjelmslev

planes to the category of affine Hjelmslev planes. If (H,g) is

a lined PH-plane, two affine points (affine lines) are neighbor

in (H,g) if and only if they are neighbor in A(H,g). If H* is

the gross structure of H, and if g* is the class of lines

neighbor to g in H, then A(H*,g*) is equal to the gross structure

of A(H,g).

*Remark. It is well known that if (H,g) is a lined PH-plane, then

A(H,g) is an AH-plane. This is stated by Lneburg [(1962), page

260, second paragraph], and is essentially proven by Klingenberg

[(1954), pages 390-392, S 1.11 and S 3.63.

Proof. Let (H,g) be a lined PH-plane and let S = S(H,g). Then,

A(H,g) = (S,ll); S is an incidence structure, and H1 is an equiva-

lence relation on the lines of S. P, a point of H, is a point of

S if and only if P g; and h, a line of H, is a line of S if and

only if h ?. g.

We use the symbol 0 to denote the relation 'affinely

neighbor' in A = A(H,g) in order to avoid confusion with the

symbol ~ which we use to denote the relation 'projectively

neighbor' in H. If 2,Q are. points of A, P Q in H if and only

if P 0 Q in A since a line is removed only if all the points on

it are also removed.

If h,k are lines of A, we wish to show that h ~ k in H

if and only if h D k in A. Assume h ~ k. Let P be any point

of A which is on h. 3y Corollary 2.34, there is a line m of H

through P such that m ?b h. Hence m -i k. Let Q = m n k in H.

Then in H*, Q* = P* = m* n k* by (OP2). Thus, C is in A; P D Q,

and Q I k. By symmetry, the corresponding statement holds for

an arbitrary point of A on k. Thus, h 0 k. Conversely, assume

h D h'. In H' there are at least two points P*,0O on h* but

not on g* by (OP2) and Proposition 2.16. By Proposition

2.31, there are points R,S on h such that R c P*, S & 0O. Let

R',S' I h' such that R' ~ R, S' S. Then in H', h* = (h')*

by (OP1); hence h h'. Hereafter we will use to indicate 'is

neighbor to' in both A and H.

Any two points of A are joined by at least one line; that

is, (AHl) holds in A.

If P I h,k; we wish to show that h n k = P if and only if

h 4 k in A. Assume that in A, h R k = P. Then h + k in H.

Hence, h 74 k in A. Conversely, assume P I h,k; h 4 k in A.

Then h + k in H and ? = h 0 k. Thus, P = b 0 k in A, and A

satisfies (AH2).

Let P and h be a point and a line of A. Let Q.= h ( g in

H. Then there is a unique line k joining P and Q in H, and we

have k 11 h in A. If k' is any line such that k' It h and P I k';

then Q I k'. Thus, k = k', and (AH3) holds in A.

32

By Proposition 2.48, A(H*,g*) is an affine plane. Define

t:A -- A(H*,g') by 9P = P', 9h = h*. Then if Ih A k( = 0 in A,

then h and k must meet in some point neighbor to g in H. Thus,

in H* the lines h* and k* meet in some point on g*. Thus,

h. II k* in A(H',g*). Hence q satisfies condition (AH4) and A is

an AH-plane. Observe that A(H*,g*) is equal to the gross

structure of A(H,g).

If w:(H,g) (H',g') is a lined PH-plane homomorphism,

it is easily seen that A(a) preserves the neighbor and parallel

relations and hence is an AH-plane homomorphism. It is also

easily seen that A(p ) = A(p)A(<) and that A(l(H,g)) 1 (Hg)

Hence, A:H -9 A is a functor. //

2.51 Definitions. Let A be an AH-plane and let (H,g) be a

lined PH-plane. If A is equal to A(H,g), then we say that A is

derived from (H,g), or we say A is derived from H (by use of the

line g), and we say A is a derived AH-plane. If A is isomorphic

to A(H,g), we say A can be extended to (H,g), or we say A

can be extended to H; we also say that (H,g) and H are

extensions of A.

*Remarks. Drake [(to appear), Corollary 6.21 states that

there is an AH-plane which cannot be extended to a PH-plane.

Drake [(1967), page 198, Theorem 3.1] states that every

finite uniform AH-plane can be extended to a finite uniform

PH-plane, and [in Bacon (to appear), Theorem 2.1] we state

that every projectively uniform AH-plane can be extended to a

uniform PH-plane, and we use the argument given by Artmann

[(1970), pages 130-1343 to show this.

2.52 Definition. An injective incidence structure homomorphism

j:S -- S' which reflects the incidence relation is called an

(incidence structure) embeddina (of S into S').

2.53 Proposition. Let (H,k) be a lined PH-plane. The map

X:G (A(H,k)) -- (H,k) defined by \(P) = P, X(h) = h for all

affine points and lines and by X(h(h)) = h ( k and X(g,) = k

for all affine lines h is a lined incidence structure

embedding. //

*2.54 Remarks. Dembowski [(1968), pages 295-296] and Artmann

[(1969), page 175, Definition 61 have given definitions of

'affine Hjelmslev plane' which they assert are equivalent to

that given by Luneburg [(1962), page 263, Definition 2.3]. In

(Bacon (1972), page 3, Example 2.11 we give an example of an in-

cidence structure and a parallel relation on the lines of the

incidence structure which satisfies the definitions given by

Dembowski and Artmann, but not that given by Luneburg. We

repeat this example here.

*Example. Take any affine plane A. Keep the same lines and the

same parallel relation. Choose one point P of A, and adjoin a

new point P' to the point set of A. Let the incidence relation

be the same for the old points and lines, and let P' be incident

with precisely the lines which go through P.

*Remarks. This example fails to satisfy the definition of

AH-plane given here (which is essentially equivalent to that

given by Luneburg). It can easily be shown that this example

cannot be derived from a lined PH-plane.

Klingenberg [(1954), page 390, D 61 calls S(Hg) an

'affine incidence plane with neighbor elements'. He then

shows [(1954), pages 391-392, S 3.63 that A(H,g) = (S(H,g),,
(H,g)
has certain properties. The example has all the properties

which A(H,g) is shown to have in Satz 3.6; although, of course,

it cannot be derived from a lined PH-plane as A(H,g) is.

2.55 Proposition. If A and A' are AH-planes and if w:A -- A'

is an incidence structure homomorphism which preserves the

parallel relation, preserves the neighbor relation on the

parallel classes and preserves the neighbor relation on points,

then w is an AH-plane homomorphism.

Proof. It suffices to show that w preserves the neighbor

relation on lines. Let g,h be lines such that g ~ h. Let

P I g and let Q be a point of h such that Q ~ P. Then g =

L(P,T(g)) and h = L(Q,T(h)). Since u(TTg) w(Wh) and wP wQ,

we see that ag ~ wh, and hence w is an AH-plane homomorphism. //

2.56 Proposition. If c:H -- H' is an AH-plane (PH-plane)

homomorphism which is a bijection on points, then r is an

isomorphism.

Proof. Let g' be a line of H'. Let rP,oQ I g' with cP + rQ.

Let g be a line through P and Q. Then og = g'; hence o is

surjective on lines. Assume TP I og. By Propositions 2.33 and

2.31, P I g. Thus, reflects incidence. Let oh = rk. If

P I h, then rP I ch; so that oP I ok and hence P I k and

conversely. Thus, by Proposition 2.34, h = k. Hence, r is

an incidence structure isomorphism. Assume
rP I rk and let k' = L(P,h). Then k' it h; ok' \ -h and

Thus, rk' = rk; k' = k and hence a reflects the parallel

relation. Since r is an incidence structure isomorphism, r pre-

-1
serves and reflects the neighbor relation. Thus r is an

AH-plane (PH-plane) homomorphism, and o is an isomorphism. //

2.57 Proposition. If ,:A -* A' is an AH-plane (PH-plane) homo-

morphism, then t induces an affine plane projectivee plane) homo-

morphism p';A* -- A'' from the gross structure of A to that of A',

and t is non-degenerate if and only if p* is non-degenerate. //

2.58 Definition. We say a lined incidence structure which is

j omorphic to the lined generalized incidence structure G (A)
-g
of some AH-plane A is a generalized AH-plane. If

w;(Ag) --~ (B,h) is a lined incidence structure homomorphism,

if (A,g) and (B,h) are generalized AH-planes, and if w preserves

the (well-defined) induced neighbor relations, then we say that

wis a generalized AH-plane homomorphism. We denote the category

of generalized AH-olanes by A .
; ... ,g

2.59 Construction of A :A -- A and A :A -- A. We define
~g g --- ~ g
A :A A by letting A (A) = G (A) and A (9) = G (w) for every
g g -g -g ~g ~g
A and i in A.

Let (A,g) be a generalized AH-plane. We let I be the
a
restriction of the incidence structure of A to the affine points

and affine lines of (A,g). If % and a are the sets of affine
a a
points and of affine lines of (A,g) respectively and if a rela-

tion II is defined on a by k 11 h 4= k,h and g have a point in

common in (A,g), then we denote (( a,1 ),ll) by A (A,g). If
a a a
w is a generalized AH-plane homomorphism, we define A (w) in the

obvious way.

2.60 Proposition. The maps A :A -- A and A :A -- A are
~g g ~'* g
reciprocal equivalences. //

2.61 Remark. Hereafter we will not distinguish between AH-planes

and generalized AH-planes except to aviod confusion. We will

say '(A,g) is isomorphic to the AH-plane B' when we mean that

(A,g) is isomorphic to the lined generalized incidence structure

of B, and so on.

2.62 Definition. An AH-plane homomorphism-' is said to be an

AH-plane embedding whenever x is injective and reflects the

incidence, neighbor and parallel relations.

*Remark. V. Corbas' argument for the validity of his Teorema

C(1965), page 375] inspired the following proposition. Corbas'

Teorema deals with surjective morphisms between affine planes.

2.63 Theorem. Let p:A* -- A* be a non-degenerate affine plane

homomorphism. Then, r is an AH-plane embedding of A' into A*;

hence p is injective and reflects the incidence and parallel

relations. Also, H* ():H* g(A*) -- H' (A') is a lined projective

plane embedding; thus, H '(r) preserves and reflects the

incidence relation. Thus, p induces a projective plane embed-

ding of the projective plane associated with A* into that

associated with A*.

Proof. Assume 4:A" -- A is a non-degenerate affine plane

homomorphism and that G*, K* and M" are points of A* whose images

under P are not collinear. We wish to show that ip is injective

with respect to parallel classes. Let P*, f* be distinct direc-

tions in A*. Let g* < r*. Since g* meets every line of '*, g*

meets every line of {Ps'l s* e '. In particular,pg* meets

LL(G',)'), L(K*,5') and pL(K*,1'). But by our assumption, at

least two of these three lines are distinct. Thus, rg* cannot

be parallel to all three; hence p' X *'. Hence r is injective

on directions.

Let P*, R" be distinct points, and let g* be the line

joining them. Let Q* be a point such thatpQ* is not onpg*;

such a point exists by our assumptions. Then P'Q*' t-R*'*; hence

P(P*Q*) R I(R*Q'); so that P* h PR'. Hence r is injective on

points.

Let yg" = rh' and let Q* be a point such that rQ* is not on

rg*. Let P* be a point of g'. Since h' 4 ,(P'Q*); h* -4 P'Q*;

hence P'Q* meets h* at some point R'. Since lpg* ( 1(P*Q*)1 = 1;

pP* = R', and P* = R'. Since 1g' Il ~ph'; g* l h* and g' = h'.

Thus, p is injective. By Proposition 2.31, reflects incidence

and hence is an incidence structure embedding. If h* 11 g*,

then either rh* = g' and h* = g* or I/h* nArg'* = 0 and

Ih* A g*9 = 0. In either case, h* It g'. Thus, p is an AH-plane

embedding.

We wish to show that Hg (P) reflects incidence. Obviously,

H* (a) reflects incidence for affine points and lines. If
~ g /

IR* I g in H* (A*), then R* I g,* in H* (A'). Let P* be a
-g ~

direction in A'; let g* 6 f* and let h* be an affine line such

that 'P I h- in H* (A*). Then, h* 1 g* in A*, and by our

earlier argument, h* I1 g* in A*. Thus, h* & Pr in A* and P" I h'

in H* (A'). Thus, H* (P) reflects incidence and is a lined
-g

incidence structure embedding. //

2.64 Corollary. If X:A -4 A' and p:A' -- A" are non-degenerate

AH-plane homomorphisms, then i is a non-degenerate AH-plane

homomorphism. //

*Remark. The following corollary was inspired by Lorimer's

argument for the validity of his Lemma 4.4 [(1973)b, page 101

which deals with surjective morphisms and the neighbor relation

on points and on lines. See Discussion A.27.

2.65 Corollary. If r:A -* A' is a non-degenerate AH-plane

39

homomorphism, then preserves and reflects both the generalized

neighbor relation and the quasiparallel relation. Thus,

P Q =* pP p.Q, and so on.

Proof. By Proposition 2.46, r preserves both relations. By

Theorem 2.63, the induced lined projective plane homomorphism

H*- (r*):H' (A*) -- H* (A'*) (where A* and A'* are the gross

structures of A and A') is a lined projective plane embedding,

and hence r reflects both relations. //

2.66 Corollary. If 1*:(H*,g*) -- (H,g*) is a lined projective

plane homomorphism, then w* is either an incidence structure

embedding or there is a line k* 4 g* such that if P* is not on

g*, then u*P* I *'k*. Thus, if w:(H,g) -- (H,g) is a lined

PH-plane homomorphism, then either there is an affine line k

such that if P is an affine point of (H,g), then F ~- wk, or

w preserves and reflects the neighbor relation (thus, P g

< wP wg, and so on).

Proof. One can easily see this by looking at A*(*). //

2.67 Proposition. If w:H* -- H* is a projective plane homomorph-

ism, and if g* is a line of H* such that wh* = og* implies h* =

g* and such that there are two points P*,Q* on g* such that

wP* 4 wQ*, then w is an incidence structure embedding or there

is a line k* 4 g* such that if P* is not on g*, then wP" I uk*.

Proof. Assume u:H* -- H* is such a morphism. If R* is not on

g*, and if wR* I wg*, then there is a point wS* on og* by our

hypotheses such that wR* 4 .S*. Thus, R(R*S*) = tg*, a contra-

diction. Hence w':(H*,g*) --* (H*, g') defined by .' =

((H*,g*),(HR,wg*),f,) is a lined projective plane homomorphism.

The result follows from the corollary above. //

*Remark. The following proposition was inspired by Lorimer's

argument for the validity of his Theorem 4.5 1(1973)b, page 103

which deals with morphisms which are surjective with respect to

points: see Discussion A.27.

2.68 Proposition. If p:A -- A' is a non-degenerate AH-plane

homomorphism such that pg 01 ph = g U h for all lines g,h; then

r is an AH-plane embedding; that is, p is injective and preserves
and reflects the incidence, neighbor and parallel relations. If,

in addition, p is surjective on points, then p is an AH-plane

isomorphism.

Proof. Assume that P:A -- A' is such a morphism and thatP =

jQ. Let g 4 PQ. Let R be a point such that R is not neighbor

to pg. Let h = PR, k = QR. Observe that b 01 k 4=4 P = Q since

h,k g, and P = g A h, Q = g n k. Henceph \yk 4.> P = Q. By

our assumption above, rP = pQ; so that h = k and hence P = 0.

Thus is injective on points. Then by Proposition 2.31, re-

flects incidence.

By Corollary 2.65, p reflects the neighbor relation.

b

Assume ,g = _h. Since rg l h; g I h. Let k be a line

such that( r k)+ T(Mg): such a line exists since r is non-degen-

erate. Then p(g n k) = pg ) rk = p(h 0 k) and thus g = h. Hence

tis injective. Thus, p is an AH-plane embedding.
If in addition t is surjective on points, then by Proposi-

tion 2.56, t is an isomorphism. //

2.69 Proposition. Let t:A A' be a non-degenerate AH-plane

(PH-plane) homomorphism. Then there are at least two points P,Q

on each line k whose images yP,Q are not neighbor in A'. Thus,

the action of is uniquely determined by its action on the

points of A.

Proof. Assume first that <:A -- A' is a non-degenerate AH-plane

homomorphism. Let k be a line of A. Let R,S,T be points whose

images under K',p (where K':A' -- (A')* is the neighbor map of A')

are not collinear. At least one of the directions pIT(RS), rTT(RT)

and rT(ST) is not quasiparallel tof k since otherwise the lines

KS(k), K'(RS), K'p(RT) and K'p(ST) would all be parallel, and hence

xK'R, K'S and K'T would be collinear. Let V be a direction such

that pf l.k. The lines L(R,P), L(S,r) and L(T,P) all meet k in

a single point: say R', S', T', respectively. Observe that at

least two of the points K'pR' KS', K>'T' are not equal, since

otherwise kIR, KrS and K'rT are collinear. Thus, there are points

P,Q on k such that pP J* Q.

Now assume p:A A' is a non-degenerate PH-plane homo-

morphism. Let k be a line of A and let R,S,T be points

whose images under K't are not collinear. Then RS, RT and ST

each meet k in, say, P, F', P". If K'pM = K'P, K'IP', KP", then

k'P(RS), K'p(RT) and K'p(ST) are copunctal. Since K',R, K'1S and

K' T are pairwise non-neighbor, we may assume pM 4 R,pS without

loss of generality. Then (K'pM),(K'rT) I ( KTS),( K'TR) and

K'tTS K'~TR; so that K'pM = x'QT. But then K'pT I KI(SR), a

contradiction. Thus, at least two of PP,pP',iP" are not

neighbor. Thus, there are points V,W I k such that pV + pW.

Thus, in AH-planes (PH-planes) the action of a non-degener-

ate homomorphism p is uniquely determined by its action on

points. //

2.70 Proposition. If m:A* -* A* is a degenerate affine plane

homomorphism, then there is a line k* such that wP* I uk* for

every point P* of A*, and exactly one of the following three

conditions holds.

(a) There is a point Q* such that -P* = wQ* for all points P"

of A'.

(b) For all lines g* of A*, ig* = wk* and there are points

P*,Q* such that wP* ; wQ*.

(c) There is a direction "* not containing k* such that wm* =

wk* for every m* j r', such that wk* f LF*, and such that wg*

wh* for some lines g*,h* e r'.

Moreover, if A* is an affine plane, there is at least one endo-

morphism of A* of each of the three types: (a), (b) and (c).

Proof. Assume that 4:A* -4 A* is a degenerate affine plane

homomorphism. Hence, the images of any three points of A* are

collinear in A*. Let G* be a point of A*. If wP* = wG* for

every point P' in A', then every line ag' goes through wG* and

hence case (a) holds and the other cases do not hold, and we can

let k* be any line of A'.

Assume that there are points G', H" such that H* / wG*.

Let k* = G*H*. Then, by our assumptions, wP* I wk' for every

point P'. There are two remaining subcases. If wg* = wk* for

every line g* of A', then case (b) holds and the other cases do

not hold. If there is a line g' such that wg* / Mk*, then,

since P* I g* implies w?" Iwg',pk' which implies wg' -W k*, we

have that w(L(H',g*)) / w(L(G*,g*)). If m'" g', then m* meets

both L(H',g') and L(G',g*); hence tm* = wk'. Hence case (c) holds

and the other cases do not hold.

Let A' be an affine plane. Let Q* be a point of A'.

Define w:A* -- A' by wP' = Q*; wg' = L(Q',g'): w is a type (a)

homomorphism. Let k* be a line and let Q* be a point on k'.

Define :A* --A" by ~P* = P* if P* I k', by P"' = Q' if P" is

not on k*, and by
a type (b) homomorphism. Let k* be a line and let F* be a

direction such that k" F*. Define 9:A* -- A* by g(P*) =

L(P*,r*) (I k* for all points P" of A', by v(g') = g' for g e *',

and by 9(h*) = k* for h* t r': 9 is a type (c) homomorphism. //

3. BITERNARY RINGS

In this section we define 'coordinatized affine Hjelmslev

plane' and 'biternary ring', construct the related categories,

and show that they are equivalent.

3.1 Definitions. Let T* be a ternary operation defined on a

set M' with distinguished elements 0* and 1* with 1' f 0". Then

(M',T*) is said to be a ternary field if it satisfies the follow-

ing five conditions:

(TFl) T'(x*,0*,c*) = T'(0O,m',c') = c" for all x*,m',c* in M'.

(TF2) T'(l',m*,0*) = T*(m',l*,0*) = m* for all m* in M'.

(TF3) For any x',m',c* in M*, there exists a unique z* in M*

such that T'(x',m*,z') = c'.

(TF4) Fcr any m',d',n*,b' in NM such that m' / n', there is

a unique x' in M' such that T'(x',m',d') = T'(x',n',b').

(TF5) For any x',c',x'',c' in M* such that x* x'*, there

exists a unique ordered pair (m*,d*) such that T'(x',mn,d*) = c"

and T'(x'',m*,d') = c''.

We say that 0' is the zero and that I' is the one of (M1*,T*).

We call the elements of M' symbols.

If (',T*) and (Q',S') are ternary fields, an ordered

triple w = ((M',T'),(Q*,S'),f ) is said to be a ternary field

homomorzhism if f :M* -- Q* is a function such that
------- = ---- u

n(T'(x*,m*,e*)) = S*(wx*,a''*,e') and such that wO* = 0*, wl* =

1*.

We denote the category of ternary fields by F.

3.2 Definitions. Let M be a set with distinguished elements 0

and 1, and with two ternary operations defined on I. Let N =

{n e MI 3 k E M, k z 0, 3 T(k,n,0) = 0, and let N' =

In 6 M : 3 k & -, k i 0, 4 T'(k,n,0) = 01. Define a relation

Son M by a ~ b (read 'a is neighbor to b') if and only if every

x which satisfies the equation a = T(x,l,b) is an element of N.

Define a relation -' on M by a ~' b if and only if every y which

satisfies the equation a = T'(y,l,b) is an element of N'. The

negation of a b is written a b and is read 'a is not neighbor

to b'. Then, (1:,T,T') is said to be a biternary ring if the

following twelve conditions are satisfied.

(BO) N = N', and a necessary and sufficient condition that

a -' b.is that a ~ b.

(Bl) The relation ~ is an equivalence relation; that is, the

relation ~ is reflexive, symmetric and transitive.

(B2) T(0,m,d) = T(a,C,d) = d for any a,n,d from N.

(B3) T(l,a,O) = T(a,l,0) = a for any a from IH.

(B4) T(a,m,z) = b is uniquely solvable-for z for any a,m,b

from M.

(B5) T(x,m, T(x,,d T(x,m',d') is uniquely solvable for x if and

only if m rr- m' for any m,d,m',d' from M.

(Bg) The system T(a,m,d) = b, T(a',m,d) = b' with a 4 a'

is uniquely solvable for the pair m,d; if a a', b b', we

have m 4 N; if a ~ a' and b + b', the system cannot be solved.

(B7) If a n a', b b', and if (a,b) / (a',b'), then one and

only one of the systems tT(a,m,d) = b, T(a',m,d) = b'3 and

fT'(b,u,v) = a, T'(b',u,v) = a' where u 6 NJ is solvable with

respect to m,d correspondingly u,v (where u e N), and it has at

least two solutions; and we have m' m", d' d" or u' ~ u",

v' v" respectively for any two solutions.

(B8) The system ty = T(x,m,d), x = T'(y,u,v)l where u & N,

m,d,v C M, is uniquely solvable for the pair x,y.

(B9) For any m,u E M, T(u,m,0) = 1 if and only if T'(m,u,O) =

1. If T(u,m,0) = 1, if T(a,m,e) = b, and if T'(b,u,v) = a for

some m,u,a,b,e,v 6 M, then (T(x,n,e) = y
every x,y 6 M.

(B10) The function T induces a function T* in M/~, and

(M/~,T*) is a ternary field with zero 0* = {z Iz ~ 01 and

one 1* = e le ~1.

(B11) Conditions (BO) through (B10) hold with T and T'

interchanged throughout; the new conditions will be called (BO)'

through (B10)'; condition (B10)' states that the function T'

induces a function T' in M/~', and that (N/~',T'*) is a ternary

field with zero 0* and one 1*; of course, N and N', and ~' are

interchanged throughout also.

Each element of N is said to be a riqht zero divisor.

3.3 Definition. If (C,T,T') is a biternary ring, then (M,T',T)

is a biternary ring by the symmetry of the definition of bi-

ternary ring: (M,T',T) is said to be the dual of (I1,T,T').

3.4 Definitions. Let (B,T,T') be a biternary ring. We will

frequently write B to denote (B,T,T'). We will frequently write

NB or simply N to denote the set of right zero divisors in B.

The elements of the set B are called svmbols; 0 is called the

zero of B and 1 is called the one of B. If NB = (01, we say

that B is a biternarv field.

3.5 Proposition. Let (B,T,T') be a biternary ring and let u e B.

Then u 0 if and only if u E N.

Proof. Assune u ~ 0. By (B3), u = T(u,l,0), and hence u E N

by the definition of neighbor in B.

Assume u e N. Then there is a k in I;, k X 0, such that

T(k,u,0) = 0. Since x = 0 and x = k are both solutions to the

equation T(x,u,0) = T(x,0,0), we have by (35) that u 0. //

3.6 Proposition. If (B,T,T') is a biternary ring, then 1 -) 0.

Proof. By (B10), 1i f 0'; hence 1 0. //

3.7 Proposition. In a biternary ring (3,T,T'), the equation

a = T(x,l,b) has a unique solution x for each pair (a,b). In

addition, a b if and only if x 6 N.

Proof. Let a and b be elements of the set B; that is, let a

and b be symbols. Since 0 1, by (B5) there is a unique

solution x to the equation T(x,l,b) = T(x,0,a). By (B2),

we have that T(x,0,a) = a. Hence, a ~ b if and only if

x E N. //

3.8 Proposition. Let (B,T,T') be a biternary ring and let m 6 B.

There is a u C B such that T(u,m,0) = 1 if and only if m N M. If

m 4 N, then the solution u is unique and u 4 N. Moreover, the

map S:M\N -- K\N defined by T(C(m),m,0) = 1 is a bijection. If

u Q M\N, then T(u,-l (u),0) = 1.

Proof. If m C N, then m 0 and there is no element u E B such

that T(u,m,0) = 1 since 0* 4 1* and T*(u*,0*,0") = 0* for every

u* in M/-. If m 4 N, then m 0 by Proposition 3.5, and by (B5)

there is a unique u such that T(u,m,0) = T(u,0,l). If u were in

N, then u 0 and, by (B5), T'(u*,m*,0*)= T4(0C,m*,0O) = 0*, a

Thus, we can define a map S:M\N -' M\N by T(5(m),m,0) = 1.

If u 4 N, then the system ZT(u,m,d) = 1, T(0,m,d) = 01 is uniquely

solvable for the pair m,d by (B6) since u 0. By (B2), d = 0,

and, since T*(u*,0*,0*) = 0*, m 4 N. Thus, S is surjective.

If m' satisfies the equation T(u,m',0) = 1, then the pair m',0

is a solution to the system above and hence m' = m. Thus, the

map 8 is bijective.

If u C M\N, then there is an m 6 M\N such that T(u,m,0)

1. Hence S(m) = u, and we have that m = S-l(u) and that

T(u,-(u),0) = 1. //

3.9 Definition. Let (B,T,T') and (M,S,S') be biternary rings.

A biternary rina homonorohism K:B -~ M is a concrete morphism

such that C<(:B) C NM; ;(0) = 0; o(l) = 1; D(T(x,m,e)) =

S(
in B.

3.10 Definitions. it is easily seen that the class of biternary

rings and their homomorphisms form a category. We denote this

category by B and call it the catecorv of biternnrv rinns. The

full subcategory of B whose objects are biternary fields we

denote by B'.

3.11 Definitions. We say C = (A,K) is a coordinatizud AH-nlane

and K = (g ,g ,E,,S:OE -- :) is a coordinatization of A
x y
whenever A is an AH-plane, gx,gy are non-neighbor lines of A,

E is a point of A not neighbor to either gx or g M is a set

with distinguished elements 0 and 1, OE is the line joining

O = gx g to E, and f:CE i M is a bijection such that (0) = 0,

(E) = 1.

Let C = (A,K) be a coordinatized AH-plane. If P is a

point of A, define 9(F) = (x,y) = (O(CE R L(P,g )),(OE r L(P,gx)));

the construction is indicated in Figure 3.1; and define O'(P) =

(y,x)'. If O(P) = (a,b), we say b is the y-coordinate of P and

that a is the x-coordinate of P; let ir = a, r P = b.
x y
If k is a line of C = (A,K), and if k g define X(k) =

[m,d] = [ir (L(O,k) L(E,g )),f (k fg )3; the construction is

indicated in Figure 3.2. Whenever k -t gx, we interchange the

roles of g and g in the definition of X to define \'(k) =

[u,v]' = [x(L(O,k) (r L(E,gx )),T (k ( g )
a a a N

L(P,g )
"x

L(0,k) = [m,0o

(y,y)

(x,x)

L(P,gy)

0 = (0,0)

Figure 3.1.

Figure 3.2.

It is easily seen that the maps 8, 6' (X, X') are well-

defined functions from (from Ik k t g y, from jk ik gx)

into M XM and that they are bijections.

If g is a line of C such that \(g) = Im,d], then Em,d3 is

said to be a representation of g; similarly, if X'(g) = -u,v]',

then [u,v'l is said to be a representation of g. If O(P) =

(x,y), then (x,y) and (y,x)' are said to be representations of P.

Since 9, 0', X, \' are bijections, we can, without fear of

confusion, identify a point or line with each of its representa-

tions or with its one representation. The line gx is called

the x-axis of C; g is called the v-axis of C; O is called the

origin of C and E is called the unit point of C. Let X denote

the point gx R L(E,g ), and let Y denote the point gy ( L(E,gx).

3.12 Definition. Any pair of statements or functions which can

be gotten one from another by interchanging the roles of gx and

g throughout are said to be xy-duals. The functions 8, 6' given

above are xy-duals, as are A and X'.

3.13 Proposition. Let g,h be lines of a coordinatized AH-plane

C, and let g = [m,e]. Then h is quasiparallel to g, h g1 g, if

and only if there are m',e' such that h = Im',e'1 and

(l,m) ~ (l,m').

Proof. Assume h 1i g. Then, since Ig* ( g *. = 1 implies

|h* A gy = 1, we have that Ih n gy = 1. Thus, for some m',e'

we have that h = [m',e']. Since h 11 g, L(O,g) II L(O,h) and

52

hence L(O,g) L(O,h). Thus, since L(E,o ) L(O,g),L(O,h), we

have that (L(O,g) n L(E,g )) c (L(O,h) A L(E,g )), and hence

(1,m) ~ (l,m').

Assume h = [m',e'] and (l,n) N (l,m'). Then

L(O,g) 11 L(O,h). and g h. //

3.14 Definitions. Let C and C' be coordin.tized AH-planes. A

coordinatized AH-plane honomorDhism or coordinatization homo-

morchism u:C -- C' is a map a which is an AH-plane homomorphism

such that w(gx) gx ', (g ) g and w(E) = E' where g gy

and E' are the x-axis, y-axis and unit point of C' respectively.

If C = (A,K) is a coordinatized AH-plane, then the

neighbor map K:A -- A* induces a coordinatization homomorphism

from C to C* = (A*,K*) where K* is the coordinatization of A'

whose x-axis is (g x) and so on; we denote this induced map by

K:C -- C* and call K the neighbor man of C.

3.15 Definition. It is easy to see that the class of coordina-

tized AH-planes together with their coordinatization homomorph-

isms form a category. We denote this category by C, and call it

the category of coordinatized affine Hielnslev planes. We denote

the full subcategory of 0 whose objects are coordinatized affine

planes by C*.

3.16 Construction of B:C -* B. Let C be a coordinatized AH-plane.

Define a ternary operation T:M M by T(x,m,e) = y if and only

if there exist a point P and a line g, P I g, such that 9(P) =

(x,y-), X(g) = Em,el. Define a second ternary operation
3
T':M -- M by interchanging the roles of g and g in the

definition of T; that is, let T'(y,u,v) = x if and only if there

exist Q, h, Q I h, such that 8'(Q) = (y,x)' and A'(h) = Cu,v]'.

Let B(C) = (M,T,T'). Given a morphism w:C -- C' in C, define a

map B(u:):B(C) -* S(C') by B(u)m = -'(1( (m))) for all m in M.

By M we mean the set of symbols of C.

*Remark. Many of the intermediate steps in the proof of the

following proposition are stated in tCyganova (1967)] (see our

Remarks A.2, A.15 and A.16 in Appendix A): she states (Lemma 1),

part of (Lemma 2), part of (Lemma 4), (Bl), (B2), (B3), (B4),

(B5), (B6), (B7) and (BS).

3.17 Proposition. The map B:C -- B defined above is a functor

from the category of coordinatized affine Hjeimslev planes to the

category of biternary rings. If C* is a coordinatized affine

plane, then B(C*) is a biternary field. If C is a coordinatized

AH-plane, then (a,b) ~ (a',b') in C ! a ~ a', b ~ b' in B(C);

[m,d] ~ [m',d'3 in C 4 m m', d d' in B(C); [u,v3]' [u',v']

in C 4 u u', v .- v' in B(C).

Proof. Assume C is a coordinatized AH-plane. Define a o b if

and only if (a,a) ~ (b,b). Let N = In e Mi n o 04. Observe that

if C is a coordinatized affine plane then N = 101.

(Lemma 1) (a,b) ~ (a',b') 44 a o a', b o b'.

(Proof) Assume (a,b) ~ (a',b'). Then

L((a,b),g ) L((a',b'),g ); so that, since CE is not quasi-

parallel to g (L((a,b),g ) r CE) (L((a',b'),g ) ( OE) and
hence (a,a) ~ (a',a'), and a o a'. Similarly, b o b'.

Assume a o a', b o b'. Then L((a,a),g ) ~ L((a',a'),g );
so that, since g is not quasiparallel to gx, (a,b) ~ (a',b).
Also L((b,b),gx) ~ L((b',b'),gx); so that (a',b) ~ (a',b').
Thus, (a,b) (a',b'). /

(Lemma 2) [m,e] [m',e'] 4= m o m', e e'; hence by
xy-duality, [u,v]' ~ [u',v']' < u u o ', v o v'.
(Proof) Assume m o m', e o e'. Then by (Lemma 1),
(l,n) ~ (l,m'); so that [m,0] 11 [m',0j. Hence Im,eo 1 [m',e'l
and, since (0,e) ~ (0,e') by (Lemma 1), [m,el- l',e'.
Assume [m,e] [m',e'). The lines are both non-neighbor
to g ; so that (0,e) ~ (0,e') and e o e'. Since [i,e7l I m',e'],

[m,0o II [m',O], and hence (l,m) ~ (l,m'). Thus, m m'. /
(Lemma 3) [m,e] I gx if and only if m o 0; hence, by

xy-duality, [u,v]' II gy u o 0.
(Proof) Assume [m,e] I gx. Then [m,0] \\ g; so that
(l,m) (1,0) and hence m o 0.
Assume m o 0. Then (l,m) (1,0) and [m,01] 1 g ; so that

Cm,e 11 g /
(Lemma 4) N = rn MI 3 k e M, k / 0, 9 T(k,n,0) = 0]; and
hence by xy-duality N = n e M 3 k e6 k 4 0, T'(k,n,0) = O].
(Proof) Assume n C N\103. Then n o 0 and (l,n) ~ (1,0);
so that [n,01 [0,01. But [n,0o] [0,0o; so that there is a
point (k,0) on both such that (k,0) Z (0,0). Hence,k 4 0 and
T(k,n,0) = 0. If n = 0, then (1,0) I (0,01; hence T(1,0,0) = 0.

L1

Assume there is an element k, k > 0, such that T(k,n,0) =

0. Then both (k,0) and (0,0) are on [n,0] and o0,0]; so that

[n,0] ~ [0,0]. Thus (l,n) ~ (1,0) and n o 0, n E N. /

(Lemma 5) a o b if and only if every x which satisfies the

equation a = T(x,l,b) is an element of N.

(Proof) Since O -- E, 0 i 1. Hence, by Proposition 3.13,

[0,a] is not cuasiparallel to [l,b]. Thus, the equation a =

T(x,0,a) = T(x,l,b) has a unique solution x.

Assume a o b. Then (0,a) ~ (0,b); so that [o,al [0,bl.

Thus (10,a \ [l,b1) ~ (0,b (\ [l,b]). Thus, if (x,a) =

[0,a] f [l,bl then (x,a) (0,b) and x o 0. Hence a = T(x,l,b)

and x E N.

Assume every x which satisfies the equation a = T(x,l,b)

is in N. .Let (x,a) = [l,b] 0 [0,al'; then a = T(x,l,b); x o 0.

Since l[l,b3 g \ = 1, (0,b) = [l,b] g g Observe that

t0,x1' 0,01' where g = [0,0]' and hence

([0,xl' [l,bl) ~ ([0,03 [l,b]); (x,a) -(0,b), and a o b. /

(BO) The xy-dual to the proof of (Lemma 5) above shows

that a o b if and only if every y which satisfies the equation

a = T'(y,l,b) is an element of N. Thus in B = B(C) the

relations ~ and ~' of the definition of biternary ring are equal

to the relation o defined here. By (Lemma 4) N ='NB = 'B so

that condition (BO) of the definition of biternary ring holds in

B(C).

(Bl) The relation o on M is obviously an equivalence

relation.

(B2) Since (0,d) I [m,d], T(0,m,d) = d. Since [0,d] lgx,

(a,d) I [O,dl. Hence T(a,0,d) = d.

56
(B3) Since (l,a) I Ca,0o T(l,a,0) = a. Since [1,0] = OE,

we have that (a,a) I [1,0] and T(a,l,0) = a.

(B4) There is a unique line parallel to [E,0o through the

point (a,b): say [E,rl. Then T(a,m,z) = b. If T(a,m,z') = b,

then (a,b) I [m,7'] ; hence z = z'. Thus, z is the unique solution

to the equation T(a,m,z) = b.

(B5) The equation T(x,m,d) = T(x,m',d') is uniquely

solvable for x if and only if there is a unique point

(x,y) I [m,d],[m',d']; hence if and only if [m,d] is not quasi-

parallel to [m',d'] and, by Proposition 3.13, if and only if

m j m '.

(B6) Look at the system [T(a,m,d) = b, T(a',m,d) = b'1.

If a a', then the unique line g joining (a,b) and (a',b') is

not quasiparallel to g Hence g = (m,d2 for some n,d and the

pair m,d is the unique solution to the system. If a a a', b o b',

then Cm,d] gx and hence by (Lemma 3) m o 0. If a o a', b b',

then, if (a,b),(a',b') I g, g I g and hence the system has no

solution Vm,d].

(B7) Let g,g' be any two distinct lines joining (a,b) and

(a',b'), (a,b) / (a',b'). Then g p g'. Observe that (g U gy

< g g y) and that g,g' II gy p g = [u,v3', g' = [u',v' for

some u,u' 4 N, v,v' e M. The lines g,g' are not quasiparallel

to g if and only if g = [m,d], g' = [m',d'] for some m,m',d,d'

in M. Hence condition (B7) holds in B(C).

(B8) Any line of the form [u,v]', u 6 N, is quasiparallel

to gy and hence meets any line [m,d] (not quasiparallel to g )

in exactly one point (x,y).

(B9) T(u,m,0) = 1 if and only if (u',l) I [m,0]. Hence

T(u,m,0) = 1 if and only if Cu,0]' = [m,0], and by xy-duality,

Cu,0]' = [m,03 if and only if T'(m,u,0) = 1. If T(u,m,0) = 1,

and (a,b) I (m,e]J,u,v]', then [m,e] = [u,v]', and hence

(x,y) I [m,e] '=> (x,y) I [u,v]'.

(B10) The construction of the ternary operation T maps,

under the neighbor map K, to the construction of T* which is the

usual construction of the ternary field of a coordinatized affine

plane--see (Hall (1959), pages 353-355, Section 20.31--and

(* (0) = 0*, K C (1) = *i, where 0* is the zero and I1

the one of (M*,T').

(Bll) Observe that since we have made no special require-

ments on C, the xy-duals of conditions (B0) through (B10) also

hold in B(C).

Thus we have shown that B(C) is a biternary ring.

Assume that w:C C' is a morphism in C, and that B(C) =

(N,T,T'), B(C') = (Q,S,S'). We wish to show that B(w):B(C) B(lC')
-I1
defined by B(C)m = V'("(C (m))) is a biternary ring homomorphism.

Define w':M -- Q by B(w)m = w'm. Then it is easily seen that

w(x,y) = (w'x,w'y), w[m,d] = [w'm,w'd] and wlu,v]' = ['u,t'vl'.

Thus since preserves incidence u'(T(x,n,e)) = S(w'x,w m,w'e)

and w'(T'(y,u,v)) = S'(w'y,w'u,w'v). Since (n,n) ~ (0,0) implies

(w'n,w'n) ~ (0,0), W'NM N N Observe that w'(0) = 0, w'(l) =

1. Hence B(w) is a morphism in B.

Obviously B(1 ) = 1 (C). If < :C1 a C2 and :C2 C3 are

morphisms in C, then if m 6 .1 B(rx)m = C3(K(l-l(m))) =

3( 0[ 2- f2 s(( (m)))) = B(p)(B(C)(m)), so that E(pc) = B(p)B(o)
and B:C --' B is a functor.

If C is a coordinatized affine plane., recall that N B(C)

{o] and hence observe that B(C) is a biternary field. //

3.18 Construction of C:; C and A:0)- ( I-x {( ),< )>')/=. Given

a biternary ring B = (C,T,T') we construct an incidence structure

SB = (P,c,I) and a parallel relation II in the following way. Let
A = M M and let cT = M M` xtO0 and T,' = M FIl. For

convenience we denote an element (m,d,0) of 0T by [m,d] and an

element (u,v,l) of qT, by [u,v]'. We define

6 : T -T M
S(g) = u>', if g = [u,v]'. We define incidence by (x,y) I Cm,d]
y = T(x,m,d) and by (x,y) I [u,v]' 4= x = T'(y,u,v). We

identify and (u>' 44> T(u,m,0) = 1 = T'(m,u,0). If

= (u)', and if there is a point (a,b) on both [m,dl and Cu,v]',

then by (B9) and (B9)', ((x,y) I [m,d] <-> (x,y) I [u,v]'), and

we identify [m,d] and [u,vl'. Denote the set of lines by &J where

(0 = (oT U T\ )/=. Define A: 0-- (M, A J >,<( )')/= in the obvious
way. We define g 11 h if and only if a(g) = &(h). We denote the

incidence structure and the parallel relation thus defined by

AB. We let B: [1,01 -- M be defined by SB(a,a) = a. Let K =

([0,0],[0,0]',(1,1),M, 5). We define C(B) to be (AB,KB).
Given a biternary ring homomorphism m:B -- B' we define

C(w):C(B) -- C(B') by C())(a,b) = (wa,wb), C(w)[m,d] = [Cm,wd]

and C(w)[u,v]' = [u,wv1]' for all points (a,b), lines [m,dl,

[u,v]' in AB. Observe that C(a) is well defined.

3.19 Lemma. Let (B,T,T') be a biternary ring. Then the

following conditions hold in the construction given above.

(a) If m,d a ', m N ;, then there is a unique line [u,v3' of

OT' such that [m,d = [u,v]'. If m 6 N, there is no such line
lu,v]'.

(b) If u,v 6 M, u N, then there is a unique line [m,d] of

9T such that [u,vl' = [m,d]. If u e N, there is no such line
[n,d].

(c) The function L: 0--- (N:xs >,x )' )/= constructed above is

well defined.

Proof. (a) Assume g = nm,d], m 4 N. Then (0,d) I g. There is

a unique u & N such that T(u,m,0) = T(u,0,l) since 0 m. Hence

T'(m,u,0) = 1, and there is a unique v such that (0,d) I Cu,v]'.

Hence [m,d] = (u,v]'. If [m,d] = [w,zl', then T(w,m,0) = 1 and

u = w. Also since 0 = T'(d,u,z), z = v. Thus [u,v]' is unique.

Assume g = [m,d], m 6 N. Then, since m ~ 0, 0 7 1, there

does not exist an element u C M such that T(u,m,0) = 1, and

hence there is no [u,v'' 6 qT' such that [m,d] = [u,v3'.

(b) This is the dual of case (a).

(c) If [m,d] = [u,v]', then &[m,d] = (m>) = (u = u,v';

hence L is well defined. //

3.20 Definition. If B = (M,T,T') is a biternary ring and if B' =

(M,T',T) is the dual of B, then (x,y) I Cm,d] in C(B) 4=

(y,x) I [m,dj' in C(B'), and (x,y) I [u,v]' in C(B) 4=

(y,x) I [u,v] in C(B'). The structure C(B') is said to be the

structure dual to C(B).

60
3.21 Prorosition. Let B be a biternary ri-ng and let B' be the

dual of B. The map 9:C(3) --C(B') defined by O(x,y) = (y,x),

[lm,d] = [m,d)' and 6[u,v]' = Eu,vl is an incidence structure

homomorphism which preserves and reflects the parallel relation. //

3.22 Proaosition. If 3* is a biternary field, then C(B*) con-

structed above is a coordinatized affine plane.

Proof. If B* = (7M,T*,T*') is a biternary field, then (M*,T*)

is a ternary field and it is easily seen that C(B*) is the

coordinatized affine plane commonly constructed over the ternary

field (M*,T*): see [Hall (1959), top of page 3563. //

3.23 Proposition. The map C:B -* C indicated above is a functor

from the category of biternary rings to the category of coordi-

natized affine Hjelmslev planes.

oof. Let B = (M,T,T') be a biternary ring. The structure AB

instructed above is an incidence structure SB with a relation

)arallel' defined on the lines of SB. We wish to show that AB

s an AH-plane. We start by proving a number of lemmas. To

void confusion, throughout the remainder of this proof we will

Ise the symbol to indicate the neighbor relation in B, and

:he symbol C to indicate the relation 'affinely neighbor' in AB:

see the definition of AH-plane, Definition 2.21.

(Lemma 1) Whenever P,Q are points, there is a line g such

61

(Proof) Let P = (a,b), Q = (a',b'). There are four cases.

(Case 1) Assume P = Q = (a,b). Then by (B2),

(a,b) I O[,bl.

(Case 2) Assume FP 0, a + a'. The system T(a,m,d) = b,

T(a',m,d) = b' has a unique solution m,d by (B6), and hence

P,Q I [m,d].

(Case 3) Assume P g Q, b b'. This is Case 2 in the

dual structure; hence there is a line Cu,v]' joining P,C.

(Case 4) Assume P Q, a a', b ~ b'. By (B7) there is

at least one line joining the two points. /

(Lemma 1) (a,b) is affinely neighbor to (a',b') 4=q a a',

b b'.

(Proof) Let P = (a,b), Q = (a',b'). If P = Q, then a ~ a',

b b' by (Bl) and P is affinely neighbor to itself since (a,b)

is on both CO,b] and [0,a' by (B2) and (B2)'. Thus, we may

assume P 1 Q.

( ) Assume P is affinely neighbor to Q. Then at least two

distinct lines g,h join P and Q. By (BS) if P,Q I g,h; then

g,h 6 T or g,h C 0 .,

(Case 1) Assume P,Q I g,h; g I h, and g,h & o Then the

system T(a,m,d) = b, T(a',m,d) = b' has two solutions for Cm,d].

Thus, by (B6) av a' and b b'.

(Case 2) Assume ?,Q I g,h; g / h, and g,h & 0T,. This

reduces to Case 1 in the dual structure.

(c-) Assume a ~ a', b b', P 1 Q. Then, by (B7) there

are at .least two lines joining P and Q; so that P [ Q. /

(Lemma 3) Let P I g,h. Then gg A hi = 1 if and only if

g 0 h.

(Proof) (4 ) Assume g 0 h, P I Q,h. We wish to show

(g hi = 1. Let P = (a,b).

(Case 1) Assume g = [m,e], h = [m',e'] and that Q I g,h;

Q x P. Let Q = (a',b'). By Ler a 2, a ~ a', b b'. Hence by

(B7), m ~ m', e ~ e'. Let (c,d) be a point on one of the lines;

say g. Then d = T(c,m,e).

Let d' = T(c,m',e').

By (E10),

d' d. Thus by Lemma 2, (c,d) O

a contradiction. Hence Ig 0 h\ =

(Case 2) Assume g,h & C ,.

structure, so Ig A hi = 1.

(Case 3) Assume g [m,e],

may assume u 6 t;, since otherwise

by writing h in the form rm',e'].

( ) Assume Ig I h\ = 1, P

(Case 1) Assume g = h. If

(0,e),(l,T(1,m,e)) I c, and since

(c',d'). By symmetry, g O h,

1.

This is Case 1 in the dual

h = Cu,vi'. By Lenma 3.19,

this can be reduced to Case

Since u 6 N by (38) Ig g

we

1

h\

I g,h. We wish to show g P h.

g = mC,e] then

0 i 1 by (B10) this case

doesn't occur, and dually if g = Lu,v '.

(Case 2) Assume g = [m,e1, h = Cm',e'], g 6 h. By (B5)

the uniqueness of the solution x = a to the equation T(x,m,e) =

T(x,mn,e') implies m i m'. Since by (B10) 0 1, and since

- is an equivalence relation there is a symbol c such that c + a.

Let d = T(c,m,e). Assume g 0 h. Then there is a (c',d') I h

such that c c', d d' by Lemma 2 and the definition of

'affinely neighbor'. Hence the equation T*(x',m*,e) =

T*(x*,m'*,e'*) has two solutions: x* = a*, x' = c', a contra-

diction. Thus g 0 h.

63

(Case 3) Assume g,h a T'' g h. This is Case 2 in the

dual structure, hence g ( h.

(Case 4) If none of the preceding cases occurs, by

Lemma 3.19 we may assume g = [m,e h = [u,v1' where u E N.

There is a symbol w such that w v. Assume g 0 h. Let z =

T(w,m,e); Q = (w,z). There is a point R = (w',z') on h such

that R O Q. By Lemma 2, w ~ w'. Since u E N, u ~ 0, and since

w' = T'(z',u,v) by (B10), w' = v'; so that w' ~ v. Thus, since

is an equivalence relation on B; w ~ v, a contradiction.

Hence g 0 h. /

(Lemma 4) If g = [m,e]; then g h if and only if h =

[m',e'3 and m ~ m', e e'. If g = Cu,v]'; then g O h if and

only if h = [u',v']' and u -u', v v'.

(Proof) (Case 1) Assume g = [m,e].

(=) Assume h = [m',e'], where m m', e e e'. Let (a,b)

be a point on one of the lines: say g. Let b' = T(a,m',e'). By

(B10), b ~ b'. Hence (a,b') I h and (a,b') O (a,b) by Lemma 2.

By symmetry, g D h.

( ) Assume g D h. If g = h we are done. Assume g P h.

By Lemma 3, g E1 h implies Ig n hi 1; hence by (ES), h / [u,vj',

u & N; hence by Lemma 3.19 (b), h = [m',e'lfor some m',e' in M.

If m + m', then by (B5), there is a unique point of inter-

section, a contradiction by Lemma 3. Hence m ~ m'. The

point (0,e) is on g. Let (u,f) I h such that (0,e) O (u,f).

Then, 0 u, e ~ f. By (B10) there is a unique z* such that

T*(0*,m*,z") = e*. Then e* = e'* = z* and e e'. Thus, m ~ m',

e ~ e'.

(Case 2) Observe Case 2 reduces to Case 1 in the dual

structure. /

(Lemma 5) The relation 'affinely neighbor' is an equiva-

lence relation on the points and lines of A .

(Proof) The relation 'affinely neighbor' is an equivalence

relation on the points of AB and is a reflexive and symmetric

relation on the lines of AB by Lemmas 2 and 4. Assume g 0 h,

h D k. If g C gT' then h 6 )T; so that k c aT by Lemna 4. Hence

g k by Lemma 4 and (Bl). Similarly, if g 6 0T,, then h,k C qT'

and g 0 k. Thus 'affinely neighbor' is an equivalence relation

on the points and lines of A /

(Lemma 6) If P is a point; g, a line, then there is a

unique line h such that P I h, h II g.

(Proof) (Case 1) Let P = (x,y) and g = Cn,b3. Then, by

(84) there is a unique symbol z such that y = T(x,m,z). Hence

[m,z3 is the unique line of aT parallel to [m,b3 containing P.

Assu-e P I [u,v]' and [u,vl' n Inr,b]. Then (u>' =
that since P I [u,v]',nm,z1, and Eu,v]' = [m,z]. Thus, [ m,z is

the unique line through F parallel to [m,bJ.

(Case 2) This case reduces to Case 1 in the dual struc-

ture. /

(Lemma 7) There is a map :A -- A which is a surjective

incidence structure homomorphism such that g n hi= 0 in AB implies

qg II fh in AB, and such that (0P =Q -C F 0 Q) and ('g = Ph

4* gO h.)

(Proof) If a & K, denote the n-equivalence class of a in

B by 8*. Then define q(a,b) = (a*,b'), q[m,d] = m*,d* and

flu,v]' = [0o,v']' when u & N. Then, since the map v:B --> B

induced by is a biternary ring homomorphism, ? preserves the

incidence relation. The remainder of the lemma is immediate by

Lemmas 2, 4 and 5. /

Thus, AB is an AH-plane. Observe that [0,01 [0,01]; that

(1,1) Z [0,0],[0,0o'; and that SB is a bijection. Thus C(B)

(AB,([0,0,[O0,0',(l,1),M,M )) is a coordinatized AH-plane.

If w:B -- B' is a biternary ring homomorphism, then C(w)

defined earlier is an incidence structure homomorphism which

preserves the parallel relation. Recall that w(NB) c NB'

Hence, if a ~ b in B, the ua ~ wb in 3'. Thus, (a,b) D (a',b')

in C(B) implies (oa,eb) 0 (wa',wb') in C(B'), and C(w) preserves

the neighbor relation on points and similarly on lines. Since

c(w) takes gx gy and E to the x-axis, the y-axis and the unit

point of C(B'), C(w) is a coordinatized AH-plane hcomoorphism.

Since C(lg) = 1(B) and C(p) C(p)C(), C:B --C is a

functor.

If B" is a biternary field, then N, = tO3 and C(B') is a

coordinatized affine plane since the neighbor map of C(B*) is

essentially the identity map. //

3.24 Proposition. The functor BC is the identity functor on B.

Proof. Let (B,T,T') be a biternary ring, and let (Q,S,S') =

BC(B). By the constructions, Q = B, and y = S(x,m,d) in BC(B)

4c C (x,y) I Em,d] in C(B) 4> y = T(x,m,d) in B. Similarly,

x = S'(y,u,v) in BC(B) 4> (x,y) I [u,v]' in C(B) <, x =

T'(y,u,v) in B. Hence BC(B) = (B,T,T').

66

Let '3:3 -- B" be a morphism in 5. Then C(w)(a,a) = (Qa,wa)
-I
for every a 6 3, and E(g(())a = (w)-l a = g"(Wa,za) = wa.

Thus BC:B -- is the identity functor on 3. //

3.25 Construction of 1C --B CB. Let C = (A,(g ,g E,,,)) be a

coordinatized AH-plane. Then CB(C) = (AB(), ([0,0o,[0c,0,(1,1),

M,nB(C))) and it can easily be seen by locking at the usual
identification of points and lines of C with their representations

that there is an isomorphism fC:C -4 C(C) which takes a point

(x,y) of C to the point (x,y) of CB(C).

3.26 Proposition. The map o: : -- C2 indicated above is a

natural isomorphism.

Proof. Let w:C C' be a morphism in C. Recall that B(w)m =

{'(W( -im)). Hence, if P is a point of OE in C, (w)(?) =

t'(wP). Thus if ? I CE in C, CB(w)(p P) = CE(w)(P,P) =

(B(w)(gP),B(w)(sP)) = (r'(wP), '(P)) = C,(wP). Thus the

diagram:

C c
c ----- ()

C' ---C (C')

commutes with respect to the x-axis of C, the y-axis of C and

all the points of GE, and hence it commutes with respect to all

the lines which are parallel to either g or g and hence with

respect to all the points of C. The image of each line of C in

CB(C') intersects either both C0,0o and [0,1i or both [0,0]' and

[0,1]' in CB(C') in points which are non-n2ichbor images of

points in C. Thus, since the diagram commutes with respect to

all the points of C, it commutes with respect to all the lines of

C. Thus P:-l CB is a natural isomorphism. //

We have shown the following theorem:

3.27 Theorem. The functors B:C -- B and C:B C are reciprocal

equivalences where C is the category of coordinatized affine

Hjelmslev planes and B is the category of biternary rings. //

3.28 Corollary. The functors B*:* -- B* and C*:B* C*

defined as restrictions of B and C respectively are reciprocal

equivalences where B* is the category of biternary fields and C'

is the category of coordinatized affine planes. //

3.29 Definition. There are reciprocal equivalences F':C" F'

and C :F' -- '* (where F* is the catecorv of ternarv fields)
-F --- -----------------------__
defined by letting F*(C) be (M,T) if B*(C) = (I,T,T'); by letting

C F(F) be the usual coordinatized affine plane constructed over
-F
a ternary field F (see the proof of Proposition 3.22); by letting

F* take a morphism w:C -r C' to (F'(C),F*(C'),f .() ) and by

letting C take a morphism o = (F,F',f) to (C '(F),CF*(F'),f ,)

where f is defined by f ,(x,y) = (f x,fy), f ,tm,d] =[p(m,ad]

and f ,0,vo = [O,av]'.

The composition of B':C' -- B and C *:F* -- is a

functor B*C *:F* -- B* from the category of ternary fields to
the category of bitrnar fields. If (,T) is a ternary field,
the category of biternary fields. If (M*,T*) is a ternary field,

Full Text

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COORDINATIZED HJELKSLEV PLANES By PHYRME YOUENS BACON A DISSERTATION PRESENTED TO TEE GRADUATE COUNCIL OF THE UNIVERSITY C? FLORIDA IN PARTIAL FULFILLMENT OP THE REQUIREMENTS FOR THE DEGREE CF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1974

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Copyright (c) 1974 by Phyrne Youens Bacon

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To my husband, Philip Bacon, to mymother , Cynthia Tanner Youens, and to the memories of my father, Willis George Youens, Sr., M.D., and my maternal grandmother, Fhyrne Claiborne Tanner.

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The white people must think paper has some mysterious power to help them on in the world. The Indian needs no writings; words that are true sink deep into his heart where they remain; he never forgets them. Four Guns, Oglala Sioux (1891) quoted in "I Have Spoken" compiled by Virginia Irving Armstrong (1971), The Swallow Press Inc., Chicago, pages 130-131.

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ACK NOWLE DGEME NTS I would like to express my thanks and appreciation to my advisor, David A. Drake, for his superior example, for his contagious enthusiasm, and for his excellent suggestions. I would like to thank the other members of my committee, Ernest E. Shult, Kermit N. Sigmon, Mark L. Teply, Mark P. Hale, Jr., and Billy Thomas, for their comments and suggestions. I would also like to thank George E. Strecker for his comments. I would like to thank Benno Artmann, N. D. Lane, and Willian J. LeVeque for having indirectly provided copies of papers referenced in my bibliography, [cygancva (1967)], LLorimer (1971)] and [skornjakov (1964)] respectively, and I would like to thank Wladimiro Scheffer for his translation of Ccyganova (1967)]. I also wish to thank the many professors and fellow students who have helped make my graduate study an interesting and challenging experience. I would especially like to thank Richard D. Present, Vj'illiam M. Bugg, >,;. Edward Deeds, E. G. Harris, Harold C. Schweinler, Don D. Miller, David R. Hayes, John G. Moore, Dennison R. Brown, George E. Strecker, W. Edwin Clark, and Charles I. Babst.

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TABLE OF CONTENTS ACKNOWLE DGEMENTS LIST CF DEFINED TERMS LIST OF FUNCTORS LIST OF SPECIAL NAPS KEY TO CATEGORIES KEY TO SYMBOLS ABSTRACT Sections 1. INTRODUCTION 1 2. HJELMSLEV PLANES 7 3. BITER NARY RINGS 4. SEMITRANSLATIONS AND GEOMETRY 5. SEMITRANSLATIONS AND ALGEBRA 6. PREQUASIRINGS AND QUASIRINGS 7. KERNELS OF QUASIRINGS 116 8. OTHER CENTRAL AXIAL AUTOMORPHISMS 123 9. AH-RINGS 139 10. HJELHSLEV STRUCTURES 150 11. DESARGUESIAN PH-FLANES 199 12. PAPPIAN CONFIGURATIONS 208 Appendices A. RESTRICTED 3ITERNARY RINGS 222 vi 44 69 36 106

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TABLE OF 20NTENTS continued Appendices continued B. QUASICONGRUSNCES 243 BIBLIOGRAPHY oco 253 BIOGRAPHICAL SKETCH 267

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LIST OF DEFINED TERMS Term Subsection natural isomorphism, functor 2.1 function, graph, compose, identity function, underlying set, concrete morphism, underlying set function, natural composition, natural identity rr.orphism, identity, concrete category, honomorphisms , map, map 2.2 generated by, generated by, of, with 2.3 surjective, injective, bijective 2.4 equivalence, isomorphism 2.5' reciprocal equivalences 2.6 preserve, reflect 2.7 incidence structure, points, lines, incidence relation, is incident with, lies on, is a point of, is on, goes through, is a line through, join, collinear, copunctal 2.8 incidence structure homomorphism 2.9 the category of incidence structures 2.10 the incidence structure induced from S by ~, the induced incidence structure 2.12 projective plane, ordinary projective plane 2.14 parallel, affine plane, ordinary affine plane 2.15 viii

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LIST OF DEFINED TERMS continued Term Subsection neighbor, projectively neighbor, protectively neighbor, projective Hjelmslev plane, PH-plane 2.17 projective Hjelmslev plane homomorphism, the category of projective Hjelmslev planes, the category of projective planes 2.19 parallel, affineiy neighbor, affinely neighbor, affine Hjelmslev plane, AH-plane 2.21 affine Hjelmslev plane homomorphism, the category of affine Hjelmslev planes, the category of affine planes 2.23 the gross structure, the neighbor map, the gross structure, the neighbor map 2.25 . is neighbor to, is not neighbor to, is not parallel to 2.26 non-degenerate, degenerate 2.30 direction 2.35 quasiparallel, quasiparallel , quasiparallel 2.37 lined incidence structure, base line, affine points, affine line, lined incidence structure homomorphism, the category of lined PH-planes, the category of lined projective planes, the category of lined incidence structures 2.39 the generalized incidence structure, the lined generalized incidence structure, 2.40

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LIST OF DEFINED TERMS continued Term Subsection generalized point, generalized line, generalized incidence structure, line, point, incidence structure 2.40 neighbor, generalized neighbor relation, neighbor relation, neighbor, neighbor relation, generalized neighbor relation 2.43 derived from, derived from, derived from H by use of, derived, extended to, extended to, extensions of 2.51 embedding, incidence structure embedding of S into S' 2.52 generalized AH-plane, generalized AH-plane homomorphism, the category of generalized AH-planes 2.58 AH-plane embedding 2.62 ternary field, zero, one, symbols, ternary field homomorphism, the category of ternary fields 3.1 neighbor, not neighbor, biternary ring, right zero divisor 3.2 dual 3.3 symbols, zero, one, biternary field -3.4 biternary ring homomorphism 3.9 the category of biternary rings, the category of biternary fields 3.10

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LIST OF DEFINED TERMS continued ,?erm Subsection coordinatized AH-plane, coordinatization, symbols, y-coordinate , x-coordinate, representation, representation, representations, X-axis, y-axis, origin, unit point 3.11 xy-duals 3.12 coordinatized AH-plane homomorphism, coordinatization homomorphism, neighbor map 3.14 the category of coordinatized affine Hjelrr.slev planes, the category of coordinatized affine pl.ines 3.15 dual 3#20 the biternary field associated with 3.29 generated by, the AH-plane generated by a biternary ring 3.31 dilatation, trace, semitranslation, gemitranslation with direction P, translation 4.1 (P,g)-endomorphism, central axial endomorphism, center, axis, (P,g) -endomorphism, central axial endomorphism, center, axis 4.3 neighbor, neighbor endomorphisms 4.6 j-angle, vertices, sides, triangle, T-related, (^j^^-related, W .g^-j-Desarguesian, Cr r g w )-H-Desarguesian 4.10 the canonical expansion of a (j-l)-angle to a j-angle 4#12 (P,Â£) -mime tic 4.13

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LIST OF DEFINED TERMS continued T erm Subsection (P,g)-transitive 4.17 order, infinite order 4.20 T-addition, T-multiplication, T'-addition, T'-multiplication, linear, linear 5.1 (k)-regular for s, (k)-regular, (k) '-regular for s, (k ) '-regular , axially regular, regular 5.10 T-regular, regular, axially regular, regular in the direction T 5.13 the category of axially regular biternary rings, the category of axially regular coordinatized AH-planes 5.21 the category of coordinatized translation AH-planes, the category of regular biternary rings 5.26 quasifield, zero, one 6.2 addition, first multiplication, second multiplication, prequasiring, right zero divisor, zero, one 6.3 quasiring 6.5 dual 6.6 skew quasiring 6.7 prequasiring homomorphism 6.11 the category of prequasirings, the category of quasirings 6.12 generated by, 6.19 xii

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LIST CF DEFINED TERMS continued Term Subsection the AH-plane generated by a prequasiring 6.19 biquasif ield, the category of biquasif ields, the category of coordinatized translation affine Planes 6#20 kernel, the kernel of a quasiring 7.1 local ring 7 # 2 trace preserving, a trace preserving endomorphisrn of the translation group 7.5 left modular for s, left modular, strongly left modular, T'-v:eakly left modular for s S.l strongly (p, g w ) -transitive 8.6 P-related, (F,gÂ«)-related, (P.g^-H-Desarguesian, strongly (P.g^J-H-Desarguesian 8.8 ((0),C0,0]')-normal for s, ( (0 ) , [0 ,0] * )-normal , T'-weakly ( (0 ) , [0,0] Â• )-norxal for s, T-weakly ( (C ) , l0,0] Â• ) -normal for s 8.10 (<0>Â» t [0 f 0l')-normal for s, (*(0) Â• ,[0,0l Â• )-normal 3.15 affine Kjelmslev ring, AH-ring, Hjelmslev ring, H-ring 9<1 AK-ring homomorphism, the category of AH-rings 9.2 kernel quasiring, the category of kernel quasirings 9.3 Desarguesian, the category of coordinatized Desarguesian AH-planes 9.6 the AH-plane generated by an AH-ring, generated by g s xiii

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LIST OF DEFINED TERMS continued Term Subsection Hjelmslev Desarguesian, strongly Hjelmslev Desarguesian 9.9 the category of division rings 9.12 the category of coordinatized Desarguesian affine planes 9.13 the Hjelmslev structure of an AH-ring S 10.3 near, near, near, near 10.6 the lined affine Hjelmslev structure of S 10.9 extended to, extended to (HtSl^g^) through yu., extended to, extended to H(S) through yu. 10.10 projectively Desarguesian 10.16 Desarguesian 10.17 (P,g)-automorphism, ( P ,g)-transitive 10.19 full j-vertex 10.26 Hjelmslev structure, near, Hjelmslev structure homomorphism, gross structure, neighbor map, the category of lined Hjelmslev structures 10.36 full, a full lined Hjelmslev structure homomorphism 10.39 basis triple 10.46 Klingenberg ccordinatization -10.47 Klingenberg coordinatization 10.49 induces x through X and X 1 , is induced through X and X' by
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LIST OF DEFINED TERMS continued Term Subsection extension of Â« through X and V 10.52 the canonical basis triple, canonical cocrdinatization, basis triple 10.59 the category of Desarguesian AH-planes with non-degenerate AH-plane homomorphisms

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LIST OF DEFINED TERKS continued Terrr ' Subsection congruence 3 2 semicongruence, projection map, quasicongruence B.3 quasicongruence homomorphism ' B.4 pointed AH-plane, base point, pointed AH-plane homomorphism, the catecory of pointed translation AH-planes, the category of pointed translation affine planes E.8 parallel 3 9 quasicongruence coordinatization, the canonical base point, the canonical quasicongruence coordinatization B.19 the category of translation AH-planes with non-degenerate AH-plane homomorphisms B.20

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LIST CF FUNCTORS Funct

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LIST OF FUNCTORS continued Functor Subsection 6.18 6.18 6.20 6.20 7.6 7.6 9.4 9.4 9.11 9.11 9.16 9.16 10.2 10.63 D:((H ) ) f -* ( D ) n 10.66 Q

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LIST OF FUNCTORS continued Functor Subsection 2 F Â«:FÂ— Z* Am7 F * :Z* A. 7

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LIST OF SPECIAL MAPS Symbol In use Subsection f w (A,A',f M ) 2.2 K k:A Â— Â• A* 2.25 K K:H Â— * H* 2.25 TT TV(g) 2.35 g g 2.40 P P(TT) 2.40 S S (H,g) 2 47 T T(x,m,b) 3.2 TÂ« T'(y,u,v) 3.2 % y.OE Â— M 3.11 + a + b 5.1 X Â„ a%b = ab 5.1 a Â• b 5.1 a-b 5.1 Z Z(x,m,a) 5.4 2* Z'(y,u,b) ( 5.4 i rx. 10.2 i y Â± s 10.2 G G (H(S),h) 10 9 G G s 10.9 Kern Kern Q 7.1

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LIST OF SPSCIAL NAPS continued Symbol In use Subsection -f

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KEY TO CATEGORIES Equivalent categories are joined by [Â• Category Subsection S incidence structures 2.10 A s lined incidence structures 2.39 A rA* affine planes 2.23 H * lined projective planes 2.39 H* projective planes 2.19 A AH-planes 2.23 c A generalized AH-planes 2.58 H PH-planes 2.19 H lined PH-planes 2.39 9 A -C coordinatized AH-planes 3.15 a ~B biternary rings 3.10 A 2 restricted biternary rings A. 6 A "~C* coordinatized affine planes 3.15 ~B* biternary fields . 3.10 A -Z* restricted biternary fields A. 6 F* ternary fields 3.1 . xxii

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KEY TO CATEGORIES continued Category Subsection T< ~2 axially regular coordinatized AH-planes 5.21 ~ B 2 axially regular biternary rings 5.21 L " v prequasirings 6.12 -Cf coordinatized translation AH-planes 5.26 _B r regular biternary rings 5.26 U Q quasirings 6.12 'Â•"^T* coordinatized translation affine Planes . 6.20 ~ B r * regular biternary fields 5.30 -Q* biquasif ields 6.20 Â•-Qp* quasifields A. 29 -C n coordinatized Desarguesian AH-planes 9.6 kernel quasirings 9.3 *-R AH-rings 9.2 ~^ D * coordinatized Desarguesian affine planes 9.13 -Qj^* kernel biquasif ields ' 9.14 a R * division rings 9.12 H s Hjelmslev structures 10.36 ( T ) translation AH-planes with nondegenerate homomorphisnis B.20 a, L M left modules 7.5 K

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KEY TO CATEGORIES continued Category Subsection -(D) Desarguesian AH-planes with nondegenerate homomorphisms 10.63 L -((H ) ) lined Hjelmslev structures with S g full lined Kjelmslev structure 10.66 homomorphisms [~K quasicongruences B.6 T p pointed translation AH-planes B.8 |~-K* congruences B.6 T* p pointed translation affine planes B.8

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KEY TO SYMBOLS Symbol Subsection Â— neighbor 2.17 2.21 3.2 6.3 9.1 A.l ^ not neighbor (see above) W parallel 2.15 2.21 tt" not parallel (see above) \\ quasiparallel 2.37 Â•H" not quasiparallel (see above) ^ near 10.6 # not near (see above) *,
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KEY TO SYMBOLS continued Symbol Subsection TV(g) the direction containing the line g 2.35 Â°" semitranslation 4.1 T Â» T , T, ,. translation 4.1 7 4 B.14 w, w* , any small Greek letter homomorphism 2.2 zero 3.1 3.2 3.11 6.2 6.3 9.1 1 one 3.1 3.2 3.11 6.2 6.3 9.1 2.8 2.8 2.8 2.21 Â°1

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A v KEY TO SYMBOLS continued Symbol Subsection A * the gross structure of A 2.25 A* affine plane 2.15 (A,K), C, C* coordinatized AH-plane 3.11 A B the AH-plane generated by a biternary ring 3 3.18 3.31 the AH-plane generated by a prequasiring V 6.19 3.31 3.18 A the AH-plane generated by an AK-ring S 9.8 6.19 3.31 3.18 B, B', (M,T,T') biternary ring 3.2 C f C Â• , (A,K) coordinatized AH-plane 3.11 D set of semitranslations 4.22 Dp set of semitranslations with direction V" 1 Â• 4.22 E unit point 3.11 E W (A), E y the ring of trace preserving endomorphisrr.s of the translation group of A 7.4 7.6

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KEY TO SYMBOLS continued Symbol Subsection g, h, k line 2.8 9* the class of lines neighbor to g 2.17 2.21 2.43 g*Â» h *Â» k* line of an affine or projective plane 2.14 2.15 g set of directions 2.35 2.40 2.43 x-axis 3.11 y-axis 3.11 PH-plane Â• 2.17 the gross structure of H 2.25 projective plane 2.14 lined PH-plane 2.39 lined projective plane 2.39 incidence relation 2.8 coordinatization 3.11 quasicongruence Â• B.4 congruence B.2 direction containing [m,0] 5.9 a line not quasiparallel to g 3.11 side of a j-angle 4.10 the set of symbols 3.1 xxviii g x

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KEY TO SYMBOLS continued Symbol Subsection M Â» M" the set of symbols continued ( M , T , T Â• )

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KEY TO SYMBOLS continued Symbol Subsection Q* the class of points neighbor to Q continued 2.21 2.43 Q*> p * point of an affine or projective plane 2.14 2.15 Q, (M,+,x,-) quasiring 6.5 R, (R,+,x) H-ring, AH-ring 9.1 R, (M,T,T") restricted biternary ring A.l (Sig) lined incidence structure 2.39 S, R, (S,+,x) AH-ring 9.1 S, (OJ,S) semicongruence B.3 T > T' ternary operation 3.2 T " partial ternary operation A.l T Â» (A,P) pointed translation AH-plane B.B (u) ' direction containing [u,o]' 5.9 Lu,v]' line not quasiparallel to q 3.11 x V, (M,+,x,-) prequasiring 6.3 W the set of translations 4.22 (W,o) the group of translations4.22 5.24 7.4 x, a, c the x-coordinate of a point 3.11 YÂ» b Â» d the y-coordinate of a point 3.11

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Abstract of Dissertation Presented to the Graduate Council "of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COORDINATIZED HJELKSLEV PLANES By Phyrne Youens Bacon June, 1974 Chairman: David A. Drake Major Department: Mathematics A coordinatization may be thought of as an isomorphism between a geometric structure and a geometric structure which has been constructed from an algebraic structure. Affine Kjelmslev planes (AH-planes) are cocrdinatized by using biternary rings; translation AH-planes by using cuasirings; Desarguesian AH-planes by using AH-rings; and Desarguesian projective Hjelnslev planes (Desarguesian PH-planes) by using H-rings. An affine plane homomorphism crf:A Â— * A' is an incidence structure homomorphism which preserves the parallel relation. Iferf is non-degenerate (that is, it does not map all the points of A into points of a single line of A') then
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If attention is restricted to AH-plane homomorphisms which take x-axis to x-axis, y-axis to y-axis and unit point to unit point, then the following pairs of categories are equivalent: biternary rings and coordina tized AK-planes, quasirings and coordinatized translation AH-planes, AH-rings and coordinatized Desarguesian AH-planes. The category of quasicongruences is equivalent to the category of pointed translation AH-planes, and the category of Desarguesian AH-planes with non-degenerate AH-plane homomorphisms is equivalent to the category of lined Hjelmslev structures with full lined Hjelsmlev structure homomorphisms. Desarguesian PH-planes are Hjelmslev structures in which every two lines meet. The directions of the x-axis and the y-axis are denoted by (0) and (0)'. The translations of a coordinatized AH-plane C are (tO^tg*,)and ( (0) Â• ,g M )-transitive if and only if the biternary ring (K,T,T') of C satisfies the following conditions: 1) T and T' are linear. 2) The Tand T'-additions are equivalent: a + b = a Â• b for all a,b in K; that is, T(a,l,b) = T'(a,l,b) for all a,b in M. 3) (K,+) is a group. 4) xm + sm = (x + s)m and x-m + s-m = (x + s)Â« m for all x,s,m in M where the two multiplications are defined by ab = T(a,b,0) and a-b = T'(a,b,0) for all a,b in M. A coordinatized AH-plane whose biternary ring satisfies the conditions listed above is a translation AH-plane if and only if the addition + is abelian. Translation AH-planes, Desarguesian AH-planes, Pappian

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translation AK-plar.es, Desarguesian i : H-planes and Pappian Desarguesian PH-planes are each characterized geometrically, in terms of algebraic properties of their coordinatizations, and (for all except the Pappian planes) in terras of properties of their endomorphisms. Algebraic characterizations are given of those coordinatized AH-planes which have a semitranslation (in an affine plane a semitranslation is a translation) with direction (k) which moves the origin to (s,sk), and of those coordinat'ized AH-planes which have a ( (0,0 ) ,g 0O )-endomorphism which moves (1,1) to (s,s). There are similar results concerning ( (0 ) , [0,0] ' )and ( (0) Â• ,[0,C]' )-automorphisms where [0,0]' is the y-axis and (0)' is the direction of the y-axis. If H is a PH-plane; if s , s , s are the sides of a triangle whose image in the gross structure of H is non-degenerate, and if each of the three AH-planes A , A , A derived from H by use of one of the sides s , s , s is Desarguesian, then H is Desarguesian. There exists a Desarguesian AH-plane which cannot be derived from any Desarguesian ?H-plane.

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1. INTRODUCTION A coordinatizaticn may be thought of as an isomorphism between a geometric structure and a geometric structure which has been constructed from an algebraic structure. Klingenberg [(1955)1 began the solution of the coordinatizaticn problem for Hjelmslev planes by constructing a projective Hjelmslev plane from an H-ring, and he showed that this constructed FH-plane has a number of properties. Luneburg [(1962)3 defined an algebraic structure (which is here called a quasicongruencc ) and showed that any translation affine Hjelmslev plane (translation AH-plane) can be coordinatized (in the sense mentioned above) by using a quasicongruence. Lcrimer L( 1971)3 continued work on the coordinatization problem by constructing an affine Hjelmslev plane from an AH-ring and then giving various theorems relating these constructed planes to the class of Desarguesian AH-planes. Cyganova C(1967)l a i so did considerable work on the coordinatization problem: she undertook to define an algebraic structure which would have essentially the. same relation to affine Hjelmslev planes that ternary fields have to affine planes. Unfortunately, her arguments contain a number of serious omissions, not all of which I have been able to repair. Her algebraic system has a ternary opertion and a partial ternary operation. In Definition A.l, a similar (but different) 1

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algebraic system is defined which is called a restricted biternary ring. In Definition 3.2, an algebraic system with two (complete) ternary operations is defined: this system is called a biternary ring. An AH-plane homomorphism which takes x-axis to x-axis, y-axis to y-axis and unit point to unit point is called a coordinatization homomorphism (Definitions 3.11 and 3.14). In Theorem 3.2 7, it is shown that the category of biternary rings is equivalent to the category of coordinatized affine Hjelmslev planes (with coordinatization homomorphisms ) . This result is used (along with others) to show that the category of quusirings (these have two multiplications) is equivalent to the category of coordinatized translation AH-planes (Corollary 6. IS), and that the category of AH-rings is equivalent to the category of coordinatized Desarguesian AH-planes (Proposition 9.11). The quasiring equivalence is used to prove the existence of a module isomorphism (in fact a natural transformation between functors) which relates the kernel of a quasiring to the ring of trace preserving endomorphisms of the translation group of the associated AH-plane (Theorem 7.7). Even without considering possible algebraic connections, there is considerable interaction between geometric properties of a Hjelmslev plane and the existence of certain types of endomorphisms of the plane. In Theorem 4.21, a geometric characterization of those AH-planes whose automorphisms are (r f g^-transitive for any given direction V is given. In Proposition 8.9, a geometric characterization of those AH-planes

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3 whose automorphisms are (P,gÂ„,) -transitive for any given point ? is given. These results are used (together with some algebraic ones) to give geometric characterizations of translation AH-planes (Theorem 5.25), of Desarguesian AH-planes (Proposition 9.10) and of Desarguesian FH-planes (Theorem 11.5). An AH-plane homomorphism w:A Â— * A 1 is required to preserve the incidence, parallel and neighbor relations. An affine plane homomorphism oc:A Â— > A' (Â«( is an AH-plane homomorphism between affine planes) is said to be non-degenerate if it does not map all the points of A onto a single line of A'. A non-degenerate affine plane homomorphism is an incidence structure embedding and reflects both the incidence and parallel relations (Â°c g \\ reflects the quasiparallel and the neighbor relations (Corollary 2.65). In Theorem 5.11, an algebraic characterization is given of those coordinatized AH-planes which have a semitranslation with direction (k) which moves the origin to (s,sk) (in an affine plane a semitranslation is a translation); in Proposition 3.2, of those coordinatized AH-planes which have a ( (0 ; 0) .g^-endomorphism which moves the unit point to the point (s,s). Propositions 8.11 and 8.16 give similar results for ( (0 ) , to ,o] Â• )and ( (0) ' ,[0,0 3' )-automorphisms respectively: here ["0,01' is the y-axis and (0) and (0) 1 are the directions of the x-axis and the y-axis respectively.

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4 Theorem 5.29 shows that the translations of a coordinatized AH-plane C are 1(0), g M )and ( (0) Â» .g^) -transitive if and only if the biternary ring (K,T,T') of C satisfies the following conditions: 1) T and T' are linear. 2) The Tand T'-additions are equivalent: a + b = a * b for all a,b in M; that is, T(a,l,b) * T'(a,l,b) for all a,b in K. 3) (M,+) is a group. 4) xra + sm Â« (x + s)m and x.rn + s-m = (x + s)-m for all x,s,m in M where the two multiplications are defined by ab c T(a,b,0) ond aÂ»b = T'(a,b,0) for all a,b in M. Theorem 5.25 shows that a coordinatized AH-plane whose biternary ring satisfies the conditions listed above is a translation AH-plane if and only if the addition + is abelian. Klingenberg [( 1555)3 attempted to characterize in terms of their automorphisms those FH-planes which are isomorphic to some PH-plane constructed from an H-ring. His argument fails however. In Theorem 11.6, such a characterization is given. This theorem also indicates a geometric characterization of these t=H-planes (they are called Desarguesian PH-planes). Lorimer [(1971)3 generalizes part of what Artin [(1957)3 calls "the fundamental theorem of projective geometry" by proving some results relating the automorphisms of a Desarguesian AH-plane which fix the origin to a set of semilinear transformations. Theorem 10.63 shows that there is a functor Y from the category of Desarguesian AH-planes with non-degenerate AH-plane homomorphisms to the category of AK-rings; if ^:A -, ais a morphism

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5 in the first category, and if (S,6) and(S',Â©') are canonical coordinatizaticns of A and A', then m. can be defined by 8VP = (Y(iOQp)Z' for some non-singular matrix Z' with first column (1,0,0) ; if there is an AH-ring homomorphism ^ and a non-singular matrix Z such that u can be defined by 0'mF = (
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6 The development here depends only on well-known results from the theories of categories, sets, algebra and affine and projective planes. Paragraphs marked with asterisks are not part of this development and are intended to relate results here to other portions of the literature. Since some of the treatment of Desarguesian AH-planes given here resembles some of the recent work of J. W. Lorimer and X. D. Lane [(1973)1 and of J. W. Lorimer [(1973)a and Q973)b], there are discussions of their recent papers and a brief history of this research in Appendix A: A. 24 through A. 23. If it is desired, Appendix A (restricted biternary rings) can be read immediately following Section 3 (biternary rings) and Appendix B (quasicongruences ) can be read immediately following Section 5 ( semitranslaticns and algebra). Nothing in any of the sections depends on anything in the appendices.

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2. HJELMSLEV PLANES In this section we give a number of definitions (some of v.'hich differ slightly from the usual ones and some of which are entirely new), prove a few basic propositions, and discuss the relationship between affine and projective Hjelmslev planes. Proposition 2.31, Theorem 2.63, and Corollaries 2.64 and 2.65 concern non-degenerate homomorphisms. Proposition 2.56 concerns isomorphisms. 2.1 Definitions . We will assume that the reader is familiar with the elementary definitions of category theory. Those of the necessary category theory definitions which are not given here are stated by Mitchell [(1965), pages 1-5, 49, 52, 59"] . We will commit a common notational inconsistency by usually denoting a category with objects A and morphisms M by simply A; our names for the various categories will reflect this notation. We will use the term ' natural isomorphism ' in plane of Mitchell's term 'natural equivalence', and the term ' functor ' in place of Mitchell's term 'covariant functor'. 2.2 Definitions . A fu nction f from a set A to a set. B is an ordered triple (A,B,G ) where G is a subset of A*B such that, if a is an element of A, there is a unique element b of B such that (a,b) 6 G . G is called the graph of f. If (a,b) Â€ G , 7

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we write f (a) = b. Two functions f = (A.B.G,) and q = (C.D.G ) r g compose to give gf = (A,D,G ) if and only if the domain C of g is equal tc the codorr.ain B of f, and, if C = B, then G is gf defined by gf(a) = g(f(a)) for every a in A. We call the function (A,A,G ) defined by 1 (a) = a for every a in A the A A identity function on A. Let A be a class such that for each A 6 A, there is a set U(A), called the underlying set of A. An ordered triple to = (A,B,f^) is said to be a concrete moronism (on A), and is written w:A Â— B, if A,B e A and f^ is a function from U(A) to U(B); that is, f^ = (U( A) , U(B) ,G ). If fja) = b for some a in U(A), we write wa = b and to(a) = b. The function f is said to be the underlying set function of to, and we define U(u>) to be f . If tf = (A,B,f^) and a = (B,C,f ) are concrete morphisms on A, we say that ( A,C , f.f^ ) , denoted /Sot, is the natural composition of & by oc. We denote (A,A,1 ) by 1 and call 1. the natural 1 U v A I A A identity morphisn on A or simply the identity on A. A concrete category A is a category whose morphisms are a subclass M of the class of all concrete morphisms on A, whose identity morphisms are the natural identity morphisms, and whose composition is the natural composition. All of the categories which we construct in what follows are concrete categories. Once we have given a class A and a subclass M of concrete morphisms called *A homomorphisms ' , we will assume that the reader will be able to identify the underlying sets of the objects in A and to verify that the given subclass of concrete morphisms is closed under natural composition and contains the natural identity

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9 morphisms. Once this has been verified, it is immediate that A is a concrete category since [a,BJ-a will always be a set; the A morphisms of A will always satisfy the requirement that *(&o0 = U&)<< where defined, and 1 where defined. Instead of saying 'the concrete morphism u = (A,B,f )', we will usually say 'the map co:A Â— *Â• B'. In an abuse of terminology, we say 'the map F:A Â— * B' when A, 3 are classes and F is a subclass of A*B such that for every A in A there is a unique B in B, written F(A), such that (A,B) Â€ F. 2.3 Definitions . If there is a map F:A Â— Â» B which 'forgets' structure, or which is the functor we construct from A to B, we frequently say that a concrete morphism oo = (AjA'jf^) on A is a B homomorphism if (F( A) , F( A ' ) ,f ) is a B homomorphism. For example, if w = (A,A',f ) is a concrete morphism on AH-planes, and if A = (S,ll), A' = (S',||) then we say uj is an incidence structure homomorphism if (S,S',f ) is an incidence structure homomorphism. We abuse our terminology in other similar respects; for example, we speak of the 'lines* of an AH-plane A = (S,t\) when we mean the lines of S. Once we have constructed a functor G:C Â— * D and shown that G is an equivalence, we will call G(C) where C is. an object in C the D-object generated by C. If F:D Â— Â»Â• Â£ is an obvious functor which 'forgets' structure, we will call FG(C) the E-object generated by C. For example, in Section 3 we construct an equivalence C:3 Â— Â» C, and if 3 is a biternary ring, we call C(B) the coordinatized AH-plane generated by B, and we call the

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10 AH-plane of C(B) the AH-plane generated by B. We will frequently say 'of' in place cf 'generated by', and ' with ' in place of 'which generates the'. 2.4 Definitions . Let *:A Â— Â» B be a concrete morphism. If the underlying set function of <* is onto, * is said to be surjective . If the underlying set function of * is one-to-one, Â«* is said to be i"jective . If Â« is both injective and surjective, it is said to be bi jective . 2.5 Definitions . A functor F:C Â— D is said to be an equivalence if and only if there is a functor G:D Â— " C together with natural isomorphisms *?:1Â— Â»Â• FG and *:G? Â— * 1Â». If an equivalence F:C Â— * D. produces a correspondence between the objects of C and D, then it is said to be an isomorphism if it is one-to-one, onto. 2.6 Definition . If F:C Â— * D and G:D Â— C are functors, and if ^ : ift Â— * Â£G and ^ : Â£Â£ Â— *1a sre natural isomorphisms, we say that F and G are reciprocal equivalences . 2.7 Definitions . If A is a class of objects such that a binary relation P is defined on the underlying set of each object A of A, then a concrete morphism w:A Â— * B is said to preserve p if whenever (a,b) Â« f for some a,b in U(A); it is also' true that dÂ»>(a) ,w(b) ) < * , We say that uj reflects p if whenever (w(a) ,w(b) ) * P.. for some a,b in U(A); it is also true that (a,b) fi P.

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11 2.8 Definitions . Let (I 8 , o],.I) be a triple of sets. Then (^,0],!) is said to be an incidence structure if *Pand o] have no elements in common and I c. *)a x 03. The elements of fare called points , the elements cf oj are called lines , and if P Â€ IP, g e <7j , we write Pig whenever (P,g) 6 I. The set I is called the incidence relation . If P I g, we say P is incident with g, P lies on g, P is a point of g, or P is on g. We also say g goes through P, or g is a line through P. We use other similar geometric language to designate incidence or non-incidence. For example, g is said to join P,Q if P,Q I g; that is, if P I g and Q I g. The capital letters P, C, G, K will be used to designate points unless otherwise indicated; similarly, the small letters g, h, k will be used to designate lines unless otherwise indicated. Points which all lie on a common line are said to be collinear , and lines which all go through a common point are said to be copunctal . We say I g A h I = n if the cardinality of the set of points incident with both g and h is n. We let gH h denote the point or set of points incident with both g and h. We define Ig (\ h A kl and g f\ h fi k similarly. We write PQ to denote the line or set of lines joining P and Q. Occasionally we write P v Q instead of PQ. 2.9 Definition . Let S = (T*,flj,I) and S' = Vp.' , oj' , I Â• ) be incidence structures. An incidence structure hcmomorphism u:S Â—+ S' is a concrete morphism which satisfies the following conditions. (1) f w is a function from I 1 U oj to V U o,'. (2) wCP) C Ti'.

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13 "Â•-equivalence class will frequently be denoted by a theretofore unused letter with an asterisk; for example, t*. Once t* has been used, however, t will denote an element of t*. Asterisks used as superscripts do not necessarily denote equivalence classes; it should be clear from the context what is meant in each case. 2.14 Definition . Let H* be an incidence structure with incidence relation I. One calls H* an ( ordinary ) projective Plane if the following three conditions hold. (0P1) Whenever P* and 0* are distinct points of H* f there is exactly one line g* such that P* I g* and Q* I gÂ». (0P2) Whenever gÂ» and hÂ» are distinct lines of HÂ», there is exactly one point P* such that P* I g* and ?* I h*. (0P3) There exist four points, no three of which are collinear; that is, no three of the four points lie on a common line. 2.15 Definitions . Let S* be an incidence structure with incidence relation I. Let \\ be an equivalence relation defined on the lines of S*. If g* \\ h* , we say g* is parallel to h*. One calls A* = (S*,\l) an ( ordinary ) affine' plans if the following'' four conditions hold. (0A1) Whenever P* and Q* are distinct points of S*, there is exactly one line g* such that P* I g* and Q* I g*. (0A2) Whenever P* is a point and g* is a line such that P* is not incident with g*, there is exactly one line h* such that

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14 F* I h* and h* and g* have no point in common. (0A3) S* has three points which are not collinear; that is, S* has three points not all on the same line. (0A4) Two lines g* and h* are parallel, g* H hÂ», if and only if g* and h* have no point in cordon or g* = h*. " Remark . It is well-known that if S* is an incidence structure satisfying conditions (OAl), (CA2) and (OA3), then there is a unique equivalence relation 11 such that (S*,\i) is an affine plane by our definition [Pickert (1955), pages 7-10]. The following result is well known. 2.16 Proposition . If H is an affine plane (projective plane), then each line of H goes through at least two (three) distinct points, and each point of H is incident with at le-st three distinct lines. Proof. This is essentially shown in Pickert [(1955), pages 7, 9-111. // 2.17 Definitions . Let H = CP,oj,I) be an incidence structure. Points P and C are said to be pro jcctivcly. neighbor and one writes.? C whenever there are distinct lines g and h such that P,C I g and P,Q I h. Lines g and h are said to be projectively neighbor and one writes g ~ h whenever there are distinct points P and Q such that P I g,h and G I g,h. One calls K a projective Hjel-slev plane (abbreviated PH-plane) whenever the

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15 following three conditions are satisfied. (PHI) If P and Q are points of K, there is at least one line g such that P,Q I g. (PH2) If g and h are lines of H, there is at least one point P such that ? I g,h. (PH3) There is a surjective incidence structure homomorphism g ~ h. If P is a point of H and if g is a line of H, one denotes the ' Â— equivalence classes containing P and g by ?* and g* respectively. 2.18 Proposition . Any projective plane is a projective Hjelmslev plane. // 2.19 Definitions .' Let H,H' be PH-planes. A projective Hjelmslev plane homomorphism w:H Â— * H Â« is an incidence structure homomorphism which preserves the relation ' pro jectively neiahbor'. We denote the category of projective Hjelmslev planes by H, and we denote the category of projective oianes with PH-plane homomorphisms by H*. 2.20 Proposition . If H is a PH-plane, and if ^:H Â— * H' is a map satisfying conditions (PH3) of the definition of PH-plane, then *? is a PH-plane homomorphism. //

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16 2.21 Definitions . Let S = (*P,oj,I) be an incidence structure, and let \\ be an equivalence relation called the parallel relation which is defined on the lines of 3. Points F and G are said to be affinely neighbor and one writes ? ~ Q whenever there are distinct lines g,h such that P,Q I g,h. Lines g and h are said to be affinely neichbor and one writes a ~ h whenever to each point of each there corresponds a point of the other which is affinely neichbor to it. One calls A = (S,U) an affine Hjelrr.slev plane (abbreviated AH-plane ) whenever the following four conditions are satisfied. (AK1) If P and Q are points of 5, there exists at least one line g such that P,Q I g. (AH2) Let P I g,h. Then P is the only point on both g and h if and only if g is not affinely neighbor tc h. (AH3) If P is a point and if g is a line, then there is exactly one line h such that F I h and g \\ h. (AH4) There is a nap ^f:A Â— Â» A' from A to an affine plane A' which is a surjective incidence structure homomorphism such that the following three conditions are satisfied. ( F ~ Q. (?2) If g,h <= 03; then^h = vfg ^ g ~ h. (h in A'. If P is a point and if g is a line of A, one denotes the "-equivalence classes containing P and g by P* and qf respectively. 'We usually denote the parallel relation of an AH-plane by U.

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17 2.22 Proposition . Any affine plane is an AH-plane. Proof. Let A* be an affine plane. By Proposition 2.16, any line g* of A* goes through at least two distinct points. In A* the relation 'af finely neighbor* reduces to the relation 'is equal to'. Using these results, one can easily show that A* is an AH-plane. // 2.23 Definitions . Let A and A' be AH-plane s. An affine Hjelmsle\ plane homomorphism A Â• is an incidence structure homomorphism from S to S ' which preserves the relations 'parallel' and 'affinely neighbor'. We denote the category of affine Hjelrr.slev planes by A, and we denote the category of affine planes with AH-plane homomorphisms by A*. ' Remark . We have shown [Bacon (1971), page 21, Corollary 3.121 that there exist two non-isomorphic AH-planes which have isomorphic incidence structures. 2.24 Propositon . If A is an AH-plane, and if
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1( relation by II*: we say g* U* h* if and only if there are lines k,m such that k ~ g; m ~ h and k II m. We call A* = (S*,H*) the gross structure of A. We call the concrete morphism K:A Â— * A* defined by *P = P*; Kg = gÂ» the neighbor map of A. If H is a PH-plane, we denote the incidence structure induced by the eguivalence relation 'protectively neighbor' by H*, and we call H* the gross structure of H. We call the concrete norphism k:H Â— Â»Â• H* defined by kP = P*; *g = g* the neighbor map of H. 2.26 Definitions . Once the gross structures have been defined, we extend the relations 'projectively neighbor' and 'af finely neighbor' in the following way. Let H be a PH-plane or an AH-plane. Let P be a point and g a line of H. We say P ~ g and g ~ P whenever the image of P is incident with the image of g in the gross structure of H. Hereafter, the symbol ~ is to be read ' is neighbor to ' except where otherwise specified. Its negation is written Â•/Â• and is read ' is not neichbor to . ' The negation of the symbol W is written tf, and is read ' is not parallel to . * 2.27 Proposition . If Â«*>:A Â— Â» A* is an AH-plane homornorphism, then <*>p I wg <Â£> 3 h such that P I h and Â«h = uig. Proof . Let h be a line through P, h l\ g; then uh = ug. // 2.28 Proposition . If w:K Â— * H ' is a surjective PH-plane homornorphism, then up I uig <Â£Â£. 3 h such that P I h and "h = uig.

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19 Proof. Assume oP I uÂ»g. Let P) is not on t'Uk). Since Â«k ^ wg, k ^ g. Since UkA *> g ) ^ Up), (k H g) + P. Thus, if we let h = (k g) v P, then ^h = u>g. // 2.29 Proposition . If H is an AH-plane (PH-plane), and if ^:H Â— * H' is a map satisfying condition (AH4) (condition (PH3)), then the gross structure H* of H is an affine plane (projective plane), and is isomorphic to H ' ; also, the neighbor map K:H Â— * H* is a surjective AH-plane (PH-plane) homomorphism satisfying condition (AH4) (condition (PH3)). Proof. Define Â©:HÂ» -^ H' by GPÂ« = P; Â©gÂ» = 4 g; then, 6 i s a well-defined bijection. If PÂ« I gÂ«, then there are Q,h, Q ~ P, h ~ g, such that Q I h; hence QPÂ« I Qg*. Since f is a surjection, 8 reflects incidence by Proposition 2.27 or 2.28: if Gp* I9g' then f P I GP' ~ 6QÂ»; gÂ« Â„ h * Â«* Gg* ~ 9hÂ«. Hence 6 preserves and reflects the neighbor relation on points (lines).' If H is an AH-plane, and if gÂ« \\* hÂ», then there are g' in g* and h' in h* such that gÂ» 11 h Â• . Then, since H>g' = 8g* and Â•fh 1 = 6h*, we have that 8g* \\ 8hÂ», and hence 8 preserves the parallel relation. Assume that H is an AH-plane and that <9 g It 4>h. Let^k be a line not parallel to ifh. Since fk-fr
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20 since h' || g, q> h Â• II Â«fg. Since <{P I h' ll i?g, we have that Â«Ph' = 4>h. Thus h' ~ h and h' II g; so that, h* HÂ« g*. Thus, Â© reflects the parallel relation. Consequently, H* is an affine plane (projective plane) isomorphic to H ' . Since k = G ^, K is a surjective AH-plane (PH-plane) homomorphism satisfying condition (AH4) (condition (PH3)). // 2.30 Definition . Let ^:A Â— A' be an AH-plane (PH-plane) homomorphism. Let K 1 be the neighbor map of A'. If there are three points R, R', R" of A whose images k'uR, k' mR * , k'mR" in the gross structure of A' are not collinear, then we say that /* is non-deoenera te; otherwise we say that u. is degenerate . 2.31 Pro position . If u. : h Â— * H ' is a non-degenerate AH-plane (PH-plane) homomorphism, then Â»*P I^g 3 Q such that Q I g and^P =^Q; and ^ P I^g <=> 3h such that P I h andMg = **h. Proof. Assume H is a PH-plane and*P I ^g. Let k' be the neighbor map of H'. Since p is non-degenerate, there is a point S such that KVS is not on KVg. Then P 4S and IPS A g| = 1, since ^P^S + ^g implies PS f g. Let Q = PS A g; then y-Q = up. Since u> is non-degenerate, there are three. lines in H whose images in (H')Â« are not copunctal. Thus, there is a' line yum in H' such that (y*P)* is not on Urn)*. Let M = m A g. Since U.P)* is not on (^m)*, ^P + ^M, and hence P' Â•/M. Let h = MP; then, j*h = l.m^P = A*g and P I h.

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21 Assume H is an AH-plane and mP Kg. Let *, w' be the respective neighbor maps of H,H*. By Proposition 2.27, there is a line h such that P I h and pq = Â»*h. Let m be a line such that K'*m -fr KVg. Let m Â• and g' be lines such that m ' l\ m, g* \\ q and P I m',g'. If lm' (\ q\ = 0, Km' = Kg'; so that g' ~ m'. Hence KVm U K.Vg, a contradiction. Thus, \m' g\ ^ 0. If m Â• ~ g, then K'^m II KVg, a contradiction. Hence lm' C\ g\ = 1. Let Q = m 1 A g. Then, if ^m ' ~y*g, nVm' II Kyg, and hence KVm II k'^g, a contradiction. Hence M.m* + A*g and Â«Q = ^P . // 2.32 Proposition . If ^:H Â— * H Â• is an AH-plane (PH-plane) homomorphism which induces an isomorphism between the gross structures, then v. is non-degenerate. // 2.33 Proposition . If u.:H Â— * H ' is a surjective AH-plane (PHplane) homomorphism, then Â«. is non-degenerate. Hence, if H is an AH-plane (PH-plane), and if P is a point and g, a line of H, then P ~ g if and only if there is a point Q on g such that P -~ Q . // ' Remark . Luneburg [(1962); pages 263, 264, 265; S'atze 2.1, 2.3, 2.4l states the first two sentences of the* following proposition for AH-planes. Klingenberg [(1955), page 101, S 5] states the first sentence of the following proposition for PH-planes. 2.34 Proposition . Let H be an AH-plane (PH-plane); then, there are at least three pairwise non-neighbor lines through each

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22 point of H, and at least two (three pairwise) non-neighbor points on each line of H. Hence each line is uniquely determined by the set of points on it. Also, each point is uniquely determined by the set of lines through it. Proof . The first two (all three) sentences of the proposition follow easily from Propositions 2.33 and 2.31. Let H be an AH-plane and let (g \ P I g] = \ g I Q I g"} . Let g,h be lines such that P,Q I g,h and g + h. Then by (AH2), Ig H hi = 1. Thus, P = Q. // 2.35 Definition . Let A be an AH-plane. A U-equivalence class of lines is called a direction . We denote the set of directions by g^. If g is a line, the direction containing g will be denoted by TUg). Arbitrary directions will be denoted by "ft, Â£, T or some other capital Greek letter. 2.36 Definition . Let A be an AH-plane. If P is a point and g a line of A, we denote the unique line of A through P parallel to g by L(P,g). If P is a point and V is a direction, we denote the unique line of P through P by L(P,D. 2.37 Definition . If g,h are lines of an AH-plane A, and their images under the neighbor map of A are parallel, we say g and h are quasiparallel , and write g U h. If h is a line and P is a direction, we write h 11 V and rih and say 'h is quasiparallel to f" and ' f is quasiparallel to h ' if there is a line g in

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23 r isuch that h \\ g. Similarly, two directions T,l. of A are said to be quasiparallel , r \\ 2. \$ if they map into the same parallel class under the neighbor map of A. The negation of the symbol W is written -It. 2.38 Proposition . Let A be an AH-plane. Two lines h,k of A have exactly one point in common if and only if they are not quasiparallel. Also, \\ is an equivalence relation. Proof. Assume h X k. Since the images of the two lines are not parallel, the lines are not neighbor. If I h f\ k \ =0, then their images would be parallel. by (AH4). Hence by (AH2), \h k\ = 1, and h and k have exactly one point in common. Assume |h kl =1. Then h 4* k. Hence h* ^ k*. But lh* fl kÂ»| ^ 0; hence h* is not parallel to kÂ». Thus, h is not quasiparallel to k. // 2.39 Definitions . If S is an incidence structure, and if g is a line of S, we say (S,g) is a lined incidence structure with base line 9Â« If (S,g) is a lined incidence structure, the points of S which are not neighbor to g are called the affine points of (S,g). Any line of S which goes through an affine point is called an affine line of (S,g). We say that ui:(S,g) Â— Â» (S',g*) is a lined incidence structure homomorphism if
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24 structure or some special type of incidence structure homomorphism, we modify our terminology accordingly. We denote the category of lined PK-planes by H , the category of lined projective planes by H* , and the category of lin ed incidence structures by S . g 2.40 Construction of G:A Â— * S and G :A Â— > S . Let A be an _ ~ ~g g AH-plane. Let g w be the set of parallel classes of A. For every parallel class TT in g , let P(TT) be a new point, and adjoin P(TT) to each line in IT. Let the P(TY)'s be different for different TT's. Let g(gj be a new line incident with each of the new points. Choose the P(TT)'s and g(g w ) in such a way that the new point set IPU 1 P (TT) I TT e gj and the new line set Â«7j U ^g(g^)} are disjoint. Let G(A) be the incidence structure obtained by adjoining the new points P(TC), the new line g(g 0O ) and the. new incidences to the points, lines and incidences of A. Define G (A) to be (G(A) ,g(g w ) ). G(A) is called the ceneralized incidence struct ure of A. G (A) is called the lined aeneralized ~g ! incidence structure of A. A point of G(A) is called a generalized point , and a line of G(A) is called a generalized line . The incidence relation of G(A) is called the generalized incidence of A. We call the original points, lines and incidence structure of A, the affine points , affine lines and affine incidence structure of A. Unless otherwise specified, line ( point ; incidence structure ) will mean affine line (affine point; affine incidence structure) in an AH-plane. If w:A Â— * A' is an AH-plane homomorphism, then u> can be

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25 extended in an obvious natural way to an incidence structure homomorphism G(u>):G(A) Â— *G(A ? ), and to a lined incidence structure homomorphism G (>*>):G (A) Â— * G (A'). ~g ~g ~g Remark . The definition of an affine point (affine line) of A agrees with the definition of an affine point (affine line) of G (A). ~9 2.41 Proposition . The map G:A Â— S constructed above is a functor from the category of AH-planes to the category of incidence structures, and G :A Â— * S is a functor from A to the ~g g category of lined incidence structures. // 2.42 Proposition . The map H* :A* Â— * H* defined by H* (A*) = ~ g g ~ g G (A*), H* (to) = G (w>) is a functor from the cateqory of affii ~g g -g a i planes to the category of lined projective planes. Proof. Pickert [(1955), page 11, Satz 7l shows that if A* is an affine plane, then G(AÂ») is a projective plane. Hence H* (A*) is a lined projective plane. Since the neighbor relation in a projective plane is trivial, if w:A* Â— * B* is a morphism in A", then H* (w) i s a g lined projective plane homomorphism. Thus, H* is a functor. // y 2.43 Definitions . Let A be an AH-plane. Let K:A Â— * A* be the neighbor map. Let R be a point or a line in G (A), and let S be a point or a line in G (A). We say R is neighbor to S, and

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26 write R ~ S, whenever G (k)R = G (k)S, G (k)R I G (k)S or ~g -g -g ~g 5 (K)S I G (k.)R in the lined projective plane H* (a*) = G (AÂ»). y 9 ~ g ~g We call the relation ~ thus defined the generalized neighbor relation of G (A), or the neiohbor relation of G (A). One g ~g can show that restricted to the affine points and affine lines of S q ( A )Â» the generalized neighbor relation agrees with the neighbor relation induced from A. Once this has been shown (see Proposition 2.44), extend the neighbor relation of A in the obvious way: we say 'R is neighbor to S ' in A (where R,S can be a point, a line, a direction or g w ) whenever R ~ S in G (A). If R is neighbor to S we v/rite R ~ S; otherwise we write R * S. We call ~ the (generalized) neiqhbor relation of A. 2.44 Proposition . Let A be an AH-plane. The restriction of the generalized neighbor relation of G (A) to the affine points and affine lines of A is the relation 'neighbor' of A. Proof. Observe that P ~ Q kP = kQ; g ~ h c^ ng=Kh; P ~ g<=> KP I Kg; g~ P <=> kP iKg. // Remark . Hereafter we will frequently not distinguish between g(TT) and TT; g(g,J and g : G (A), G(A) and A; G (w) , g(u>) and oo. ~g Â»g Â» 2.45 Proposition . Let A be an AH-plane. (1) Let h be a line and let T be a direction of A. The following are equivalent. a) h II T.

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27 b) h ~ r. c) If P I h, there is a line g <= P such that g 11 h and P Â€ g A h. (2) Let Â£ ,V be directions in A. The following are equivalent. i) ijr. ii) l ~ r. iii) If h el, then h U T. Proof. Part (1). Let h be a line, and let T be a direction of A. Assume h 1\ ?. Then there is a line g in T such that g \\ h. Hence Kg U Kh in A*, and TT(kq-) I wh in HÂ» (A*). Thus, h ~ T. g Assume h ~ T. Then k(T) I K(h). Let P I h, and let g = L(P,D. Then since Kg = kP v k(T) h, we have that g 1\ h. Assume that if P I h, there is a line g fc T such that g l\ h and Pig. Let P I h. Then P I g, g ft h; hence h 11 T . Part (2). Let Â£ ,T be directions in A. Assume I II P. Then *(Â£) = k(D, and hence 1 T. Assume Â£ T. Let h eÂ£. Let P I h. Let g = L(P,D. Then since k(Â£) = k(H), we have that Kh = Kg; hence h H g. Thus, h u r. Assume that for every h 6 Â£ , h II P . Let h Â£ Â£. Then there is a g 6 T such that h II g. Thus, kKTTCg)) = K (TT(h)), and 2.46 Proposition . If <* is an AH-plane homomorphism, <* preserves the generalized neighbor relation defined above as well as the quasiparallel relation.

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28 Proof . Since Â« is an AH-plane homomorphisra, * preserves the Â•affinely neighbor' relation. Hence by Proposition 2.34, Â« preserves the relations P ~ g and g ~ P. If T ~ Â£ , then there are lines g fc P , h t 1 such that g ~ h. Hence *g .Â«h, and Â«P ~
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29 [(1955), pages 9-10] shows that S is the incidence In ,g ; structure of an affine plane. Two lines of A*(HÂ»,g*) are parallel if and only if they fail to meet in S. (and hence (H* ,gÂ» ; meet at a point on gÂ».) Thus, A*(H*,gÂ») = (S. > ft) is an ~ In , g ) affine plane. If uj:(H*,g*) Â— Â» (H*,g*) is a morphism in H* , then uÂ»{p*i P* I g*~\ c {?* \ p* i g*"5, and u> maps the affine points (lines) of H* into the affine points (lines) of H* . Hence A*(ui) is an affine plane homomorphism. It is easily seen that A* is a functor. Let A* and A* be affine planes. Observe that A*H* (A*) = Â— g A*. If w:A* Â— *Â• A* is an affine plane homomorphism, then A*H* (uj) = w. Define y to be the identity map on A*. Then ~ g a T:A*H* Â— Â» 1* is a natural isomorphism. ~ ~ g -A* c If (H*,gÂ») is a lined projective plane, define a map ^ (H , j:(HÂ«,gÂ«) Â— * HÂ» A*(H*,g*) by letting it be the identity on the affine points and the affine lines of (H*,gÂ»); by letting it take a point P* on g* to POT) where TT is the set of affine lines through P% and by letting it take g* to g(g o0 ). It is easily seen that OC. t is an isomorphism. If I n , g ) w:(H*,gÂ») Â— > (H*,gÂ») is a morphism in H* , then, since a lined projective plane homomorphism is completely determined by its action on the affine points and the affine lines, we have that g natural isomorphism. Thus, H* and A* are reciprocal equivalences. //

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30 2.49 Proposition . If (G,gJ is the lined generalized incidence structure of an AH-plane A, then A can be obtained from (G.a ) by a construction identical to that used to obtain the affine plane A(H*,gÂ») from a lined projective plane (K*,g*). // 2.50 Proposition. The map A:H Â— * A constructed in Construction 2.47 is a functor from the category of lined projective Hjelmslev planes to the category of affine Hjelmslev planes. If (H,g) is a lined PH-plane, two affine points (affine lines)' are neighbor in (H,g) if and only if they are neighbor in A(H,g). if HÂ« is the gross structure of H, and if g* is the class of lines neighbor to g in H, then A(H*,gÂ») is equal to the gross structure of A(H,g). Remark . It is well known that if (H,g) is a lined PH-plane, then A(H,g) is an AH-plane. This is stated by Luneburg 1(1962), page 260, second paragraph], and is essentially proven by PClingenberg [(1954), pages 390-392, S 1.11 and S 3.6], ' Proof. Let (H,g) be a lined PH-plane and let S = S, x . Then (h,g) A(H,g) = (S,l|); S is an incidence structure, and II is an equivalence relation on the lines of S. P, a point of H, is a point of S if and only if P + g; and h, a line of H, is a line of s if and only if h Â•jt g. . Â• We use the symbol to denote the relation ' af finely neighbor' in A = A(H,g) in order to avoid confusion with the symbol ~ which we use to denote the relation 'projectively

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31 neighbor' in H. If P,Q are. points of A, P ~ Q in H if and only if P D Q in A since a line is removed only if all the points on it are also removed. If h,k are lines of A, we wish to show that h ~ k in H if. and only if h k in A. Assume h ~ k. Let P be any point of A which is on h. By Corollary 2.34, there is a line m of H through P such that m f h. Hence m f k. Let Q = m (1 k in H. Then in H*, C* = ?Â• = m* A k* by (0P2). Thus, C is in A; P Q, and Q I k. By symmetry, the corresponding statement holds for an arbitrary point of A on k. Thus, h D k. Conversely, assume h h'. In H* there are at least two points P*,Q* on h* but not on g* by (CP2) and Proposition 2.16. By Proposition 2.31, there are points R,S on h such that R Â£ P*, S Â€-0*. Let R',SÂ» I h' such that R' ~ r, s< ~ S. Then in H*, h* = (h')Â» by (0P1); hence h ~ h'. Hereafter we will use ~ to indicate 'is neighbor to' in both A and H. Any two points of A are joined by at least one line; that is, (AH1) holds in A. If P I h,k; we wish to show that h k = P if and only if h Vk in A. Assume that in A, h H k = P. Then h ^ k in H. Hence, hf k in A. Conversely, assume F I h,k; h ^ k in A. Then h + k in K and ? = h (\ k. Thus, P = h k in A, and A satisfies (AH2). Let P and h be a point and a line of A. Let Q. =' h g in H. Then there is a unique line k joining F and G in H, and we have k II h in A. If k Â• is any line such that k* U h and P I k'; then Q I k'. Thus, k = k', and (AH3) holds in A.

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32 By Proposition 2. 43, A(H*,gÂ») is an affine plane. Define Â«f:A Â— * A(HÂ»,gÂ») by fP = PÂ», ) preserves the neighbor" and parallel relations and hence is an AH-plane homomorphism. It is also . easily seen that A(aÂ«) = A(a)AU) and that A(l,Â„ .) = 1 , ~~ I "Â• I ~ ~ ( H , g ) A ( H , g ) Hence, A:H Â— * A is a functor. // 2.51 Definitions . Let A be an AH-plane and let (H,g) be a lined PH-plane. If A is equal to A(H,g), then we say that A is derived from (H,g), or we say A is derived from H ( by use of the line g), and we say A is a derived AH-plane. If A is isomorphic to A(H,g), we say A can be extended to (H,g), or we say A can be extended to H; we also say that (H,g) and H are extensions of A. Remarks . Drake [(to appear), Corollary 6.2] states that there is an AH-plane which cannot be extended to a PH-plane. Drake [(1967), page 198, Theorem 3.1] states that every finite uniform AH-plane can be extended to a finite uniform PH-plane, and [in Bacon (to appear), Theorem 2.l] we state that every protectively uniform AH-plane can be extended to a

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33 uniform PH-plane, and we use the argument given by Artmann [(1970), pages 130-134] to show this. 2.52 Definition . An injective incidence structure homomorphism Â«o:S Â— S' which reflects the incidence relation is called an ( incidence structure ) embedding ( of s into 5*). 2.53 Proposition . Let (H,k) be a lined PH-plane. The map ^ : i?g ( Â£ (H ' k)) * (H Â» k} defined by X(P) = P, X(h) = h for all affine points and lines and by X(TUh)) = h A k and X(g ) = k for all affine lines h is a lined incidence structure embedding. // Â» 2.54 Remarks . Dembowski [(1968), pages 295-2961 and Artmann [(1969), page 175, Definition 6] have given definitions of Â•affine Hjelmslev plane 1 which they assert are equivalent to that given by Liineburg [(1962), page 263, Definition 2.3]. In [Bacon (1972), page 3, Example 2.1] we give an example of an incidence structure and a parallel relation on the lines of the incidence structure which satisfies the definitions given by Dembowski and Artmann, but not that given by Luneburg. We repeat this example here. ' Example . Take any affine plane A. Keep the same lines and the same parallel relation. Choose one point P of A, and adjoin a new point Pto the point set of A. Let the incidence relation be the same for the old points and lines, and let P* be incident with precisely the lines which go through P.

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34 Remarks . This example fails to satisfy the definition of AH-plane given here (which is essentially equivalent to that given by Liineburg). It can easily be shown that this example cannot be derived from a lined PH-plane. Klingenberg [(1954), page 390, D 61 calls S, , an (H,g) 'affine incidence plane with neighbor elements'. He then shows [(1954), pages 391-392, S 3.6] that A(H,g) = (s, , , H) ~ (TTg) ~ w (tfh) and u, P ~ W Q, we see that ^g ~ wh, and hence w is an AH-plane homomorphism. // 2.56 Proposition . If
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35 Proof. Let g Â• be a line of HÂ». Let = L(P,h). Then k' \\ h ;
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36 2.59 Construction of A :A Â— > a and AÂ„ :A Â— A. We define ~g g ~ II g A :A Â— Â» A by letting A (A) = G (A) and A (oo) = G (to) for everv 9 g -g -g -g ~g cvciy A A and w in A. Let (A,g) be a generalized AH-plane. We let I be the a restriction of the incidence structure of A to the affine points and affine lines of (A,g). If 1* and (* are the sets of affine points and of affine lines of (A,g) respectively and if a relation II is defined on oj^ by k II h 4=* k,h and g have a point in common in (A,g), then we denote ( (*Â£ ,o, ,1 ) , i\ ) by A (A. a). If a J a a ~ll y w is a generalized AH-plane homomorphism, we define A (w) in the obvious way. 2.60 Proposition . The maps A :A Â— + A and A ,:A Â— Â» A are ~ g g ~ u g reciprocal equivalences. // 2.61 Remark . Hereafter we will not distinguish between AH-planes and generalized AH-planes except to aviod confusion. We will say '(A,g) is isomorphic to the AH-plane B* when we mean that (A,g) is isomorphic to the lined generalized incidence structure of B, and so on. 2.62 Definition . An AH-plane homomorphismÂ•< is said to be an AH-plane embedding whenever tx is injective and reflects the incidence, neighbor and parallel relations. ' Remark . V. Corbas' argument for the validity of his Teorema C(1965), page 375 1 inspired the following proposition. Corbas'

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37 Teorema deals with surjective morphisms between affine planes. 2.63 Theorem . Let u:A* Â— Â» A* be a non-degenerate affine plane homomorphism. Then, u. is an AH-plane embedding of A* into A*; hence i*. is infective and reflects the incidence and parallel relations. Also, K* OO :Â£Â• (A*) Â— * H* (!Â•) is a lined projective plane embedding; thus, H* (u) preserves and reflects the incidence relation. Thus, u induces a projective plane embedding of the projective plane associated with A* into that associated with A*. Proof. Assume j*:A* Â— * A* is a non-degenerate affine plane homomorphism and that G", KÂ» and M* are points of A* whose images under yu are not collinear. We wish to show that yu is injective with respect to parallel classes. Let P* f Â£Â• be distinct directions in A*. Let g* 6 C Â• . since g* meets every line of Â£Â•, ug* meets every line of {j*s*\ s* e 1Â»"J. In particular , u g* meets |Â»L(G*,2Â»), Â«.L(K*,?*) and y*L(MÂ»,2Â»). But by our assumption, at least two of these three lines are distinct. Thus, i*g* cannot be parallel to all three; hence -T* 4 w%* . Hence i* is injective on directions. Let P* f R* be distinct points, and let g* be the line joining them. Let Q* be a point such that^Q* is not onM-g'; such a point exists by our assumptions. Then P*QÂ« JrRÂ»Q*; hence y-(P*Q*) tt^(R*Q*); so that mP* ^ j*R* . Hence u, is injective on points. Let pg m = ^h* and let Q* be a point such that **Q* is not on

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38 /*g*. Let P* be a point of g*. Since ^h* -ft i*(P*QÂ»); h* -K P*Q*; hence P*C* meets h* at some point R*. Since l^gÂ» A i*(P*Q*)l = 1; ^PÂ« = /*R*, and PÂ« = R*. Since ^gÂ» il^h*; g* l\ h* and g* = h*. Thus, p is infective. By Proposition 2.31, j^ reflects incidence and hence is an incidence structure embedding. If uh* II /*g* f then either ^h* =^g* and h* = g* or Lh' A ug'l =0 and |hÂ« l\ g*( , 0. In either case, h* \\ g*. Thus, j* is an AH-plane embedding. We wish to show that H* ( u ) reflects incidence. Obviously, H* (/*) reflects incidence for affine points and lines. If h R ' J 9Â«o* in H* (A ~*), then R* I g Â• in HÂ« (A*). Let V* be a ' g "Â• ~ g direction in A*; let g* 6 r* and let h* be an affine line such that^f Iy*h* in H* (A*). Then, y-h* 11 j+ gÂ« in A*, and by our earlier argument, h* l| g* in A*. Thus, h* * V in A* and r* I h* in S* g tA * J * Thus, {j* U) reflects incidence and is a lined incidence structure embedding. // 2.64 Corollary . If X:A -+ A' and u:A' Â— ^ A" are non-degenerate AH-plane homomorphisms , then yuX is a non-degenerate AH-plane homomorphism. // ' Remark . The following corollary was inspired by Lorimer's argument for the validity of his Lemma 4.4 ni973)b, page 10"] which deals with surjective morphisms and the neighbor relation on points and on lines. See Discussion A. 27. 2.65 Corollary . If **:A Â— * A Â• is a non-degenerate AH-plane

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39 homomorphism, then m. preserves and reflects both the generalized neighbor relation and the quasiparallel relation. Thus, P ~ Q <=> mP ~ y^Q, and so on. Proof . By Proposition 2.46, ^ preserves both relations. Bv Theorem 2.63, the induced lined projective plane homomorphism H* (u*):H* (A') Â— * H* (A'*) (where A* and A" are the cross g ; g g Â» structures of A and A 1 ) is a lined projective plane embedding, and hence Â»*. reflects both relations. // 2.66 Corollary . If co*:(H*,g*) Â— + (H*,gÂ«) is a lined projective plane homomorphism, then wÂ» is either an incidence structure embedding or there is a line k* 4 g* such that if P* is not on gÂ», then wÂ«PÂ« I *i*k*. Thus, if w:(H,g) Â— > (H,g) is a lined PH-plane homomorphism, then either there is an affine line k such that if F is an affine point of (H,g), then <*>P ~ w k, or u> preserves and reflects the neighbor relation (thus, P ~ g <=^>u i P ~ ujg, and so on). Proof . One can easily see this by looking at A*(u>Â»). // 2.67 Proposition . If w:H* Â— * H* is a projective plane homomorphism, and if g* is a line of H* such that toh* = ug' implies h* = g* and such that there are two points P*,Q* on g* such that uiP* Â£ ioq* } then io is an incidence structure embedding or there is a line k* 4 g* such that if P* is not on g*, then uPÂ« I wk*.

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40 Proof. Assume Â»:HÂ» Â— * H* is such a morphism. if r* i s not on g', and if wR* I Â«ogÂ», then there is a point wsÂ« on wg* by our hypotheses such that wR* Â£ wS*. Thus, Â«:(H*,gÂ«) Â— * (HÂ» , gÂ«) defined by us' = ( (H*,g*) , (H*,wgÂ«) , f ^ ) i s a lined projective plane homomcrphism. The result follows from the corollary above. // Remark. The following proposition was inspired by Lorimer's argument for the validity of his Theorem 4.5 C(1973)b, page 101 which deals with morphisms which are surjective with respect to points: see Discussion A. 27. 2.68 Proposition. If ^:A Â— A' is a non-degenerate AH-plane homomorphism such that ^g II ph =5g U h for all lines g,h; then p is an AH-plane embedding; that is, ^ is injective and preserves and reflects the incidence, neighbor and parallel relations. If, in addition, p is surjective on points, then ^ is an AH-plane isomorphism. Proof. Assume that ^:A Â— A' is such a morphism and that -P = r Q. Let g Â€ PQ. Let R be a point such that yuR is not neighbor to jÂ»g. Let h = PR, k = QR. Observe that h ft k c=* P = Q since h,k + g, and P = g (\ h, Q = g (\ k. Hence ^h U Â«*k <Â£Â» P = Q. By our assumption above, r P = ^Q; so thatch = ^k and hence P = Q. Thus p. is injective on points. Then by Proposition 2.31, i* reflects incidence. By Corollary 2.65, yu reflects the neighbor relation.

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41 Assume /-g = ^h. Since^g lUh; g \\ h. Let k be a line such that TU^k) l> TY(^g): such a line exists since ^ is non-degenerate. Then ^(g f\ k) = ^g A ^k = j^ih (\ k) and thus g = h. Hence p. is infective. Thus, ^ is an AH-plane embedding. If in addition ^ is surjective on points, then by Proposition 2.56, p. is an isomorphism. // 2.69 Proposition . Let yu.:A Â— A' be a non-degenerate AH-plane (PH-plane) homomorphism. Then there are at least two points P,Q on each line k whose images y*P,*Q are not neighbor in A'. Thus, the action of yu. is uniquely determined by its action on the points of A. Proof. Assume first that ^*:A Â— * A is a non-degenerate AH-plane homomorphism. Let k be a line of A. Let R,S,T be points whose images under k'^Â» (where *' :A* Â— > (A')* is the neighbor map of A') are not collinear. At least one of the directions ^TT(RS ) , r TT(RT) and ^TT(ST) is not quasiparallel to^k since otherwise the lines xy(k), K y(RS), K'f(RT) and k'^(ST) would all be parallel, and hence k'^R, kVS and k'^T would be collinear. Let P be a direction such that r H n^k. The lines L(R,D, L(S,D and L(T,D all meet k in a single point: say R', S', TÂ», respectively. Observe that at least two of the points k'^R ' , k'^S Â• , *VT' are not equal, since otherwise kVR, kVS and K yT are collinear. Thus, there are points P,Q on k such that Â«*P f uQ. Now assume ^:A Â— > A' is a non-degenerate PH-plane homomorphism. Let k be a line of A and let R,S,T be points

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42 whose images under k'/* are not collinear. Then RS, RT and ST each meet k in, say, P, P', P". If K ' r K = tyP, k'^P Â• , k'^P", then Ky(RS), K'^(RT) and Ky(ST) are copunctal. Since K Â»yuR, kVS and k'^T are pairwise non-neighbor, we may assume ^M 4uR,uS without loss of generality. Then (k'/.M), (x'^T) I ( K VTS),( *y.TR) and K'^TS ^ KyTR; so that K'^M = n'yuT. But then k'mT I kV(SR), a contradiction. Thus, at least two of ^P,~P Â• , uPÂ» are not neighbor. Thus, there are points V,W I k such that kV ^ i*W. Thus, in AH-planes (PH-planes) the action of a non-degenerate homomorphism * is uniquely determined by its action on points. // 2.70 Proposition . If w:A* Â— > A* is a degenerate affine plane homomorphism, then there is a line k* such that ooP* I Â«ik* for every point P* of A*, and exactly one of the following three conditions holds. (a) There is a point Q* such that uÂ»PÂ» = W Q* for all points P* of A*. (b) For all lines g* of A', tog* = wk* and there are points P*,Q* such that wP* A wQ*. (c) There is a direction ?Â• not containing k* such that wm* = wk* for every m* \$ V*, such that wk' Â£ u>V , and such that cog" 4 wh* for some lines g*,h* e r*. Moreover, if A* is an affine plane, there is at least one endomorphism of A* of each of the three types: (a), (b) and (c). Proof. Assume that ui:A* -> A* is a degenerate affine plane

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43 homomor phi sin. Hence, the images of any three points of A* are collinear in A*. Let G* be a point of A*. If Â«*}?Â• = ooG* for every point P* in A*, then every line ug* goes through toG* and hence case (a) holds and the other cases do not hold, and we can let k* be any line of A*. Assume that there are points G*, H* such that wH' / wG'. Let k* = G*H*. Then, by our assumptions, wF' Iwk' for every point P*. There are two remaining subcases. If cog* = wk* for every line g* of A*, then case (b) holds and the other cases do not hold. If there is a line g* such that wg* / tok*, then, since P* I g* implies toP* Iu>g*,ok* which implies tog* 4f iok* , we have that w(L(H*,g*)) ^ w(L(G*,g*)). If m* Vr g*, then m* meets both L(H*,g*) and L(G*,g*); hence iurn* = tok*. Hence case (c) holds and the other cases do not hold. Let A* be an affine plane. Let Q* be a point of A*. Define u>:A* Â— * A* by uoP* = Q*; u>g* = L(Q*,g*): to is a type (a) homomorphism. Let k* be a line and let Q* be a point on k*. Define k :A* Â— > A* by Â« P* = P* if P* Ik*, by Â«P* = Q* if P* is not on k*, and byocm* = k* for all lines m* of A*. Then <*. is a type (b) homomorphism. Let k* be a line and let P* be a direction such that k* ^ P*. Define V:A* Â— * A* by v(F*) = L(P*,P*) A k* for all points E* of A*, by v(g*) = g* for g* e P*, and by \>(h*) = k* for h* ^ T*: C is a type (c) homomorphism. //

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3. BITERNARY RINGS In this section we define 'coordinatized affine Hjelmslev plane' and 'biternary ring 1 , construct the related categories, and show that they are eguivalent. 3.1 Definitions . Let T* be a ternary operation defined en a set M* with distinguished elements 0* and 1* with 1* Â£ 0*. Then (H*,T*) is said to be a ternary field if it satisfies the following five conditions: (TF1) T*(x*,0*,cÂ») = TÂ»(0*,m*,cÂ») = cÂ» for all x*,m*,c* in M*. (TF2) T'(l',m*,0Â«) = TÂ» (mÂ« ,1* ,0* ) = m* for all m* in H". (TF3) For any x*,m*,c* in M", there exists a unique z* in M* such that T*(x*,m*,z*) = c*. (TF4) For any m*,d*,n*,b* in N* such that m* 4 n*, there is a unique x* in M* such that T*(xÂ»,m*,dÂ») = T* (x* ,n* ,b* ) . (TF5) For any x* ,C , x ' * ,c * * in K* such that x* ^ x'Â», there exists a unique ordered pair (m*,d*) such that T* (x* ,m* ,d* ) = c* and T*(x' * ,mÂ« ,d* ) =Â• c * * . We say that 0* is the zero and that 1* is the one of (M*,T*). We call the elements of H* symbols . If (MÂ»,TÂ») and (Q*,S*) are ternary fields, an ordered triple uj = ( (M*,TÂ») , (Q*,s*) ,fj is said to be a ternary field homomorphisrr, if f^M* Â— * Q* is a function such that 44

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45 u>('T*(xÂ»,m*,e*)) = S* (wx*,uim* ,we* ) and such that wO* = 0*, u>l* = 1*. We denote the category of tern ary fields by F* . 3.2 Definitions . Let M he a set with distinguished elements and 1, and with two ternary operations defined on M. Let N = {n 6 Ml 3 k Â€ M, k t 0, 3 T(k,n,0) = o"5 , and let N' = (n'Hia k t, K, k ^ 0, * T'(k,n,0) = 0"5. Define a relation ~ on M by a ~ b (read 'a is neighbor to b') if and only if every x which satisfies the equation a = T(x,l,b) is an element of N. Define a relation Â«*' on M by a ~' b if and only if every y which satisfies the equation a. = T'(y,l,b) is zv. element of ."."Â•. The negation of a ~ b is written a Â£ b and is read 'a is not neighbor to b'. Then, (!:,T,T f ) is said to be a biternary ring if the following twelve conditions are satisfied. (BO) N = N', and a necessary and sufficient condition that a Â— ' b is that a ~ b. (Bl) The relation is an equivalence relation; that is, the relation ~ is reflexive, symmetric and transitive. (B2) T(0,m,d) = T(a,C,d) = d for any a,m,d from K. (B3) T(l,a,0) = T(a,l,0) = a for any a from M. (B4) T(a,m,z) = b is uniquely solvable -for z for any a,m,b from M. (B5) T(x,m,d) = T(x,m',d') is uniquely solvable for x if and only if m /m' for any m,d,m',d' from M. (B6) The system T(a,m,d) = b, T(a',m,d) = bÂ» with a ^a' is uniquely solvable for the pair m,d; if a 4> a ' , b ~ b 1 , we

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46 have m Â€ N; if a Â« a 1 and b + b', the system cannot be solved. (B7) If a ~a', b~ b', and if (a,b) 4 (a',b'), then one and only one of the systems Â£T(a,m,d) = b, T(a',m,d) = b Â• "5 and \$T'(b,u,v) = a, T'(b',u,v) = a* where u <= N*} is solvable with respect to rn,d correspondingly u,v (where u fe N) , and it has at least two solutions; and we have m' ~ m", d' ~ d" or u Â• ~ u" v' ~j v" respectively for any two solutions. (B8) The system ly = T(x,m,d), x = T'Cy.u.v)"} where u e N, m,d,v (, M, is uniquely solvable for the pair x,y. (B9) For any m,u G M, T(u,m,0) = 1 if and only if T'(m,u,0) = 1. If T(u,m,0) = 1, if T(a,rr.,e) = b, and if T'(b,u,v) = a for some n,u,a,b,e,v Â€ M, then (T(x,m,e) = y <=\$> T'(y,u,v) = x) for every x,y 6 M. (BIO) The function T induces a function T* in M/~ , and (K/~,TÂ») is a ternary field with zero 0* = {z lz ~ 0] and one 1* = \ e I e ~ 1^ . (BID Conditions (30) through (BIO) hold with T and T' interchanged throughout; the new conditions will be called (BO)' through (BIO)'; condition (BIO)' states that the function TÂ« induces a function T'* in MA', and that (K/~',T'Â») is a ternary field with zero 0* and one 1*; of course, N and N', ~ and ~* are interchanged throughout also. Each element of N is said to be a right zero divisor . 3.3 Definition. If (M,T,TÂ») is a biternary ring, then (M,T',T) is a biternary ring by the symmetry of the definition of biternary ring: (M,T*,T) is said to be the dual of (M,T,TÂ»).

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47 3.4 Definitions. Let (B,T,T<) be a biternary ring. We will frequently write B to denote (B,T,?'). We will frequently write N B or simply N to denote the set of right zero divisors in B. The elements of the set 3 are called sv-bcls ; is called the zero of 3 and 1 is called the one of 3. If N = {o] we say B . that B is a biternary field . 3.5 Proposition. Let (2,T,T') be a biternary ring and let u e B. Then u if and only if u 6 N. Proof. Assume u ~ 0. By (33), u = T(u,l,0), and hence u fe N by the definition of neighbor in 3. Assume u Â€ N. Then there is a k in K, k 4 0, such that T(k,u,0) = 0. Since x = and x = k are both solutions to the equation T(x,u,0) = T(x,0,0), we have by (35) that u ~ 0. // 3.6 Prop osition . If (E,T,T') is a biternary ring, then 1*0. Proof. By (E10), 1* ^ 0Â»; hence 1*0. // 3.7 Proposition. In a biternary ring (3,T,TÂ«), the equation a = T(x,l,b) has a unique solution x for each pair (a,b). In addition, a ~ b if and only if x 6 N, Proof . Let a and b be elements of the set 3; that is, let a and b be symbols. Since 0*1, by (B5) there is a unique solution x to the equation T(x,l,b) = T(x,0,a). By (32),

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48 we have that T(x,0,a) = a. Hence, a ~ b if and only if x G N. // 3.8 Proposition . Let (3,T,T') be a biternary ring and let m 6 B. There is a u Â€ B such that T(u,n,0) = 1 if and only if m Â£ M. If m ^ NÂ» then the solution u is unique and u ^ N. Moreover, the map S:M\N Â— * M\N defined by T(J(m),m,0) = 1 is a bisection. If u Â€ M\N, then T(u, i" 1 (u) ,0) = 1. Proof . If m Â€ N, then m ~ and there is no element u e B such that T(u,m,0) = 1 since 0* ^ 1Â« and T*(u*,0*,0*) = 0Â» for every u* in MA,. If m ^ N, then m -f by Proposition 3.5, and by (B5) there is a unique u such that T(u,m,0) = T(u,0,l). If u were in N, then u ~ and, by (B5) , T* (u* ,m* ,0Â» ) = T* (0* ,mÂ« ,0* ) = 0*, a contradiction. Thus, we can define a map S :M\N Â— M\N by T(S(m),m,0) = 1. If u <Â£ N, then the system (T(u,m,d) = 1, T(0,m,d) = 0l is uniquely solvable for the pair m,d by (B6) since u y0. By (B2), d = 0, and, since TÂ»(u*,0*,0Â«) = 0*, m j N. Thus, I is surjective. If m' satisfies the equation T(u,m',0) = 1, then the pair m',0 is a solution to the system above and hence m' = m. Thus, the map S is bijective.' If u Â£ M\N, then there is an m fe KAN such that T(u,m,0) = 1. Hence S(m) = u, and we have that m = %" (u) and that T(u,J>~ 1 (u),0) = 1. // 3.9 Definition . Let (3,T,TÂ») and (M,S,S') be biternary rings.

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49 A biternary ring ho-onorohism *:B Â— * M is a concrete morphism such that *(N B ) Q N^-, *(0) = 0; oUl) = 1; *(T(x,in,e) ) = S(Â«x,*m,e(e) and tf(?'Â£y,u,v)) = S ' (oty ,*u , *v) for all x,n,e,y,u,v in B. 3.10 Definitions . It is easily seen that the class of biternary rings and their homomorphisms form a category. We denote this category by 3 and call it the category of biterr.a rv rings. The full subcategory of B whose objects are biternary fields we denote by B*. 3.11 Definitions . We say C = (A,K) is a coordinatized AH-nlane and K = (g^g ,E,K,^:CE Â— * K) is a coordinsticaticn of A whenever A is an AH-plane, g^ ,g are non-neighbor lines of A, x y E is a point of A not neiahbor to either a or g M is a set x 3 y' with distinguished elements and 1, CS is the line joining = 9 V ft g,. to E, and f :CE -^ M is a bisection such that (0) = 0, X y ) " (E) = 1. Let C = (A,K) be a coordinatized AH-plane. If ? is a point of A, define 6(P) = (x,y) = C^(0S A L(P,g )),f;(CE fl L(P,g ))); y x the construction is indicated in Figure 3.1; and define fi'(P) = (y,x)'. If 0(F) = (a,b), we say b is the y-ccordinate of P and that a is the x-coordinate of P; let 1r F = a , -It P = b. x y If k is a line of C = (A,K), and if k -tf g , define X(k) = ~ " Y rm,d] = [ir y (L(0,k) ft LCE.g )),Â«* (k f\ g )3 ; the construction is indicated in Figure 3.2. Whenever k -fr g , we interchange the roles of g and g in the definition of X to define V (k) = x y [u,v]' = fir (L(C,k) A L(E,g )),1T (k f\g )]'.

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50 Figure 3.1 . g y

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51 It is easily seen that the maps 9, &< (X, X') are welldefined functions from 1* (from {k \ k -tf g } . from fk Iklo]) ~ y J ~ y x J into MXT-1 and that they are bisections. If g is a line of C such that X(g) = tm,d], then Um,d3 is said to be a representation of g; similarly, if \'(g) = Uu,vl', then [u,vl' is said to be a representation of g. If 0(?) = (x,y), then (x,y) and (y,x) Â• are said to be representations of F. Since 0, 6', A, X* are bijections, we can, without fear of confusion, identify a point or line with each of its representations or with its one representation. The line g is called x the x-axis of C; g is called the y-axis of C; is called the origin of C and E is called the unit point of C. Let X denote the point g f\ L(E,g ), and let Y denote the point g (\ L(E,g ). x y y x 3.12 Definition . Any pair of statements or functions which can be gotten one from another by interchanging the roles of g and x g throughout are said to be xy-duals . The functions 0, 0' given above are xy-duals, as are' A and X'. 3.13 Proposition . Let g,h be lines of a coordinatized AH-plane C, and let g = [m,e]. Then h is quasiparallel to g, h 11 g, if and only if there are m',e' such that h = tm',e'] and (l,m) ~ (l,m' ) . Procf_. Assume h 11 g. Then, since Ig* g *1 = 1 implies lh* H g *1 = 1, we have that lh g 1 = 1. Thus, for some m',e' y y we have that h = [m'.a']. Since h 11 g, L(0,g) II L(0,h) and

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52 hence L(0,g) ~ L(0,h). Thus, since L(E,g ) ^ L(0,g) ,L(0,h) , we have that (L(0,g) t\ L(E,g )) v (L(C,h) A L(E,g )), and hence y y (l,m) ~ (l,mÂ«). Assume h = [m ',e'] and (l,m) ~ (l,m'). Then L(0,g) 11 L(0,h). and g \\ h. // 3.14 Definitions . Let C and C* be coordinatized AH-planes. A coordinatized AH-plane honomorohism or coordinatization homomorphism u:C Â— * C is a rr.ap uo which is an AH-plane homomorphism such that w(g ) = g ' , *>(q ) = c < and 10(E) = E' where q '. g Â• x x y ~y xy and E' are the x-axis, y-axis and unit point of C respectively. If C = (A,K) is a coordinatized AH-plane, then the neighbor map K:A Â— A* induces a coordinatization homomorphism from C to C* = (A*,?;*) where K* is the coordinatization of A* whose x-axis is (g )'Â• and so on; we denote this induced map by K:C Â— C* and call K the neichbor man of C. 3.15 Definition . It is easy to see that the class of coordinatized AH-planes together with their coordinatization homomorphisms form a category. We denote this category by <:, and call it the category of coordinatized affine Kjelmslev planes . We denote the full subcategory of t whose objects are coordinatized affine planes by C* . 3.16 Construction of B:C Â— * B. Let C be a coordinatized AH-plane. Define a ternary operation T:iM Â— * M by T(x,m,e) = y if and only if there exist a point P and a line g, Pig, such that fl(P) =

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53 (x,y), X(g) = Cm, el. Define a second ternary operation T':M Â— * M by interchanging the roles of q and q in the x y definition of T; that is, let T'(y,u,v) = x if and only if there exist Q, h, Q I h, such that 8'(Q) = (y,x)* and h' (.h) = Cu,v"}'. Let B(C) = (X,T,T'). Given a morphism us:C Â— Â•> C * in C, define a map B(ui):B(C) Â— * B(C) by B(uj)m = Â£' (u( " (m))) for all m in M. By M we mean the set of symbols of C. Â• Remark . Many of the intermediate steps in the proof of the following proposition are stated in Lcyganova (1967)] (see our Remarks A. 2, A. 15 and A. 16 in Appendix A): she states (Lemma 1), part of (Lemma 2), part of (Lemma 4), (Bl), (B2), (B3), (B4), (B5), (B6), (B7) and (B3). 3.17 Proposition . The map B:C Â— B defined above is a functor from the category of coordinatized affine Kjeimslev planes to the category of biternary rings. If C* is a coordinatized affine plane, then B(C*) is a biternary field. If C is a coordinatized AH-plane, then (a,b) ~ (a',b') in C ^ a ~ a 1 , b ~ b' in B(C); [m,dl ~ [m'jd'l in C ^ m ~ mÂ», d ~ d' in B(C) ; [u,v) ' ~ [u',v'] ' in C <=Â£ u ~ U', v ~ v' in B(C) . Proof . Assume C is a coordinatized AH-plane. Define a o b if and only if (a, a) ~ (b,b). Let N = \n e m\ n o o}. ' Observe that if C is a coordinatized affine plane then N = ^.o} . (Lemma 1) (a,b) ~ (a', fa') <Â£> a o a', b o b'. (Proof) Assume (a,b) ~ (a',b'). Then

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54 L((a,b),g ) ~ L((a',b'),g ); so that, since OE is not quasiparallel to g , (L((a,b),g ) H OE) ~ (L((a',b'),g ) H OE) and hence (a, a) ~ (a', a'), and a o a'. Similarly, b o b'. Assume a o a', b o b'. Then L((a,a),g ) ~ L((a',a'),g ); so that, since g is not auasiparallel to q , (a.b) ~ (a'.b). Y x ' Also L((b,b) t g ) ~ L((b',b'),g ); so that (a' ,b) ~ (a',b'). Thus, (a,b) ~ (a' ,b' ) . / (Lemma 2) [m,e"J ~ [m',e'] <=} m o m', e o e'; hence by xy-duality, [u,v]' ~ [u',v']'
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55 Assume there is an element k, k Â£ 0, such that T(k,n,0) = 0. Then both (k,0) and (0,0) are en [n,o] and [o,o]; so that [n,0] ~ [o,o]. Thus (l,n) ~ (1,0) and n o 0, n Â€ N. / (Lemma 5) a o b if and only if every x which satisfies the equation a = T(x,l,b) is an element of N. (Proof) Since -yE, 4> 1. Hence, by Proposition 3.13, [0,a] is not quasiparallel to [l,bj. Thus, the equation a = T(x,0,a) = T(x,l,b) has a unique solution x. Assume a o b. Then (0,a) ~ (0,b); so that [o,a] ~ [0,b]. Thus ([0,al f\ [l,b]) ~ ([0,bl f\ [l,b]). Thus, if (x,a) = Co, a"] H [l,b] then (x,a) (0,b) and x o 0. Kence a = T(x,l,b) and x Â£ N. Assume every x which satisfies the equation a = T(x,l,b) is in N. . Let (x,a) = [l,bl [o,a]'; then a = T(x,l,b); x o 0. Since l[l,bj (\ g \ =1, (0,b) = [l,b] A g . Observe that l0,xl' ~ 10,0]' where g = [o,0]' and hence (l0,xl ' fl [l,b]) ([0,0] ' A [l,b]) ; (x,a) ~(0,b), and a o b. / (B0) The xy-dual to the proof of (Lemma 5) above shows that a o b if and only if every y which satisfies the equation a = T'(y,l,b) is an element of N. Thus in B = B(C) the relations ~ and ~' of the definition of biternary ring are equal to the relation o defined here. By (Lemma 4) N = 'N = N* so B B that condition (B0) of the definition of biternary ring holds in B(C). (Bl) The relation o on M is obviously an equivalence relation. (B2) Since (0,d) I [m,d], T(0,m,d) = d. Since [o,d] llg , (a,d) I Co,d]. Hence T(a,0,d) = d.

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56 (B3) Since (l f a) I [a,0'J, T(l,a,0) = a. Since [l,0] = OE, we have that (a, a) I [l,cl and T(a,l,0) = a. (B4) There is a unique line parallel to [m,o] through the point (a,b): say [m,z]. Then T(a,m,z) ,b. If T(a,m,z') = b, then (a,b) I fm,zO; hence z = z'. Thus, z is the unique solution to the equation T(a,m,z) = b. (B5) The equation T(x,m.d) -_Kx.m'jd') is uniquely solvable for x if and only if there is a unique point (x,y) I Cm,d],r m ',dO; hence if and only if ["m.d] is not quasiparallel to [mÂ»,d'] and, by Proposition 3.13, if and only if m 4> m' . (B6) Look at the system \$T(a,rr.,d) = b, T(a',m,d) = b'^. If a Â£ a', then the unique line g joining (a,b) and (o',b') is not quasiparallel to g y . Hence g = [m,d3 for some m,d and the pair m,d is the unique solution to the system. If a cS a' bob* then [m,d] 11 g^ and hence by (Lemma 3) m o 0. Ifa oa'.b/b', then, if (a,b),(a',b') I g, g li g and hence the system has no solution fm,d]. (B7) Let g,gÂ» be any two distinct lines joining (a,b) and (a',b'), (a,b) /. (a',b'K Then g ~g'. Observe that (g (\ q ~ s y ^ 9' ! V and that g.g' !g y <^ g [u.v]-, g Â« = [u', V ']< for some u,u' 6 N, v,V e M . The lines g,g' are not quasiparallel to g y if and only if g = [m,d], gÂ« = [m',d-] for some m,m',d,din M. Hence condition (B7) holds in B(C). (B8) Any line of the form [u,v]' f u Â£ N, is quasiparallel to g y and hence meets any line fm,d] (not quasiparallel to g ) in exactly one point (x,y).

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57 (39) T(u,m,0) = 1 if and only if (u",l) I [m.O']. Hence T(u,m,0) = 1 if and only if [u,0"]Â» = [m,0l, and by xy-duality, fu,0]' = [m,0] if and only if T'(m,u,G) = 1. If T(u,m,0) = 1, and (a,b) I [m,e] , [u, v] ' , then [m,e~\ = [u,v]', and hence (x,y) I [m,e] <^> (x,y) 1 [u,v3*. (BIO) The construction of the ternary operation T maps, under the neighbor map K, to the construction of T* which is the usual construction of the ternary field of a coordinatized affine planeÂ— see [Hall (1959), pages 353-355, Section 20.3]Â— and \$***," (0) = 0Â», ^'K^ (1) = 1*, where 0* is the zero and 1* the one of (M*,TÂ»). (BID Observe that since we have made no special requirements on C, the xy-duals of conditions (BO) through (BIO) also hold in 3(C) . Thus we have shown that B(C) is a biternary ring. Assume that u>:C Â— * c ' is a morphism in C, and that 3(C) = (;-;,T,T'), B(C) = (Q,S,S'). We wish to show that B(w) :B(C) Â— *B(C) defined by B(<-j)m = t,'^ 10 ^ (m))) is a biternary ring homomorphism. Define us' :M Â— > Q by B(ui)m = uj'm. Then it is easily seen that w(x,y) = (w'x, m'y) , u>Cm,d3 = [co'm,*>'d3 and uÂ»fu,v]' = r^'u , * (T(x,m,e) ) = S(u)'x, u>'m, w'e) and u)' (T' (y,u,v) ) = S ' (w'y, w'u.w 1 v) . Since (n,n) ~ (0,0) implies (w'n,w'n) ~ (0,0), w'N C N . Observe that ui'(0) = 0, ui* ( 1 ) = 1. Hence B(<*Â») is a morphism in B. Obviously B(l c ) = 1 B(C) If Â«-C 1 -^ C 2 and p:C 2 Â— C 3 are morphisms in C, then if m 6 M , 3(AÂ«)m = \$ Ce*(\$ Â„ ~~ n ; 3 ^ 3 ( p ( f 2 t2 U( S!~ CmJ>))) = B(j5)(B(oO(m)), so that B(pec) = B(/5)B(Â«) and 3:C Â— * B is a functor.

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58 If C is a cocrdinatized affine plane-, recall that N , . = B(C) {0] and hence observe that 3(C) is a biternary field. // 3,18 Construction of C:5 Â— Â» C and A : 0] Â— * (M x { < >,< >Â•])/=. Given a biternary ring B = (K,T,T') we construct an incidence structure 5 = CP,oj,I) and a parallel relation W in the following way. Let V= Mx M and let Oj = M * M *tOl and 0} = M < M * til. For convenience we denote an element (m,d,0) of 0] by [m,d] and an element (u,v,l) of m by [u,v]'. We define U ji A -g T ^ Â«5 T , -*H*H >,< Yl by A F (g) = , if g = [m,d] f and 6 (g) = ', if g = [u,vl'. We define incidence by (x,y) I [m,d] <=} y = T(x,m,d) and by (x,y) I Cu,v]' <Â£> x = T'(y,u,v). We identify and ' 4=> T(u,m,0) = 1 = T'(m,u,0). If = ', and if there is a point (a,b) on both [m,d"] and [u,v]', then by (B9) and (B9)Â«, ((x,y) I [m,d"] <=> (x,y) I [u,v]'), and we identify [m,d] and [u,v]'. Denote the set of lines by a] where 0j = (0] T U 0j T ,)/=. Define A : 0] Â— * (M Â» t< >,< >'})/= in the obvious way. We define g II h if and only if &(g) = A(h). We denote the incidence structure and the parallel relation thus defined by A fi . We let 5 B :[l f ol Â— * H be defined by \$ g (a,a) = a. Let K = ([0,0],[0,0]',(1,1),K,C, ). we define C(B) to be (A.K ). o /v B B Given a biternary ring homomorphism uj:B Â— > B' we define C(w):C(B) Â— *C(BÂ«) by C(u>)(a,b) = (oja.cob), C(w)[r.,d] = [>om,ujd] and C(lo)[u,v1' = [ w u,wvl' for all points (a,b), lines [m,d], [u,vl' in A . Observe that C(uj) is well defined. 3.19 Lemma . Let (B,T,T') be a biternary ring. Then the

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59 following conditions hold in the construction given above. (a) If m,d 6 K, m Â£ N, then there is a unique line Cu,v3' of oj T , such that [m,d] . [u,v]'. If m 6 N, there is no such line Cu,v]'. (b) If u,v 6 M, u ^ N, then there is a unique line [m,d] of 0] T such that [u,v]' = [m,d]. If u 6 N, there is no such line [m,d]. (c) The function &:0]-^ (HÂ«t( >,<. >'})/= constructed above is well defined. Proof. (a) Assume g = [m,d], tn 4 N. Then (0,d) I g. There is a unique u Â€Â• N such that T(u,m,0) = T(u,0,l) since + m. Hence T'(m,u,0) = 1, and there is a unique v such that (0,d) I Cu,v]'. Hence [m,d] = [u,v]Â«. If [m,d] = [w,z]', then T(w,m,0) = 1 and u = w. Also since = T'(d,u,z), z = v. Thus [u,v]' is unique. Assume g = [m,dl, m 6 N. Then, since m 0, 0^1, there dees not exist an element u e M such that T(u,m,0) = 1, and hence there is no [u,vl' fe oj t , such that [m,dl = [u.v]'. (b) This is the dual of case (a). (c) If [m,d] = [u,v]', then &[m,d] = (m*> = Â» = &[u,v]Â«; hence A is well defined. // 3.20 Definition. If B = (M,T,T') is a biternary ring and if B= (M,T',T) is the dual of B, then (x,y) I [m,d") in C(B) <=^ (y,x) I [m,dlÂ« in C(B'), and (x,y) I [u,v]' in C(B) <=> (y,x) I [u,v] in C(B'). The structure C(BÂ») is said to be the structure dual to C(B).

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fcviC 60 3.21 Proposition . Let 3 be a biternary ri-ng and let 3' be the dual of 3. The map 6:C(3) -*c(B'> defined by 0(x,y) = (y,x), 9[ra,d] = tra.dl' and 0[u,v]' = [u,v] is an incidence structure homoraorphism which preserves and reflects the parallel relation. // 3.2? Proposition . If B* is a biternary field, then C(B*) constructed above is a cccrdinatized affine Diane. Proof. If 3* = (M*,T*,T*Â«) is a biternary field, then (M*,T*) is a ternary field and it is easily seen that C(3*) is the coordinatized affine plane commonly constructed over the ternary field (K*,T*): see [Hall (1959), top of pace 356]. // 3.2 3 Prop osition . The map C:B -* C indicated above is a functor from the category of biternary rings to the category of coordinatized affine Hjelmslev planes. oof. Let B = (K,T,T') be a biternary ring. The structure A 3 .nstructed above is an incidence structure S D with a relation Parallel' defined on the lines of s_. We wish to show that A Â» B s an AK-plane. We start by proving a number of lemmas. To void confusion, throughout the remainder of this proof we will ise the symbol ~ to indicate the neighbor relation in B, and :he symbol Q to indicate the relation 'af finely neighbor' in A : B see the definition of AH-plane, Definition 2.21. (Lemma 1) Whenever P,Q are points, there is a line g such that P,Q I g.

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61 (Proof) Let P = (a,b), Q = (a',b'). There are four cases. (Case 1) Assume P = Q = (a,b). Then by (E2), (a,b) I [0,bl. (Case 2) Assume P 4 Q, a ^ a ' . The system T(a,m,d) = b, T(a',m,d) = b 1 has a unique solution m,d by ( B6 ) , end hence P,Q I [m,d].

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62 (Proof) (<= ) Assume g h t P I g,h. We wish to show Ig A hi = 1. Let P = (a,b). (Case 1) Assume g = Â£m,el, h Â«= [m^e'], and that Q I g,h; Q ^ P. Let Q = (a',b' ). By Lemma 2, a ~a', b -v b ' . Hence by (B7), m ~ m', e ~ e ' . Let (c,d) be a point on one of the lines; say g. Then d = T(c,m,e). Let d' = T(c,m',e l ). By (BIO), d' ~ d. Thus by Lemma 2, (c,d) D (c',d'). By symmetry, g h, a contradiction. Hence Ig A h\ = 1.

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63 (Case 3) Assume g,h e 0J T ,Â» g ^ h. This is Case 2 in the dual structure, hence g j3h. (Case 4) If none of the preceding cases occurs, by Lemma 3.19 we may assume g = [m,e~], h = [u,vl' where u Â£ N. There is a symbol w such that w -/v. Assume g h. Let z = T(w,m,e); Q = (w,z). There is a point R = (w',z') on h such that R Q. By Lemma 2, w ~ w'. Since u Â€ N, u ~ C, and since w' = T'(z',u,v) by (BIO), w" = v* ; so that w' ~ v. Thus, since ~ is an equivalence relation on B; w ~ v, a contradiction. Hence g \$ h. / (Lemma 4) If g = [m,e]; then g h if and only if h = [m',e'] and m m', e ~e'. If g = [u,v]'; then g h if and only if h = [u'jV'l' and u ~ u Â• , v ^v'. (Proof) (Case 1) Assume g = [m,e]. (Â£=) Assume h = [m'je'l, where m m', e ~ e'. Let (a,b) be a point on one of the lines: say g. Let b' = T(a,m , ,e'). By (BIO), b ~ b'. Hence (a,b<) I h and (a,b') D (a,b) by Lemma 2. By symmetry, g h. ( =^) Assume g D h. If g = h we are done. Assume g I h. By Lemma 3, g h implies Ig hi 4 1; hence by ( E3 ) , h 4 [u,vV, u 6 N; hence by Lemma 3.19 (b) , h = [m',e']for some m',eÂ» in K. If m * m', then by (B5), there is a unique point of intersection, a contradiction by Lemma 3. Hence m ~ m*. The point (0,e) is on g. Let (u,f) I h such that (0,e) D (u,f). Then, ~ u, e ~ f. By (BIO) there is a unique z* such that T*(0*,m*,zÂ«) = e*. Then e* = e'* = z* and e e'. Thus, m ~ mÂ», e ~ e ' .

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64 (Case 2) Observe Case 2 reduces to Case 1 in the dual structure. / (Lemma 5) The relation 'af finely neighbor' is an equivalence relation on the points and lines of A . 3 (Proof) The relation 'af finely neighbor' is an equivalence relation on the points of A and is a reflexive and symmetric relation on the lines of A_ by Lemmas 2 and 4. Assume q D h B 3 1 h P k. If g e oj t , then h 6 0j t ; so that k 6 oj by Lemma 4. Hence g D k by Lemma 4 and (31). Similarly, if g 6 Oj ( , then h,k Got and g D k. Thus 'af finely neighbor' is an equivalence relation on the points and lines of A . / B (Lemma 6) If P is a point; q, a line, then there is a unique line h such that P I h, h II g. (Proof) (Case 1) Let P = (x,y) and g = [m,b]. Then, by (B4) there is a unique symbol z such that y = T(x,m,z). Kence lm,2 J is the unique line of 0| parallel to [m,bl containing F. Assume P I [u,vl' and [u,v]' (l I>.,b]. Then * u y =^m> so that since P I [u,v"]Â« ,U,zl f and Cu,v]' = [m,z"]. Thus, [ m,z] is the unique line through F parallel to [m,b], (Case 2) This case reduces to Case 1 in the dual structure, / (Lemma 7) There is a map ^:A^ Â•Â— Â» A_. which' is a surjective incidence structure homomorphism such that Ig f\ hi in A implies E Â«fg H ^h in A fi< , and such that (Â»? F = if Q <=^ p D Q) and ( g O h.) (Proof) If a & M, denote the Â— equivalence class of a in B by aÂ». Then define Â«?(a,b) = (a*,bÂ«), *ffm,d] = m%d* and

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65 *f[u,v"]' = [0*,v*]' when u Â€ N. Then, since the map v:B Â— Â» B* induced by ~ is a biternary ring homomorphism, ^ preserves the incidence relation. The remainder of the lemma is immediate by Lemmas 2, 4 and 5. / Thus, A g is an AH-plane. Observe that [0,01^ Co,Ol'; that (1,1) [o,0],[o,0 "]'; and that %_ is a bijection. Thus C(B) = (A B , ( [0,01, [0,0]' , (1,1) fM,Â§ ) ) is a coordinatized AH-plane. If w:B Â— *Â• B ' is a biternary ring homomorphism, then C(ui) defined earlier is an incidence structure homomorphism which preserves the parallel relation. Recall that u>(N ) C N B B ' Hence, if a bin 3, the uia Â«-wb in 3'. Thus, (a,b) D (a',b') in C(B) implies (u>a,) is a coordinatized AH-plane homomorphism. L c(B) and S^/ 30 ^ = x ( /^Â£ (<<) ' 2 : ^ ~^ ^ is a Since C(1J = ].Â„,Â„, and C(ftx) = C(ft)C(Â«), C:B ~ B functor. If B* is a biternary field, then N = to! and C(B*) is a coordinatized affine plane since the neighbor map of C(3*) is essentially the identity map. // 3.24 Proposition . The functor BC is the identity functor on B. Proof. Let (B,T,T') be a biternary ring, and let (Q,S,S') = BC(B). By the constructions, Q = 3, and y = S(x,m,d) in 3C(B) O (x,y) I [m,dl in C(3) <Â£=\$> y = T(x,m,d) in 3. Similarly, x = S'(y,u,v) in BC(B) <*=> (x,y) I [u,vl* in C(B)<\$=> x = T'(y,u,v) in B. Hence BC(B) = (B,T,T').

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66 Let ui:3 Â— *Â• 3" be a morphism in 2. Then CU)(a,a) = (*ja,wa) for every a e 3, end B(cU))a = -r"CU)Â« a = {"(we,wa) = wa. Thus BC:B Â— Â•> 3 is the identity functor on 3. // 3.25 Construction cf p:l~ Â— Â•> CB. Let C = (A, (g ,a ,E,M,c)) be a coordinatized AK-plane. Then CB(C) = ( A_ ( ) , ( LO ,0l ,[0, 0] ' , ( 1 , i ) , M, ^r(D^ an " ^" t can eas ^-^y be seen b Y looking at the usual identification of points and lines of C with their representation; that there is an isomorphism P r :C Â— C3(C) v.'hich takes a point (x,y) of C to the point (x,y) of CB(C). 3.26 Proposition. The map Â»:l*s 1 t r _ c Indicated above is natural isomorphism. Proof . Let u>:C Â— * C ' be a morphism in C. Recall that B(w): ^'(uj(<^ m)). r.ence, if P is a point of CE in C, EU)(*P) = ^Â•UP). Thus if ? I CE in C, CB(u>)(o P) = CBUM^P^P) = (B(w)(^P),S(w)(\$P)) = C^'UP) , Â£,' UP)) = * (up). Thus the diagram: CB(m) -* CB(C') commutes with respect to the x-axis of C, the y-axis of C and all the points of GE, and hence it commutes with respect to all the lines which are parallel to either g or c , and hence with 'x ' y respect to all the points of C. The image of each line of C in CB(C') intersects either both Co,Ol and [0,ll or both [0,o]' and

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67 lOjlJ 1 in CB(C') in points which are non-neighbor images of points in C. Thus, since the diagram commutes with respect to all the points of C, it commutes with respect to all the lines of C. Thus fsljs, Â— * CB is a natural isomorphism. // We have shown the following theorem: 3.2 7 Theorem . The functors B:C Â— * B and C:3 Â— Â» C are reciprocal equivalences where C is the category of coordinatized affine Hjelmslev planes and B is the category of biternary rings. // 3.26 Corollary . The functors B*:C* Â— * 3* and C*:B* Â— Â» C* defined as restrictions of B and C respectively are reciprocal equivalences where B* is the category of biternary fields and C* is the category of coordinatized affine planes. // 3.29 Definition . There are reciprocal equivalences F*:C* Â— -> F* and C p *:F* Â— * C* (where F* is the category cf ternary fields) defined by letting F*(C) be (M,T) if B*(C) = (H,T,T'); by letting C F *(F) be the usual coordinatized affine plane constructed over a ternary field F (see the proof of Proposition 3.22); by letting F* take a morphism w:C Â— > C Â» to (F* (C) ,P* (C ' ) ,f _ , J and by letting C * take a morphism <* = (F.F'.f ) to (C Â•(F). C *(F').f ) -r ' Â•>. Â«. F Â»~p ' i Â«' where f ^ , is defined by f -t (x f y) = (f^x.f^y), f^^.d! = Um,Â«d] and f , rO,vl' = fdO,etv"]' . The composition of B*:C* Â— Â» B* and C *:F* Â— * c* is a functor B'C F *:FÂ» Â— B* from the category of ternary fields to the category of biternary fields. If (H*,T*) is a ternary field,

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63 then BÂ»C F *(M*,TÂ») = (MÂ»,T*,T") is called the biternary field associated with the ternary field (M*,T*). 3.30 Proposition . If u>:A Â— * A' is a non-degenerate AH-plane homomorphism; that is, if there are three points of A whose images under k'u> are not collinear (where k' denotes the neighbor map of A 1 ), then there are coordinatizations K, K' such that w" = ( (A,K) , (A' ,K' ) , f w ) is a coordinatizaticn homomorphism with 0, X and Y as three such points; hence B(w") is a biternary ring homomorphism. Conversely, if w":(A,K) Â— Â» (A', K ') is a morphism in C, then the images of P = 0, P = X and P = Y are not collinear in the gross structure of A ' , and hence u> = (A,A*,f f| ) is non-degenerate. // 3.31 Definition . If B is a biternary ring, we call A , the B AH-plane we constructed in Construction 3. IS, the AH-plane generated by B. Remark . Appendix. A can be read at this point if desired. However, nothing treated there is used in any of the proofs in the sections which follow.

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4. SEMITRANSLATICNS AND GEOMETRY In this section we define the concepts of semitranslation, j-angle, (P,g w )-j-Desarguesian, ( P,g w )-endomorphism, (F, A) -mime tic, translation and ( P, g^-transitive and prove that an AH-plane A is ^jgÂ«,i-H-Desarguesian if and only if the ser.itranslations of A are (PÂ»9 fc )-transitive as well as various other results. 4.1 Definitions . A homomorphism
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70 shown to be equivalent to ours by using Prepositions 4.5 and 4.7 (4) below. 4.3 Definitions . Let A be an AH-plane. An endonorphism
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71 4.5 Definition . V.'e say that an endomorphism
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72 If P ~ tp, then in the above argument L(F,PR) ~ L(*P,PR)-, and hence R ~
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73 is' not the identity, then all the directions of and (Q^J^J are ( T , g>> )related if they are T-relsted and * i(i+1) Â« k i(i+1) for each i, 1 Â•Â£ i < j and m i(i + l) * P f Â° r each i ' 1 Â£ i < J* We sa >' A is (rSgoJ-jPesarouecian , if whenever two j-angles (P.;m .) and l ef (Q i ;k ef } are (P >9jr elated, then Q I LCC^m^). We say that A is (r i ,g,)H-Desarauesiar. , if A is ( P,gÂ„)j-Desarguesian for every j Â£ 2. We will sometimes put a dash in place of m in gh J = ^ P i' P 2' " * ' F j ;n l2' " * * ' m il* if m ah Can be any appropriate line between ? and P h ; for example, the direction may be specified by some requirement on J. We sometimes write a direction Â£ in place of ra to indicate that m , = L(P .1) . gh gh 9 4.11 Prop osition . If an AH-plane A is (P,gJ-j-Desarguesian,

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74 j > 2, then A is (V,g _)-( j-1 )-Desarguesian. Proof . Let K = (P ..., P j-l ;rn l2'"* ,rn (j-l)l ) be a tj-D-angle. Then K. . 1^^ ^i^,. ) is a j-angle. j-1' 12'" 12'*"'"'(j-l)l Using this expansion, we see that the proposition follows easily. // 4.12 Definition . The expansion given above will be called the canonical expansion of a (j-l) -angle to a j-angle. 4.13 Definitions . An AH-plane A is said to be (P, 2.) -mime tic if whenever g, g Â« ,h,h Â« ,k,k ' are lines of A such that g,g' are in P; h 11 hÂ»; k,k' are in Â£ and 1 k A h A g | /Â„ , |k A h A g Â• 1 4 0, IkA h'ngl/O, then IkA h' A g'l 4 0. See Figure 4.1. We say A is ( P , g^ ) -mimetic if it is ( P,2 ) -mimetic for every direction 1. Fiaure 4.1.

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75 4.14 Propcsitlon . Let A be an AH-plane. Then A is (P, geo )-2Desarguesian if and only if it is (P,l)-mimetic for every Â£ 4F . Â£Â£Â£Â£| Â« <=>) Look at the biangles (g n k,g' A k;k,h) and (g n k Â• , g Â• k Â• ; k ' , k Â• ) . (Â«=) Let (P Â± ;m ef ) and (Q^k^) be (r.g. )-related biangles such that m 12 * 2 and i * T. Then, by (r,l)m i m i cry there is a point R I S 2 Â» k 12 .MQ 1? m 21 ). Hence R = Q^ Thus Q 2 I LCO^.m^). and A is (P,g 4<> )-2-Desarguesian. // 4.15 Proposition . Let A be an AH-plane and let PC H> L(P,P) for some points K,P and some direction P. Then there is a unique line KP joining K and P, and KP + P. Proof. Observe that K * L(P,P) implies K + p; hence K and P are joined by exactly one line. If KP ~ T, then k(KP) & k(D; hence K(KP) = K(L(?,D), a contradiction. Thus KP * T. // 4.16 Proposition . Let A be an AH-plane. If A is (P,l)-mimetic for some P, P <+Â• 1 , then every semitranslation with direction Â£ is a translation; that is, every (1 , g#o ) -automorphism is a translation. Proof. The identity map is a translation. Assume A is (P,Â±)mimetic for some P, P 4 g. , and let
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76 We wish to show that h' is a trace of
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77 L(K',k) = Jc' and cr(H) I g 2. By Proposition 4.18, A is (PÂ»9oJand (.i.,gÂ„) -mime tic. Let cr be a semitranslation with direction TT. Then A is (A, ID-mimetic for some /\ + IT. Thus, by Proposition 4.15, 0 )-4Desarguesian, then the semitranslations of A are (r,g (>0 )-transitive. If the order of A is greater than two and if A is (P,gÂ„)-3Desarguesian, then the semitranslations of A are (P,gÂ») -transitive. If the semitranslations of A are (P,gÂ») -transitive, then A is (n Â»gÂ«, ) H D esarguesian; that is, A is ( P,g eo )-j-Desarguesian for

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78 every integer j greater than or equal to two. Proof. Assume that the semitranslations of A are (T.g^J-transitive, that (F ;m ) and (Q ;k ) are any two (P ,g^)-related jangles, and that r is a (Pjg^-semitrar.slation taking P to Q . Thena-m^ = LCQ^m^) = k^; L(K,D and define T .by using the construction above with S, T (S) in place of K, K'. We wish to show that f (P) = T (?) for every ooint P in D H D . Let * Â•> K S P ^ Â°K A D S* If p = K or P = S, the equality is immediate. If P / S,K; look at Figure 4.2. Let k 6 K'T (P). We have that KS -f P, SP * P. Using the fact that A is (P.g^-O-Desargueslan,

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79 fÂ„ (K) . K' K P = K T K (S) = Q 2 S = P. Figure 4.2. we have that T (P) I L(K',PK). Hence T and T agree on S K , S D A D . Define L(Q,T) . Thus T US Q coincides with oon all of D . If Q + L(S,r), then Q L(K, ) and the above argument with the roles of K and S interchanged shows that T coincides with Â«" on D . 4 v_<

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so s-P Figure 4.3 . Now assume that A has order greater than two and that A is at least (r , ,g i>0 )-3-De3arguesian. This is the remaining case. TÂ„ (r Pick a point W such that W + L(K, D ,L(S,D . Then T and W w and T g ) agree on D, ; (\ D ?; (D {j (\ D^ by the argument for T K and V And hence Â°" P = T W (P) on D w Thu s,
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81 Let h be a line of A, and let G be a given point on h. Assume h + V. Then T_(H) =
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82 Thus
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83 Proof. Let g 6 T and Pig. Let h Â€ P such that o-F,-rcrP I h; such a line exists since T e D-. Since N g 6 r, g't Â£ and g" Â£ A such that F I g,g',g" for some point P. Then P + X P by Proposition 4.7 (2). Thus, g" = P(X?) and g" >,L g,g'. We wish to show that crX has no traces neighbor to either g or g*. By our hypotheses, cr X e D for some direction A.. If A~r,i,then h = L(?,Jl) is a trace of crX. Since Xcr =

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84 h,g', we have that o"P /X(irP), a contradiction to X + 1 by Proposition 4.7 (2). Hence, XL -/i -1 ,! and (\
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85 P which is not neighbor to h. Let R,S be points on k; R,S /. P. Let Â£ be a direction net neighbor to h,k. Let R' = L(R,Â£) A h; S' = L(S,1) H h; then since R,S + ?; and 1 + h,k, we have that R',S' -Y P. Let obe a ( P,g )-endomorphism which takes R' to S'. Then cr takes L(R,Â£) to L(s,l) and hence R to S. By repeating the argument, we can see that the endomorohisms of A are (P.q )transitive. Assume g is affine and that P = TT where TV Vg. Let A,l be directions; A ,1 4TT ,TT(g) . Let Q I g; Q + h. Let R' = L(G,M n h; S' = L(G,1) h; then, R',S' + g. Let cr be a (TT,g)endomorphism taking R' to S*. Then g. Let Gig; RÂ« = GR A h; S* = GS A h; then, R*,S' 4g. Let
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5. SEMITRANSLATIONS AND ALGEBRA In this section we relate properties of the biternary ring of a coordinatized AH-plane to the existence of certain semitranslations. We also show that the category of regular biternary rings is equivalent to the category of coordinatized translation AH-planes. 5.1 Definitions . Let (M,T,T") be a biternary ring. We define the Tadditi on + :M* M Â— * M by a + b = T(a,l,b). We define the Tmultiplication n:Hx M Â— K by axb = T(a,b,0). The T-multiplication is usually denoted by juxtaposition; that is, we usually write ab instead of axb. We define the T'addition Â»:MxM Â— Â»M by a Â• b = T' (a,l,b). We define the T 'multiplication Â• :Mx M Â— * M by a-b = T'(a,b,0). We say T is linear if T(a,m,b) = (am) + b for all a,m,b eM. We usually assume that multiplications occur before additions and write (am) + b as am + b, and write (a. in) * b as a.m * b. Similarly T" is said to be linear if T'(a,m,b) = a-m Â• b for all a,m,b 6M. Observe that the definitions of the Tand T'-additions are dual; as are the definitions of the Tand T '-multiplications. 5.2 Proposition (Compare Lorimer [(1971) , page 166, Theorem (6.1.2)1). Let (K,T,TM be a biternary ring with T-additon +. 86

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87 Then (K, + ) is a loop; that is, it is a binary algebra with a unit such that any equation x + a = y can be solved uniquely for any one of x, a, y once the other two are specified. Remark . If a loop is associative it is a group. Proof. By (B2), + a = a, and by (B3), a + = a for all a in M; hence is a unit of (:!, + ). Given x, a in M, x + a is welldefined. Given x,y in M, by (B4) there is a unique a such that x + a = y. Given a, y in M, by (B5) there is a unique x such that x + a = T(x,0,y) and by (32) T(x,0,y) = y. If z + a = y, then T(z,0,y) = y by (32); so that z = x, and the solution is unique. Hence (K, + ) is a loop. // 5.3 Definition . Let (K,T,TÂ») be a biternary ring whose T-addition (M,+) is a group. We will denote the T-additive inverse of an element b in M by b and write a b in place of a + (b). Also we will write a + b + c to mean a + (b + c). 5.4 Definitions . Let B = (M,T,T') be a biternary ring. If a, m, c are elements of M, then by condition (34) there is a unique z in K such that T(a,m,z) = c. Define a function Z:M 3 Â— * M by letting Z(a,m,c) = z whenever T(a,m,z) = c. Let Z' be the dual function whose definition has T 1 in place of T above. 5.5 Proposition. Let (M,T,TÂ») be a biternary ring, and let s be an element of H. If s * 0, then L :M Â— * M defined by L (m) =

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88 T(s,m,0) is a bijection which preserves and reflects the neighbor relation. Proof. Let t,s 6 M, s f 0. Since s Â± 0, there is an m such that t = T(s,m,0), = T(0,m,0) by (B6). Hence L is surjective. Assume L (z) = L (w) , z Â£ w. Let t = L (z). Then (z,0), s s s ' ' (w,0) are both solutions to T(s,m,d) = t, T(0,m,d) = ; so that t ~ 0, s ~ 0; a contradiction. Thus L is biiective. s If m ~ n, then sm ~ sn by (BIO); hence L preserves the s neighbor relation. If sm ~ sn, and if m /n , then by (BIO) s Â— 0, a contradiction. Hence L reflects the neighbor relation. // 5.6 Proposition . Let (M,T,TÂ») be a biternary ring, and let s 6 M. If s f 0, then the map R :M Â— M defined by R (m) = s s T(m,s,C) is a bijection which preserves and reflects the neighbor relation. Proof. Assume s ^ . Let k e M. There is a unique x such that T(x,s,0) = T(x,0,k) = k by (B5). Hence R is surjective. Assume R s (m) = R s (n). Let y = T(m,s,0) = T(n,s,0). Observe that y = T(m,0,y) = T(n,0,y).. Thus by ( B6 ) , m ~ n, and by.(B7) either m = n or s ~ 0. Thus, m = n and R is injective. By (B10), R preserves the neighbor relation. If R (m) ~ R (n), m f n, then by (B10), s ~ 0; hence R reflects the neighbor relation. //

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89 Remark . The following proposition closely resembles the first result that I proved in connection with ccordinatization. 5.7 Proposition . Let (A,K) be a coordinatized AH-plane. (Part I) If ois a dilatation on A, then the action of
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90 following are equivalent. a) <* induces a translation .on A. b) Conditions (D), (DÂ«), (T) and (T-) are satisfied. c) Conditions (D), (DÂ») , (T) and (?Â») are satisfied. Proof. (Part I) Assume
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91 *fu,v]Â« where u Â£ N to be [ u, Z ' ( f ( ) ,u, f ( v)] ' , then < preserves incidence and Â« g I! g for all lines g of A. We wish to show that el preserves the neighbor relation. Since Â«< preserves the neighbor relation for parallel classes, it suffices to show that * preserves the neighbor relation with respect to points. Ler (x,y) ~ (x',y'), and let h and k be distinct lines joining (x,y) and (x',y*). Then Â« h -ft' tik and hence
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92 5.S Corollary . If f ' is bijective, then condition (D) of Proposition 5.7 is equivalent to: (D) # f'(T(x,m,(f)" 1 (T(f(0),m,t))) = T(f(x),r.,t) for all x,m,t in K. So, by xy-duality, if f is bijective, then (D') is equivalent to: (D') f (T 1 (y,u,f~ 1 (T'(f '[0),u,s))l = T'(f (y),u,s) for all y,u,s in h. Similarly, if f is bijective, then (D") is equivalent to: (DÂ») f(T'(y,u,f -1 (T'(f '(C), u,s))) = T Â• ( f Â• ( y) , u , s) for all y,s in M and all u in N. Proof. Assume f is bijective. Then t = Z(f ( ) ,rr,f Â• (b) ) if and only if T(f (0) ,m,t) = f'(b), and if and only if b = (f) (T(f (0) ,m,t) ) for any m in K. Thus, it is easily seen that (D) and (D) are equivalent, and by xy-duality it is easily seen that if f is bijective, then (U') and (DÂ») are equivalent. The proof that if f is bijective, then (D") is equivalent to (D") 1 ' is similar to the dual to the first part of the proof given above. // 5.9 Definition . Let (A,K) be a coordinatized AH---lane. We denote the direction containing [m,0] by (m) , and, we denote the direction containing [u,o]' by (u)' for every m,u in M. 5.10 Definition. Let (M,T,T') be a biternary ring and let s and k be elements of M. We write T(s,x,0) as sx. We say (M,T,T') is (k)recular for s whenever (M,T,TÂ») satisfies the followinq

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93 three conditions for s and k, and for all x,m,b,y,u,v in M. 1) T(s,k,T(x,~,b)) = T(T'(T(s,k,x),l,Z'(sk,l,s)),m,Z(s,m,T(s,k,bÂ») 2) T'(T(s,k,T'(y,u,v)),l,Z'(sk,l,s)) = T Â• ( T ( s , k , y ) , u , Z ' ( Â£ k , u , T Â• ( T ( s , k , v ) , 1 , Z Â• ( s k , 1 , s ) ) ) J . 3) T(T'(T(s,k,x),i,Z'(sk,l,s)),k,d) = T(s,k,T(x,k,d)). We say (K,T,T') is (k)regular if it is (k)-regular for all s in M. We define (k) 'regular for g and (k) 'reoular dually. We say (H,T,T') is axially regular whenever it is both (0)and (0) '-regular. We say (M,T,T') is regular if it is' both (k>and (k) '-regular for all k in M. 5.11 Theorem . Let (A,K) be a coordinatized AH-plane with biternary ring (M,T,T'), and let s,k be elements of Z. Then A has a semitranslation with direction (k) taking (0,0) tc (s,sk) if and only if (Z,T,T") is (k)-regular for s. If tr is a se-.itranslation with direction (k) which takes (0,0) to (s,sk), then
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94 f (T(x,k,d) ) = T(f(x),k,d). With x = 0, this becomes f'(d) = T(s,k,d) for any d in M. with d = 0, this becomes f (0) = t = sk. From condition (D) of Proposition 5 . 7 we obtain: T(s,k,T(x,:r:,b)) = T (f (x) ,m, Z( s ,m,T( s,k ,b) ) ) for all x ,T.,b in K. And from condition (D-) we obtain: f(T'(y,u,v)) = T*(T(s,k,y),u,Z'(sk,u,f (v))) for all y,u,v in M. Letting v = 0, u = 1, we have: f(y) = T Â• (T( s,k,y) , 1 ,Z Â• ( s k, 1 , s ) ) for all y in M. Putting this result in the equations we cot from conditions (D) and (D') we get conditions 1) and 2) above. Given any x,d in M, there is a y in M such that (x,y) I [k,dl. Then since [k,d] is a trace, f'(T(x,k,d)) = T(f(x),k,d). Thus condition 3) holds for ail x,d in M. (Â«=) Assume that (M,T,TÂ») is (k)-regular for s. Define a map
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95 Hence, by Proposition 5.7, gÂ»'H Dss a-5uesian by Theorem 4.21.

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96 5.15 Corollary. Let A be an AH-plane. The foil owinq are equivalent. 1) A is regular. 2) For some ccordinatizaticn K, the biternary ring of (A,K) is regular. 3) For every cc: rdinatization K, the biternary ring of (A, JO is regular. 4) A is isomorphic to the AH-plane AÂ„ generated by a regular biternary ring B. Proof . 11 4*> 2) <Â£\$ 3) Obvious. 2) =^7 4) Assume the biternary ring B of (A,K) is regular. Then (A,K) is isomorphic to C(3) by Proposition 3.26; hence A is isomorphic to the AH-plane A of C(B). / S ~ 4) =^2) Assume that 0:A_ Â— * A is an isomorphism from B the AH-plane generated by the regular biternary ring (B,T,T') to A. Let K B = ([0,0l,U0,03',(l,l),B,C ) where % is defined by 5 (a, a) = a fcr all points (a, a) on (0,0)v (1,1). Observe that (A D ,K ) = C(B) and that E(A p ,K) = (B,T,TÂ«). Let K = (etO.Ol.SCo.Ol'jOd.D.B^gG" 1 ). Then B(A,K) = (B.T.T'J. Hence A is regular. // Remark . Recall that we have defined a%b = ab = T(a,b,0); a-b = T'(a,b,0); a + b = T(a,l,b); and a Â» b = T'(a,l,b) for all a,b in M. 5.16 Corollary . A coordinatized AH-plane (A,K) has a semitrans-

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97 lation with direction (I) taking (0,0) to (s,s) if and only if the biternary ring 3(C) = (K,T,T') satisfies the following three conditions (here C = (A,K)): 1) s + T(x,m,b) = T(s + x,m,Z(s,m,s + b)) for all x,x,b in M. 2) s + T'(y,u,v) = T'(s + y,u,Z'(s,u,s + v) ) for all y,u,v in M. 3) (s+x)+d=s+(x+d) for all x,d in M. If
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98 condition: 1)" T(x,m,T(s,m,w) ) = T(x * s,rn,w) for ail x,m,w in M. Letting m = 1, w = in 1)", we get a) above. If condition a) holds it is easily shown that condition 1)" is equivalent to condition b) and that condition 2)' is equivalent to c). // 5.18 Corollary . The semitranslations of a coordinatized AK-plane (A,K) with biternary ring B(A,K) = (M,T,T') are ( (0 ) ,g rt )-transitive if and only if the following three conditions hold for all s,x,m,b in K: a) x+s=x*s. b) T(x,m,T(s,m,b)) = T ( x + s,m,b). c) T' is linear; that is, T"(x,m,b) = x-m * b. d ) ( M , + ) is a group:. Proof . (=Â») Condition c) of Corollary 5.17 with m = 1 implies the T'-addition is associative; with b = it implies that T* is linear; that is T'(x,n,s) = (x-m) * s. Since the Tand T 1 additions are equal by Corollary 5.17 condition a), by Proposition 5.2, (M,+) is a group. (4=) By c) and d) above, we have that T*(x,m,b) + s = ((xÂ«m) + b) + s = x-m + (b + s) = T"(x,m,b + s) for all s,x,m,b in M. Hence conditions a), b) and c) of Corollary 5.17 are satisfied for every s in H. Let (a,b) and (d,b) be any points on a line [0,b], Then for some s in M, T(a,l,s) = d and hence there is a "semitranslation Â«r taking (a,b) to (a + s,b) = (d,b); hence the semitranslations of A are ( (0 ) ^^-transitive. //

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99 5.19 Corollary . Let (A,K) be a coordinatized AH-plane with biternary ring (M,T,TÂ»). The semitranslations of A are ((0),g )and ( (0) 'jg^J-transitive if and only if the following four conditions are satisfied: 1) T and T" are linear; that is, T(x,m,b) = xm + b and T'(x,m,b) = x m * b for all x,m,b in M. 2) The Tand T'-additions are equivalent; that is, x + a = x * a for all x,a in M. 3) CM,+) is a grouc . 4) xm + sm = (x + s)m and x.m + s-ra = (x + s).m for ail x,s,m in M. Proof. (-Â£) Assume the semitranslations of A are ((0),gÂ„)and ( ( Â°J ',g w )-transitive. Then, by Corollary 5.18 and its xy-dual, the first three conditions above follow immediately. The remaining condition follows from condition 5.13 b) with b = and' its xy-dual. / (<=) Assume that conditons 1), 2), 3) and 4 3 hold in the biternary ring (H,T,TÂ«). Then apply conditions 1), 3), 4) and 1) to. get: T (x,m,T( s ,m, b) ) = xm + ( sm + b) = (xm + sm) + b = ((x + s)m) + b = T(x + s,m,b) for all x,s,m,b in M. Then, by Corollary 5.18, we havethat the semitranslations of A are ( (0) ,gÂ„)-tra .sitive, and by an xy-dual argument, they are also f(0) l t g_)-transitive. . // 5.20 Corollary . Let (A,K) be an axially regular coordinatized AH-plane. Then, any semitranslation of A is a translation, and

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100 if we let U = i or 1 v and T are translations with directions (0) and (0)' respectively!, then U = \
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101 5.24 Definition . Let A be an AK-plane. We say that A is a translation AK-olane if A has a group of translations which is transitive on the points of A. 5.25 Theorem . Let A be an AH-plane. The following are equivalent. 1) A is a translation AH-plane. 2) A is regular; that is, the semitranslaticns of A are (r,g^)-transitive for every direction V, 3) A is T-regular for three pairwise non-neighbor directions r. 4) For some cocrdinatization K, the biternary ring (!Â•'., T,T') of (A,F) is axially regular and (K,+) is abelian (see Corollary 5.23 for a characterization of axially regular). 5) The set of translations of A forms an abelian group which acts transitively on the points of A. 6) A is regular in two non-neighbor directions and the set of semitranslaticns of A is closed under composition. 7) A is regular in two non-neighbor directions and the set of translations of A is closed under composition. 8) For every coordinatization K of A, the biternary ring (M,T,T') of (A,K) is regular (hence, axially regular) and (M,+) is abelian; the set of translations W of A is an abelian group under composition and W = {
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102 Proof . 1) =Â£ 2). Assume A is a translation AH-plane. Let V be a direction; let g e V and let P,P' I g. There is a translation
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103 4) =Â£ 5). Assume that for some coordinatization K, the biternary ring (M,T,T I ) of (A,K) is axially regular and that (K,+) is abelian. Then Z(x,m,c) = -xm + c and Z'(y,u,w) = -y-u + w for all x,m,c,y,u,w in M. Using these two equalities, we can easily show that conditions 1), 2) and 3) of the definition of "(k)-regular for s" are satisfied for every k,s in M. Thus, every cr defined by (a,b) = (a + s,b + sk) for all points (a,b) in A and for some s,k in M is a ( (k ) ,g e0 )-semitranslation. By Corollary 5.12, these semitranslations are ( (k ) ,g oo )-transitive. By Proposition 4.19, each such cr is a translation. By xy-duality, each cr ' defined by
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104 1) and 2) => 7). Obvious. / 1) and 2) => S). Assume A is a regular translation AK-plane. Then A is axially regular, and the set of translations is an abelian group under composition. Let K be any coordinatization of A. If (;-'., T, 7') is the biternary ring of (A,K), then (MjT.T 1 ) is regular by Corollary 5.15. The remainder follows by Corollary 5.20. / S) => 4) . Obvious. / 9) <=> 2). This is immediate by Theorem 4.21. // 5.26 Definitions . Let C denote the category of coordinatized translation AH-pla.nes and let E denote the cateqory of rccular ~~ r * biternary rinqs. Let 3 :C m Â— * B and C :B Â— -Â»Â• C b'=> the i 2Â— ~r T r ~T r T restrictions of 3 and C to C m and B respectively. T r 5.27 Corollary . The maps B :C Â— * B and C :B Â— * C defined above are reciprocal equivalences such that B C = 1. // ~r~T ~B 5.28 Corollary . A biternary ring (M,T,T') is regular if and only if (M,+) is abelian and (M,T,T') is axially regular; that is, (K,+)"is abelian and (M,T,T') satisfies the four conditions of Corollary 5.23. // 5.29 Theorem . The translations of a coordinatized AH-plane C are ((0),g M )and ( (0) Â• ,g w ) -transitive if and only if the biternary ring (M,T,TÂ«) of C satisfies the following conditions: 1) T and T' are linear.

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105 2) The Tand T '-additions are equivalent; that is, a + b = a * b for all a,b in M. 3) (M,+) is a group. 4) xm+sm=(x+s)m and x-m + s-m = (x + s)-m for all x,s,m in M. Proof . This follows easily from Proposition 4.19 and Corollaries 5.12 and 5.23. // 5.30 Proposition. The maps B *:C * Â— Â» B Â• and C *:B* Â— C Â• i ^ ~r T r ~T r T defined by restricting B and C to the categories of coordinatized translation affine planes C * and regular biternary fields B * T r respectively are reciprocal equivalences. //

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6. PREQUASIRINGS AND CUASIRINGS In this section we define prequasiring , quasiring, and ccordinatizc-d translation AH-plane and prove that the category of quasirings is equivalent to the category of coordinatized translation AH-planes. 6.1 Construction of E:B 2 Â— > V. Let 5 = (M,T,T") be an axially regular biterr.ary ring, and let + , * , Â• be defined as usual; that is, a + b = T(a,i,b); axb = ab = T(a,b,0); a-b = T'(a,b,0) for all a,b in K. Then we denote (M,+,X,*) by E(B). Let w:B Â—* B' be s morphism in B , and define E(oo):E(B) Â— E(B') by E(u>)m = wm. 6.2 Definition . Let M be a set with distinguished elements and 1 , ^ 1 , on which two binary operations Â© , Â® are defined. We say that (M,Â©,Â®) is a quasifield if the following conditions are satisfied: (QF1) (M,Â©) is an abeiian group with identity 0. (QF2) (MM03,Â®) is a loop with identity 1; 0Â®x = xÂ®0 = V x. (GF3) (a Â© b)Â®m = (aÂ®m) Â© (bÂ®m) for all a f b,m from M. (CF4) For all r,s,t from M such that r ^ s, the equation zÂ®r = (zÂ®s) Â© t has a unique solution z. We say that is the zero and 1, the one of (J",Â©,Â®). 106

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107 6.3 Definitions . Let M be a set with two distinguished elements 0, 1, and let three binary operations +, /, Â• be defined on M. We will write ab in place cf axb. The operation + is said to be the addition of M; X is said to be the first multiplication of M, and Â• the second multirlication of M. Let N = {n Â« H| 3 k Â« M, k 4 0, 3 kn Â» 0] and let N= {n fe K \ 3 k e M, k 4 0, -* k-n = ol. Define a ~ b if and only if every x such that a = x + b is in N and define a ~' b if and only if every x such that a = x + b is in N*. We say (M,+,X,Â«) is a precuasirlng if end only if the following twelve conditions are satisfied: (VWO) N = N';_ a b <-=\$> a ~* b for all a,b in M. (VW1) (M,+) is a group with identity 0, and (N, + ) is a subgroup of (M,+). (VW2) 0m = mO = for all m in M. (VW3) I'm = ml = m for all m in M. (VW4) xm + sm = (x + s)m for all x,s,m in M. (VW5) xm = xm' + d is uniquely solvable for x if and only if ( m m ) ^ N . (VW6) The equation am = b with a Â± N is uniquely solvable for m. (VVJ7) if a Â€ N, b e N with a,b not both zero, then one and only one of the equations (am = b, b-u = a (with u 6 N)] is solvable for m Â€ M, u e M. If am = b, then there is an m' Â£ m such that am' = b. If. b-u = a , u <= N, then there is a u Â• & N, u' ^ u, such that b-u' = a. (VW8) The system (y = xm + d, x = y.u + v where u 'e n3 is uniquely solvable for the pair x,y.

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108 (VV, T 9). For any m,u Â€ H, up = 1 if and only if m.u = 1. if urn = 1, if am. + e = b and if b.u + v = a; then (xm + e = y y-u + v = x ) for all u,m, a,e,b, v,x,y in M. (WHO) The binary operations +, x , . in M respectively induce binary operations Â©, Â®, Â© in K/Â„ f and if 0* and 1* are the images of and 1 respectively, then (KA, ,Â©,Â©) and (M/Â„ ,Â©,Â©) are quasifields with zero 0* and one 1*. (VW11) Conditions (Vv/0) through (V'-ilO) with x replaced by . and by x hold in (M,+,X,.); these conditions are called (VWO)' through (VW10) ' . Each element of N is called a right zero divisor . The element is called the zero and 1 is called the one of (M + A .). 6.4 Proposition. Let (M,+,X,.) be a prequasiring. Then a ~ b if and only if (a b) e M. // 6.5 Definition. We say a prequa siring (K,+,x,0 is a cuasirinc if and only if (M,+) is abelian. 6.6 Definition. If (M,+,x,-) is a prequasiring, then (H,+,-,x) is a prequasiring; (M, +,Â«,*) is said to be the dual of (M,+,X,-) 6.7 Definitions . If we replace conditions (VW9) and (VW9) * in the definition of prequasiring by: (VW9s) If m,u are elements of M such that either urn = 1 or m.u = 1, then (xm). u = x and (y-u)m = y for all x,y in M. then we say that the new structure (M,+,x,.) thus defined is a

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109 skew quasirinq. If (K,+,x f .) is a skew quo siring, then <;Â•:, + ,-, x) is also a skew quasiring and it is called the dual of (M,+ x .). Remark. Condition (VW9s) is self-dual; that is, if we interchange the X and the in (Vâ€¢9s), the result is equivalent to (VW9s). 6.8 Proposition. Let (M,+,X,.) be a prequasiring or a skew quasiring; then -(ya) = (-y)a and dually -Cy-a) = (-y).a. Proof. Observe that ya + (-y)a = (y + (-y ) ) a = 0. // Remark. Letting y = 1 , a = -1 we observe that in a prequasiring or skew quasiring (-D(-l) = 1 and dually (-l)-(-l) = 1. 6.9 Proposition. The definitions of prequasiring and skew quasiring are equivalent. Proof. 1) Assume V = (M, +,*;Â•) is a prequasiring. If urn = 1 , then since 0m + = and -u + = , we have by (VW9) with e = 0, v = that xm = y if and only if y. u = x. If urn = 1, then xm = (xra) and hence (xm)-u = x for all x in H; similarly, (y.u)ra = y for all y in M. By duality, the same results hold if m-u = 1. Thus (VW9s) holds in V and V is a skew quasiring. / 2) Assume V = (M,+,X,.) is a skew quasiring. If um = 1, then (lm).u = 1 by (VW9s)j so that m-u = 1. Dually if m-u = 1, then um = 1. Assume um = 1, am + e = b and b. u + v = a f or some

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110 a,b,e,v in M. By (VW9s), (am)-u = a = (b e).u = b-u e-u. Thus v = -e-u. Similarly e = -vm. If xm + e = y for some x,y in M, then xm vm = y = (x v)m = y. Hence x v = y.u and y-u + v = x. 3y duality if y-u + v = x, then xm + e = y. Thus (VV/9) holds in V. 3y duality (VW9)' holds in V and V is a prequasiring. // 6.10 Proposition . In a prequasiring if m Â«Â£ N then there is a u Â£ N such that urn = 1; as one can see by applying' (VW5) to the equation xm = xO + 1. Dually, if u Â£ N, there is an m \$ N such that m.u = 1. // 6.11 Definition. Let (V l+f *,.), ( V Â• , + Â• , x Â• , .Â• ) be prequasirings. We say ^:V Â— V is a pre qua siring homomorphism whenever u>(0) = io(l) = l, M (N V )
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Ill the conditions of Corollary 5.23, T(x,rr.,b) = X m + b, T(y,u,v) = y u + v, a + b = a * b, (M,+) is a group, xm + sm = (x + s)m and y u + s u = (y + s) u for all x,m,b,y,u,v in K. We wish to use these properties and the definition of biternary ring to show that E(5) is a precuasir inq. Observe that r, ! = M as sÂ°ts* hence (BO) implies (VWO). The element is the identity of the group (M, + ). Let N = N E(B) Observe that a b in 3 (a b) is in N in E(B).. Let a,b e N in E(B). Then a ~ 0, b ~ and hence by (Bl) a ~ b in B. Thus (a b) e N and ( N', + ) is a subgroup of (F., + ), and (VWl) holds. With d = 0, ( E2 ) implies (VW2) . (B3) implies (VW3). We know that (VW4) holds by the conditions on E(B) given above. ( B5 ) implies (VW5). With a' = 0, b* = (B6) implies (VWG). With a' = 0, b' = 0, (B7) implies (VW7). (B8) implies (VW8). (B9) implies (VW9). (B10) implies that (M/~,T*) is a ternary field. In fact if we let B* = (M/~ ,TÂ» ,T' * ) , we can see by the constructions that B* is isomorphic to B(kC(B)) = (Q,S,S') the biternary ring of the coordinatized affine plane (C(E))Â» associated with C(B). Since the neighbor map is surjective, (C 1 ,g w )-translations in C(B) get taken by K to (P Â• ,g,Â„) -translations in (C(B))Â». Thus the translations of (C(B))Â» are ((C), g Jand ( (0 ) Â• ^-transitive and by Corollary 5.20 and the remarks following it, (C(B))* has a transitive group of translations. Kence by another well-known theorem [Hall (1959), page 362, Theorem 20.4.61, the algebraic system which is associated with (C(B))* and which is naturally isomorphic to (M/~ ,Â©,Â®) is a cuasifield. Hence (VW10*) is satisfied. Since (Ell) implies (VW11J, E(I^) is a prequasiring.

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112 Let w: D be a homomorphi; wnere a = (K,T,TM and D = (Q,R,R'). Then, u>(T(a,m,b)) = R(coa,oÂ»m,u>b) and w(T' (a,m,b) ) = R' (wa,:V Â— Â» w be a prequa siring homomorphism. Define EÂ»(io) by (E'(Â«o))m = u>m for all m in K. 6.15 Proposition . The map E':V Â— B defined above is a functor from the category of prequasirings to the category of regular biternary rings. Moreover E*E and ES' are identity functors. If V is a prequa siring, then N = N , as sets. Proof. Let (V,+,x,-) be a prequasiring . We wish to show that E'(V5 is an axially regular biternary ring. Let N be the set of right zero divisors of V. Then N = (n fe v|3 k, 'k / 0, i T(k,n,0) = C] = N E , (V) and similarly N' y = N' Pf( } . Let V

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113 Observe that a ~ b if and only if a b in N; hence, a ~ b if and only if the unique x which satisfies the equation a = T(x,l,b) is in N. Thus a v b if and only if every x which satisfies the equation a = T(x,l,b) is in N. Similarly, a ~ b if and only if every x which satisfies the equation a = T*(x,l,b) is in N. (VWO) implies (30). (VW ) implies (Bl) as can easily be checked. (VW1) and ( VW2 ) imply (D2). (VW3) implies (53). (VWl) implies (E4). (VW5) and (VWl) imply (E5). (VW6), (VWl) and (V./4) imply (B6). (VW7), (VWl) and (VW4) imply (E7). (VW3) implies (S3). (VW9) implies (B9). (VWlC) implies (BIO), and (VW11) implies (311). Hence E*(V) is a biternary ring. By Corollary 5.2 3, E'(V) is (0)and ( ) '-regular . It is easily seen that if u>:V Â— > W is a precua siring homomorphism, then E'(w) is a biternary ring hemomorphism, that S' (1 V ) = 1 E'(V) and that E Â• C are reciprocal equivalence such that W = l r -. //

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114 6.18 Corollary . The r^aps Q:C Â— Â» Q and QÂ»:Q Â— * c defined by restricting V and V' to C T and Q respectively are reciprocal equivalences such that CC ' = 1 where c" is the cateqory of ** *Â• Q T 3 coordinatized translation AH-planes and Q is the category of qua siring s . Proof. If (A,?.) is a coordinatized translation AH-plane, then B(A,K) is axially regular and (M, + ) is abelian by Theorem 5.25. Hence Q(C) = EB(A,K) is a qua sir inc. If Q is a quasiring, then S'(Q) is axially regular and (M, + ) is abelian. Thus, by Theorem 5.25 since B(C(E'(Q))) = Â£'(Q), Q'(Q) = C(E'(Q)) is a coordinatized translation AH-plane. // 6.19 Definition . If V is a prequasiring, we call A the E ' ( V) AH-plane generated by V, and denote it by A . 6.20 Definitions . If Q is a quasiring such that N = {0~\ , we say that Q is a biquasif ield . We denote the cateqory of blquasif ields by C and the category of coordinatized transla tion affine planes by C *Â• 6.21 Propos ition. The maps Q*:C * Â— * Q* and Q'*:0* Â— * C Â• ' * 1 Â— Â— i * i -i . ^ T x. T defined by restricting. Q and Q' to C Â• and Q* respectively are reciprocal equivalences such that Q*Q'* = 1-.. Proof. Assume Q is a biquasif ield. Then N = {0~\, and hence

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115 E'(Q) is a biternary field. Thus, C = CE'(Q) is a coordinatized affine plane, and, since Q(Q) = CE'(Q), C is a coordinatized translation affine plane. Assume C is a coordinatized translation affine plane. Then B(C) is a regular biternary ring with N = lo]. Thus EB(C) is a quasiring with N eb(c) = {o"j. Hence Q(C) is a biquasif ield. //

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7. KERNELS CF QUASIRINGS In this section we define the kernels of a quasiring; we define the ring of trace preserving endorr.orphisms of the translation group of a translation Hjelnslev plane; and we prove a theorem relating these concepts. 7.1 Definition . Let (C,+,x,Â«) be a quasiring. Define 'Kern C as follows: Kern Q = {k 6 QJk(xm) = (kx)m, k(x-m) = (kx).m, kx = k.x, and k(x + m) = kx + km for all x,m (, q] . We say (Kern Q,+,X) is the kernel of the quasiring Q. 7.2 Definition . Let (R,+,X) bo a ring with 1^0. Then (R,+,K) is said to be a local ring if all the elements of R which do not have two-sided multiplicative inverses form an ideal of R. 7.3 Proposition . If (Q,+,X,0 is a quasiring, then the kernel of Q is a local ring and if we define N to be N (\ Kern Q, then N K consists of a11 th e elements of Kern Q which do net have twosided multiplicative inverses. If q 6 Kern Q, and if b e. Q, then q(-b) = -(qb). Proof Â• If s,b e Kern Q, then it is routine to verify using the 116

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117 axioms for a quasiring that ab and a + b are in 'Kern Q. Obviously and 1 are in Kern Q. 3y Proposition 6.8 if b,a Â€ Q then -(ba) = (-b)a and -(b-a) = (-b).a. Assurr.e b 6 Kern Q: we wish to show that -b Â€ Kern Q. Observe that (-b)(xm) = -Cb(xm)) = -((bx)ra) = (-(bx)m) = ( (-b)x)m; that (-b)x = -(bx) = -(b.x) = (-b).x; that (-b)(x + m) = -(b(x + m) ) = -(bx + bm) = (-bm) + (-bx) = (-b)x + (-b)rr. and dually so that -b 6 Kern Q. Since (Kern C,+,X) satisfies both distributive laws, is closed under + and x, has and 1, has associate multiplication and is an abelian group under addition, it is a ring. Define N to be N (\ Kern Q. Let q Â€ (Kern Q)\N . Then since q Â•/Â• 0, there is a unique q 1 6 Q such that qq ' = 1 by (VV. r G). Since 1 40, q' + 0. Thus there is a unique q" such that q*q" = 1. But q Â£ Kern Q so that q = q(q'q") = (qq')q" = q Â»; S o that q = q", and q is invortible in Q. We wish to show that q' is in Kern Q. Observe that q(q'(xm)) = (qq'Hxn) = X m = ((qc')x)m = (q(q'x))m = q((q'x)m). Since = q(b + (-b)) = qb + q(-b) ; q(-b) = -(qb). Thus = q(q'(xm)) q((q'x)m) = q[q'(xm) (q'x)m]. Hence since q ^ N, q'(xm) (q'x)m = 0; so that q'(xm) = (q'x)m and dually q'.(x.m) = (q'.x).m. Since q(-b) = -(qb), q(q'x q'.x) = q(q'x) q(q'.x) = x x = 0, and since q j N, q'x q'.x * and q*x = q'.x. Similarly q(q'(x + m) q'x q'm) = 0; so that q'(x + m) = q'x + q'm. Thus q' Â€ Kern Q. Observe that q e N implies that q does not have an inverse. Since (N,+) is a subgroup of (C,+), (N ,+) is a subgroup of Kern 0If q 6 Kern Q, n e N , then cn,n'q 6 KernQ,N and hence qn,nq e N ? . . Thus, N R is an ideal of Kern Q and N

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118 consists of all the elements of Kern Q which do not have twosided multiplicative inverses. Since 1/0, Kern Q is a local ring. // 7.4 Defi nitions . Let C be a coordinatized translation AH-plane with quasiring Q(C) = (Q,+,X,.). Denote the translation which takes (0,0) to (s,t) by T (g fc) . Then if (x,y) is a point of C, T (s t) (x ' y) = (x + s ' y + t) b y Th eorem 5.25. Let A be a translation AH-plane with translation group (W,o). we say an endomorphism J of W is trace preserving if whenever g is a trace of a translation
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119 Definition 7.4 and X is defined by \(a)T T 7 (t,f) (at, at') If u>:C Â— *Â• C is a surjective morphism in C m , define T K Q (us): K Q (C) Â— * K Q 2 (C) by K Q 2 Go) (q, (m,k ) ) = (Q(w)q, (Q(
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120 Remark . Once this result has been shown, we have as an immediate consequence that the module 'Kern Q acting left multiplicatively on (Q,+)' is a 'geometric invariant'; that is, it is independent (up to isomorphism) of the particular coordinatization of the AH-plane A. ' Remark . The fact that the set of trace preserving endomorphisms of the translation group of an AH-plane is a ring was shown by Klingenberg [(1955), page 103, S 13] and Luneburg T(1962), page 279, Satz 5.2 J. Proof. Let C = (A,K) be a coordinatized translation AH-plane. Let E W be the Â£et of trace Preserving endomorphisms of the translation group of C. Let two operations +,x be defined on E as W follows: for each 01, \r i E and for each T in W, let (a + lr)r = oi r oirT and let (oiL-)t = ot(Jb-T). We wish to show that E is closed W under each of these operations. If g is a trace of T, then g is a trace of o\T and of L-T and hence of
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121 Hence, since [0,0] is a trace of T. and [0,0]' is a trace of r (0,f) 5 0ir (t,t') = T (f (t),0)Â° r (0,g(t')) = T (f (t),g(f )) f Â° r some functions f,g from Q to Q where (Q,+,X,Â«) is the quasiring of C. Since [l,0] is a trace of T ., f = g. Since [t,0] is a trace of T ( ^ t) , Â«T (ljt) ^ (f(1)jf(t)) r ( f ( i ) ,f ( i ) t ) ; S Â° that f(t) = f(l)t, and xy-dually, f(t) = f(l)-t. Let a = f(l). Then flit. . = r, , ,, and a(xm) (ax)m; and xy-dually (x,xm) (ax, a (xm) J a-(x-m) = (a-x)-m. Since T. . = t. . o T. ,, a(x + y) = (x + y,0) (x,0) Cy,0) ax + ay. Since at = a-t for all t in G, a Â£ Kern Q. Conversely, if a & Kern Q, the map X(a):(W,o) Â— Â» (V.',o) defined by \(a)r /1 _ . . . = T. . , , . is a trace preserving endo1 ( t,t ' ) (at, at ' J moronism of (W, o) since X(a)f = X. . , r ' ( t, tm) (at, (at)m) Ma)(T. t , , S Â«T. ,J = X(a)T, . ., , , (t,f) (s,s') (t + s,t' + s') T (at + as, af + as') = ( X(a) T ( t,t Â• ) ) ' ( X(a) T (s, s ) > and ^ (a)T (t..u,t.) " r ((at.).u,atT CbserVe th3t X:Kern C E W is bijective. It is easily seen that X is a ring homomorphism and hence that X is a ring isomorphism. Since T : (Q, + ) * (C , + ) Â— C.;,o) is a group isomorphism and since (t,f) = r (at,at<)Â» ^cVi lC) ~* E~* isomorphism. Let r be a direction of C. Assure T = (m): if P m (u)', u 6 N, the argument' will be xy-dual to this. If W Â£ E and TÂ»j_ Â». ^ fe '^r.Â» then oir.. . = T. . . , > Â€ W . Kence the (t,tm) n' (t,tm) (at,(at)m) P module Â„(W_) is well defined. The map (WÂ„,o) is a l r qrouD isomorphism. Since X(a)
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122 then Q(w) is a quasiring homomorphism, and, since to is surjective, Q(Â«o)(Kern Q) C Kern C. Thus, K Q 2 (^) is easily seen to be a module homomorphism. Since y^ and rj are nodule isomorphisms, we have immediately that W(w) = n ( o 2 (to) )Â« " 2 is a module homomorphism. If co:C Â—C' is a morphism in (C ) S , then 2 1c^K~ ^ W ^ = E'l^^lr and hence ^ is a natural isomorphism. // 7.8 Proposition . Let (C ,1) be the category of coordinatized translation AH-planes each with a base direction and with surjective morphisms (which take base direction to base direction) The obvious maps K Q:(C T ,S) S Â— ^ and Â£ (w Â£ J : (C T ,tJ S Â— M are functors and the map P: Â— * F ('Â£'<) is a natural isomorphism. Proof. If co:(C,D Â— * (C',V) is a morphism in (C ,2) , define E (W Â£ )( W ) to be / ( C , j p. ) ( K Q ( w)(| (c pj" 1 )). The proposition then follows from Theorem 7.7. //

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8. OTHER CENTRAL AXIAL AUTOMORPHISMS In this section we give both geometric and algebraic characterizations of those coordinatized affine Hjelmslev planes whose automorphisms are ( (0,0) ,gj -transitive and those whose endomorphisms are strongly ( (0,0) ,g,.)-transitive. We give algebraic characterizations of those classes of coordinatized AHplanes whose automorphisms are respectively ( (0 ) ,[ , C] ' )and ( (0) Â•, [0,0]') -transitive. 8.1 Definition . Let (K,T,T') be a bi ternary ring and let s e H. We say (M,T,T') is left modular for s whenever (M,T,TÂ«) satisfies the following three conditions for all x,m,d in M. a) sx = s-x; that is, T(s,x,0) = TÂ»(s,x,0). b) s(T(x,m,d)) = T(sx,m,sd). c) s(TÂ»(x,m,d)) = T'(sx,m,sd). We say (M,T,T') is left modular if it is left modular for all s Â€ M\N. We say (K,T,T') is strongly left modular if it is left modular for all s fe. M. We say (M,T,T') is T'weal-lv left modular for s whenever (M,T,T') satisfies the following two conditions for all x,m,d in M and. all u in N. a)' s(T'(x,u,d)) = T' (sx,u,sd). b) *' = condition b) above. ' 123

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124 Remark . If a ternary field satisfies condition b)' for s, its associated biternary field satisfies condition a)' trivially; hence by Proposition 5.2 to follow its associated biternary field is left modular for s. e.2 Proposition . Let C = (A,K) be a coordinatized AH-plane; let (K,T,1") be the biternary ring of C and let s 6M. Then the follov.-ing three conditions are equivalent . 1) A has a ( (0,0) ,g^)-endoinorphism taking (1,1)' to (s,s). 2) (M,T,T') is left modular for s. 3) (K,T,T*) is T'-weakly left modular for s. If
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125 ((0,0), g)-endomorphism. .By b)', ( x ,y) I [n,b] implies
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126 an automorphism. If s + C, then ) obvious by Proposition Â£.2. / (4=) Assume that (K,T,T') is left modular; that is, that conditions a), b) and c) hold in C for all s 6 M\N and for all x,m,d Â£ M. Then t
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127 8.7 Proposition. Let C = (A,K) be a coordinatized AH-plane with biternary ring (M,T,TÂ«). The endomorphisms of A are strongly ((0,0) .g.J-transitive if and only if (K,T,TÂ«) is strongly left modular. If (M,T,T-) is strongly left modular, then the set of ((0,0),g )-endomorphisms is equal to { = (x',x'm). Since x e K\N, there is an s e H such that sx = x'. By condition b) , s(xm) = (sx)tn; so that -Â«lated triangles and F 2' Q 2 + F ' then Q l T Li Q 3 Â» m 31 ^'.'e say A is strongly (P,g^)-HDesarguesian if whenever (P ;m ) and (Q ;k .) are two (P,g )x er i er ' J oo related triangles and P + P, then Q I L(c ,m

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128 8.9 Proposition . An AH-ple " is (P.g^-H-Desarguesian (strongly (Pjg^-H-Desarguesian . if and only if the automorphisms (endomorphisms) of A are ( P,gÂ„) -transitive (strongly (p,g )transitive ) . Remark . Lorimer and Lane C(1973), page 40, Theorem 5. Ill state a result similar to the second of these two equivalences: their configuration property is not the same as ours. Proof. (=Â£>) Assume A is (P.g^J-H-Desarguesian (strongly (P,g )K-Desarguesian) and let Q be a point not neighbor to P. Let Q' be any point on PC which is net neighbor to P; (let Q* be any point on PQ.) We wish to construct an endoir.orphism such that g and hence C.R + g and QR # g. Let = g HlCC'jQR). Then If R = Q, then L(Q',GS) = L(S',SQ) and hence *r(R) =
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129 for all points of A. If g is a line of A and if R I g, define ?) . There is a (P,g w )-endomorphism
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130 Remark . The prepositions in the remainder of this section are not used in any of the proofs in the other sections or in the appendices. 8.10 Definitions . Let (M,T,i") be a biternary ring, and let s e M. We say (M,T,TÂ») is ( (0 ) , Co, 0] ' ) ncrrr.al for s whenever (M,T,T') satisfies the following five conditions for all x,m,b in M. a) xs = x-s. b) (xs)m = x(sm) . c) s has a two-sided (M, x)-multiplicative inverse s~ . d) T(xs,s~ m,b) = T(x,m,b). e) (T'(x,r-.,b))s = T'(x,ms,bs). If (M,T,I') is ((0),t0,0]')-normal for all s 6 KM.', we say (M,T,T') is ((0,0, Co, 0] Â»)-normal. We say (M,T,T') is T 'weakly ( (0) ,[0,0]' )nor:nal for s whenever (M,T,T') satisfies the following two conditons. a)' There is an s' such that T(xs,s'm,b) = T(x,~,b) for all x , m , b in M . b) ' (T'(y,u,v))s = T'(y,us,vs) for all y,v in M and all u in N. We say (M,T,TÂ») is Tweakly ( (0) ,[0,0") Onormal for s whenever (K,T,T') satisfies the following two conditions. a)" There is an s ' f such that T( x .s, s'm,b) = T(x,m,b) for ail x,b in M and all m in N. b)" (T' (y,u,v) )-s = T' (y,u.s,v.s) for all y,u,v in M.

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131 8.11 Proposition . Let C = (A,K) be a coordinatized AH-plane with biternary ring (l-\,?,T>). Then, the following four conditions are equivalent. 1) A has a ( (0 ) ,g ) -automorphism which takes (1,0) to (s,0). 2) (M,T,T-) is ( (0),C 0,0] ') -normal for s and s 4 0. 3) (:Â•;,T,T , ) is T' -weakly ( ( ) , fo, o] ' )-normal for s and s + 0. 4) (M,T;TÂ«) is T-weakly ( (0 ) , [o ,o] Â• )-ncrmal for s and s + 0. If A has a ( (0) ,g )-automorphism ,, 0] = \^,0~\ and by M be defined by: Vx = T'(y,S'u,Vv). Observe that V0 = 0. Let v = 0, then x = T'(y,u,0) <=> Vx = T'(y,S'u,0) or x =-y-u <^ Vx = y.S'u. Let u = 1; then Vx = x-S'l. Let x = 1; then S'l = s; so that Vx = x.s. Now, y= T(x,m,e) <-, y = T(x s , Sm,e ) , and hence (*) T(x,m,e) = T(x-s,Sm,e) .

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132 . Let e = 0, x = 1, m = s. Then s = s(Ss). , r i-jt L :M Â— > l\ defined s by L s (m) = sm is a bijection; hence Ss = 1. Let e = , m = s in (*); then xs = x-s fcr all x e M and condition a) holds in (K,T,TÂ»). Let e = 0, m = sy, x = 1 in (Â«); then sy = s(S(sy)). Thus S(sy) = y. Let e = 0, m = sy in (*); then x(sy) = (xs)y and condition b) holds. Since R , L are bijective, there are s*, s" such that s's = 1, ss" = 1. Then s' = s'(ss") = (s's)s" = s". Let s denote s'. Obviously s~ is the unique two-sided (M,X}multiplicative inverse of s. Let e = , x = 1 in ( * ) ; then m = s(Sm) and s'm = s*(s(Sm)) = (s's)Sm = Sm. Thus, (*) becomes T(x,m,e) = T(xs,s m,e) and conditions c) and d) hold. Mow, x = T'(y,u,v) if and only if Vx = T * (y, S 'u , Vv) ; hence (TÂ»(y,u,v))s = T'(y,S'u,vs). Let v = 0, y = 1; then us = S'u. So that (T'(y,u,v)s) = T*(y,us,vs) and condition e) holds. Hence (K,T,T') is ( (0) ,[0,0]' )-nor:nal for s, and "Pby Â«r(x,y) . (xs,y). We v/ish to show that
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133 quasiparallel to g y ) to lines quasiparalle-1 to g^ (not quasiparallel to g y ). Since R g and L g , preserve the neighbor relation, preserves the neighbor relation on parallel classes, and by Proposition 3.17, 1). Assume s * and that (M,T,T') is T-weakly ((0), [0,01') -normal for s. Lot s' be an element satisfying condition a)". Define 1) shows that
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134 biternary ring (M,T,T'). The automorphism-s cf A are ((0),q )y transitive if and only if C:,T,T') is ( (0) , [0,0l Â» ) -normal. If the automorphisms of A are ((0),q )-transitive, then the set of y ((0),g )-3utorr.orDhisms of A is the set f o\ s Â€ MAN "5 v/her^ ) Obvious by the preceding proposition; also observe that t
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135 associated biternary field satisfies condition b) for s trivially (let n = 0, w = b.) 6.16 Proposition . Let (A,K) be a coordinatized AH-plane with biternary ring (K,T,T"). Then A has a ((C) ',g ) -automorphism moving (1,0) to (l,s) if and only if (H,T,T') is ( (0 ) Â• ,[0 ,0 ] Â• )normal for s. If A has a ((0)',g )-automorphism cr moving (1,0) to (l,s), then ) Assume A has a ( (0) ' ,Eo,0} Â• }-automorphism moving (1,0) to (l,s). Since (x,y) = [o,y"] f\[0,xVi (x,T(x,s,y)) I [T(l,s,m),b]. Hence T(x , s,T(x,m,b) ) = T(x,T(l,s,m) ,b) and condition a) holds in (M,T,T"). Assume u e 11; then (x,y) 1 [u,vl ( <^> (x,T(x,s,y)) I [n,v:]' where [n,w]' = ( v,T ( v, s, ) ) y(u ,T (u, s , 1 ) ) . Since
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136 (l,s). Define
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137 (<=) Assume (K,T,T') is ( (0) ' , [0,0] Â» ) -normal. Lst g = [0,xlÂ« be a line of (0)'. Let Q,Q' be points of g not neighbor to g y ; then x f 0. Lot C = (x,y), C= (x,z). By (36), since X + 0, there .is a unique s such that z = T(x,s,y). By the preceeding proposition
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138 44a one-to-one map . Let G = L(T',21') f\ g. Then, if T = L(T',TT) (\ L(G,S), then rT a T'. Thus, cr is an automorphism. Assume g is affine and that P = TT; TT ~TT(g). Let S be a point such that Q =
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9. AH-RINGS In this section we construct equivalences relating the categories of AH-rings, of Desarguesian AH-planes and of kernel quasirings, and we give various characterizations of Desarguesian AH-planes. 9.1 Definitions . A ring (S,+,x) is said to be an affine Hjelmslev ring or AH-ring if and only if it has the following five properties. (Rl) S is an associative ring with 1^0. (R2) Every zero divisor is a two-sided zero divisor. (R3) The set of zero divisors N is a maximal ideal of S. (R4) The elements of S which are not in N have multiplicative inverses. (R5) For every pair n,m fe N , there is an s e S such that ns = m or ms n. A ring R is said to be a Hjelmslev ring or H-rinq if and only if it is an AH-ring satisfying the following additional property. (RH6) For every pair n,m 6 N , there is an r Â€ R such that rn = m or rm = n. 9.2 Definitions . Let S,S' be AH-rings, and let u,:S -^* S' be a ring homomorphism such that w>(0) Â» 0, u>(l) Â« 1. We say w: S Â» S 139

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140 is an AH-ring horororyhi sn if oo(", : ) R and X':R Â— Â» 6 coined K ~ K bove are reciprocal equivalences. If Q is a kernel cuasiring, then N x(c) = N Q . If S is an AH-ring, then N ( > = N . Also, XX' and X*X are identity functors. Proof . Let (0, +,X,-) be a kernel quasiring. Let N = N , and let (S,+,X) = X(C) = (Q,+, X). Since, by Proposition 7.3, Kern Q is a local ring with 1^0, condition (Rl) holds in S.An element n of Q is a right zero divisor in S if and only if n 6 N. If m is a left zero divisor in* S, then, for some k ^ 0, mk = 0. Since m0 = and mk = 0, we have by (VW6) that m fe N. If u & K, then since u,0 6 N, by (VW7) there is a w Â€ Q such that uw = or u.w = where w / 0. But Kern Q = Q; so that u-w = uw, and u is

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141 a left zero divisor. Thus, (R2) holds in S. If s Â€ Q\N, then by (VV:6) there is an m such that sm = 1. Since s j N, by (VV/5) there is s unique x such that xs = xO + 1 = 1. Thus, since x = x(sm) = (xs)ir. = m, each s Â€ Q\N has a two-sided multiplicative inverse and (R4) holds in 3. Since (Q/~,Â©,Â®) is a divison ring, N is an ideal of (Q,+,X). Since N is a maximal ideal of S, (R3) holds in S. Observe that N = N . If n,m Â£ N, then, by (VW7), there is an s such that ns = m or m-s = n. F3ut mÂ»s = ms since Kern Q = Q and hence (R5) holds in S. Thus,' S is an AHrina with N = N . S Q If io:Q Â— * c ' is a kernel quasiring homomorphism, then w(N ) C N , and it is easily seen that X(u>) is an AH-ring homomorphism. It is now easily seen that X : Q Â— Â» R is a functor. ~ K Let (S,+,K) be an AH-ring. Let = (S,+,X,x) and let N = N . Observe that if we define N_ , N' according to the definii> Q Q t.ions oiven in the definition of prequasirinq, then N = N' = N Q as sets, and that N x , (s j = ^' s Observe that (VWO), (VWl), (VV/2), (VW3), (VW4), (VW5), (VW6), (VV/3), (V.v'10) and their duals hold in Q. Observe that mu = 1 if and only if urn = 1 . A routine calculation shows that (VW9) and hence (VW9) * hold in 0. Let a,b 6 N, not both zero. Assume both am = b, and bt = a for some m fe S, and some t Â€ N. Then a = amt and a(l mt) = 0, where (1 mt) fe S\N. Then a = Q, b = 0, a contradiction. Since S is an AH-ring, there is an s fr S such that as = b or bs = a. If bs = a, and if s Â« S\N, then as" = b. Thus (VW7) and (VW7) ' hold in. C. Since (VWll) holds also, and since (S, + ) 'is abelian, is a quasiring. Also, Kern Q = Q, and hence Q is a kernel quasiring.

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142 If to:S Â— * S' is an AH-ring homcmorphism, then ui(N ) C N s _ s , Thus, it is easily seen that X'(w) is a quasiring homomorphism and that X':R Â— Q is a functor. "' K It is also easily seen that X'X = 1, and that XX* = 1^ . -Q K ~~ -R Hence X and X' are reciprocal equivalences. // 9.6 Definitions . Let A be a translation AH-plane. If r is a translation of A such that for some point P, rP Â•/P, and if t' is a translation such that every trace of t is a trace of T Â• , and if for every such pair r, t* there is a trace-preserving endomorphism of the translation group such that St = t', then A is said to be Desarquesian . We denote the category of coordina tized Desarquesian AH-planes with coordinatized AH-plane homomorphisms by C . 9.7 Lemma . Let (Q, + ,*,Â•) be a prequasiring such that s(x + m) = sx + sm for all s e Q\N and for all x,m e Q. Then (Q,+) is abelian; (Q,+,x,.) is a quasiring, and s(x + m) = sx + sm for all s,x,m Â€: Q. Proof . Assume (Q,+,x,.) is a prequasiring such that s(x + m) = sx + sm for all s 6. Q\N and for all x,m e Q. For. (a + b) ^ N Â» (a + b)(l +1) =a+b+a+b; also, (a + b)(l +1) = a + a + b + b. Since (Q,+) is. a group, a + b = b + a when (a + b) 4 N Â« If (a + b) Â€ N, a & N; then b e N, (1 + a) 4 N and hence 1+ (a + b) = (1 + a) +b = b+ (1 + a) =b+ (a + l)' = (b + a) + 1 = 1 + (b + a) ; so that a + b = b + a when a,b t N.

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143 If (a + b) C M, a Â£ N, then b 4 N, and -a <Â£ N; so that -a + (a + b) = b . b + (a a) = (b + a) a = -a + (b + a); hence a + b = b + a. Thus, (C, + ) is abeiisn. Let n 6 N; then (n 1) ^ N and n(x + m) = ( (n 1 ) + 1 ) (x + m) = (n 1 ) (x + m) (X + m) = (n l)x + (n Dm +x + m=nx-x+nm-m+x+m= nx + nm. Thus, s(x + m) = sx + sm for all s,x,m Â£ Q. // 9.8 Definition. If S is an AH-ring, we call A r , . , . .. (see Cont IX IS)) struction 3.13) the AH-plane generated by S, and denote it by A . i. i s 9.9 Definitions . We say an AH-plane of order > 2 (of order 2) is Hjelmslev Desarguesian if it is (P^)-!-:and (T,g )-3Desarguesian ((P.g^J-Hand (r ,g M )-4-Desarguesian) for every point P and every direction V. We say A is stroncly Hjelmslev Desarguesian if it is strongly ( P.gJ-H-Desarguesian for every point P, and ( r,g eo )-H-Desarguesian for every direction V. 9.10 Proposition . Let A be an AH-plane. The following are equivalent. 1) There is a point P and two directions T, 1 , r
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144 K of A, the quasiring of (A.,K) is a kernel, quasiring: this represents an algebraic characterization: see Theorem 5.25, 1) <Â£3> 4) <Â£> 8). 5) A is a translation AH-plane, and there is a coordinatization K of A such that the quasiring of (A,K) is a kernel quasiring: this also represents an algebraic characterization. 6) A is isomorphic to the AH-plane A generated by some AHring S, and for every AH-ring R such that A is isomorphic to A , R R is isomorphic to the ring E (A) of trace preserving endomorphisms of the translation group of A. 7) A is isomorphic to the AH-plane generated by some AH-ring S. 8) A is strongly Hjelmslev Desarguesian: this is a geometric characterization. 9) A is Hjelmslev Desarguesian: this is a geometric characterization. 10) There is a point P and two directions T,Â£ ; P i1 ; such that A is (P,g w )-, (Pjg^)and (Â£ ,g -5 )-H-Desarguesian: this is a geometric characterization. Remark . In the next section we will show that, if A is a Desarguesian AH-plane, then the automorphisms of A are (T,g)-transitive for every direction T and every line g of A. Proof . D -=\$2). Assume condition 1) holds in A. Let K be a coordinatization of A with = P, g =. L(P,D, g Â„ L(P,-Â£). x y Then the automorphisms of A are ((0),g )-, ((0)*,g )and

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145 ( (0,0) jg^) -transitive and hence A is axially regular. Let (Q,+,X,-) be the prequasiring of (A,K). By Proposition 3.2, s(x + d) = sx + sd for all x,d in Q and all s in Q\N. By Lemma 9.7, Q is a quasiring and s(x + d) = sx + sd for all x,d,s in Q. Since (Q,+) is abelian, A is a translation AH-plane with quasiring (Q, +,*,Â•). By Corollary 3.3, if s 6 Q\N, then s eKern Q, since the automorphisms of A are ( (0,0) ,9^) -transitive. And since any n e N can be written as n = (n 1) + 1 where (n 1),1 e q\n, we see that Kern Q = Q. We wish to show that A is Desarguesian. Let T be a translation of A such that for some point P, rp 4, p. Then r = T (a, am) or r = r (, m .) f Â°r some a,m in Q. Assume T = T. ,. Since vaÂ«m, a; (a, am) >P -f P, a i0. Let T' be a translation such that any trace of T is a trace of r' . Since [m,0] is a trace of T, it is a trace of T'. Hence, T' = T (c for some c in Q. Since a 4 , by (VW5) there is an x in Q such that xa = xO + c = c. Hence, by Theorem 7.7, there is a trace preserving endomorphism of the translation group, which we denote by \(x), such that X(x)t = t* . If T a r we use the dual argument to get V = T ((x.al.m x.a)' and the remainder follows as before. Thus, A is Desarguesian. / 2) =^> 3). Assume A is Desarguesian. Then A is a translation AH-plane. Let K be a coordinatization of A, and let (Q,+,X,Â«) be the quasiring of (A,K). Let X be the translation T= T (l,0)* Then T(0 ' 0) = (1 Â»Â°> and (0,0) + (1,0), since 1 40. Let q be any element of Q, and let T* = T ... if k' is a trace ( q , ) of r, then k = [o,a~] for some a 6Q, and hence k is a trace of

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146 T'. Since A is Desarguesian, there is a trace preserving endomorphism (of the translation group) S> such that St = T Â« . By Theorem 7.7, there is an x Â£ Kern Q such that T. % = T, (q,0) (xl,xO) Thus, q = x and Q = Kern Q. We have just shown that if K is any coordinatization of A, then the associated quasiring is a kernel quasiring. Thus, by appropriate choice of coordinatizations , we can see that, by Corollaries 8.3 and 8.4, the automorphisms (endomorphisms) of A are ( Pjg^J-transitive (strongly (p,g w )transitive) for every point P, and, by Theorem 5.25, the automorphisms of A are (T,gJ -transitive for every direction P. / 3) =5 1). Obvious. / 3) =?? 4). Let K be a coordinatization of A. By 3), the endomorphisms of A are ((0,0)^)-, ((0),g )and ((0)',g )transitive and the first part of the proof of 1) =Â£> 2 ) then shows that A is a translation AH-plane and that the quasiring of (A,K) is a kernel quasiring. / 4) =^5). Obvious. / 5) =?> 1). Assume A is a translation AH-plane and that K is a coordinatization of A such that quasiring of (A,K) is a kernel quasiring; then, the automorphisms of A are ((O^g^)-, ((0)',g )and ( (0,0) jg^J-transitive by Theorem 5.25 and Proposition 8.3. / .5) =^> 6). Assume A is a translation AH-plane, and that K is a coordinatization of A such that the quasiring (G,+,x,.) of (A,K) is a kernel quasiring. Observe that (A,K) is isomorphic to (A B ,K B ) where B = E'(Q) by Corollary 6.18. Let (S,+,x) = X(Q). Then S is an AH-ring, and A is isomorphic to A_ where B = E'X'(S). B Â•Â— Â«Â» Thus, A is isomorphic to the AH-plane generated by S.

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147 Assume 6:A Â— * A is an isomorphism from A to the AH-olane ft * A generated by some AH-ring R. Let K be the usual coordinatization of A . Let K be the corresponding coordinatization of A; that is, let g = S~ [o,0~] and so on. Observe that T(x,m,b) in B(A,K) is equal to xm + b in R, and T*(y,u,v) = yu + v. Hence Q(A,K) is a kernel quasiring and by Theorem 7.7, R is isomorphic to Â£ (A), the ring of trace preserving endomorphisms of the translation group of A. / 6) => 7). Obvious. / 7) =^> 5). Let 6: A Â— v A be an isomorphism from A to the AH-plane A generated by an AH-ring (S,+,X). Let K be the obvious coordinatization of A; that is, let g = Q~ [.0,0^, and so on. Let C = (A,K). Then it is routine to show that C(B(C)) = (A ,K ) where K is the usual coordinatization of A . Obviously A is a translation AH-plane. Observe that the quasiring of C is equal to (S,+,X,x) and hence is a kernel quasiring. Thus, 7) =\$ 5). / 3) => 8). This is immediate by Proposition 3.9 and Theorem 4.21. / 8) =^ 9). Obvious. / 9) => 10). Obvious by Theorem 4.21. / 10) ^ 1). This is immediate by Proposition 8.9 and Theorem 4.21. // 9.11 Proposition . The. maps R:C Â— * R and R':R Â— * C defined by R(C) = X(0(O), RUÂ»J = X(Q(oo)) and R'(S) = QÂ«(XÂ«(S)) f R'U) = Q'CX'Cuj)} are reciprocal equivalences where R is the 'category of AH-rings and C is the category of coordinatized Desarguesian AH-planes. Also RR' = 1^.

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148 Proof. If C is a cocrdinatized Desarguesian AH-plane, then its quasiring is a kernel quasiring; hence X(C(C)) is an AH-ring. If S is an AH-ring, then X'(S) is a kernel quasiring and, since QQ ' = lg, Q'(x'(S)) is Desarguesian by Proposition 9.10. The remainder of the proof in immediate. // 9.12 Definition . When we say division ring, we nean associative division ring: these are sometimes called skew fields. Observe that the category of division rings is equal to the category R* where R* is the category of AH-rings such that N = \o]. 9.13 Corollary . The restrictions of R and R' to R* :C * Â— Â» R* and R*':R* Â— > c * (where C " is the category of coordi natized Desarguesian affine planes and R' is the category of division rings) are reciprocal equivalences, and R*R* Â• = 1~ . // 9.14 Proposition . The category of division rings R* is equivalent to the category of kernel biquasif ields 6 * v K is defined in the obvious way. Proof . This is immediate by Proposition 9.5. // The following is an example of a coordinatized AH-plane which is not a translation AH-plane, but whose first ternary operation T is the same as that of a Desarguesian AH-plane; hence T is linear and (>:, + , >0 is an AH-ring. Also, (M,x) is abelian for this example.

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149 9.15 Example . Let (A S Â» K S ) b e the coordinatized Desarguesian AH-plane generated by the H-ring S = Z 3 [x]/(X 2 ), and replace the parallel classes (X) ' and (2X) ' by Ux,n1'| n e N] U t[2X,ra]'l m 4 N"J and l[2x,n]' | n Â« N| (J ![x,m]' | ra t NJ respectively. The new coordinatized AH-plane (A ,K ) has the property that the first ternary operation of its biternary ring is the same as the first ternary operation of the biternary ring of (A ,K ). We wish to show that A is not ( (0) ,gÂ«,)-H-Desarguesian. In A , the triangles J = ((0,0),[l,0]a [2,1 + x],(X,l);(l),(2),U,0]Â«) and j, . ((1,0),L((1,0),(1)) H L((l + X,1),(2)),U + X,l);(l),(2),[x,l]') are ( (0) , gj-related. In A* the triangles J and J' are still ( (0),g oo )-related, but since [x,l]' is the unique line through (1,0) and (1 + X,l), the point (1 + x,l) is not on the line L( (1,0) ,[x,0] '). Hence A is not ( (0) .g^-H-Desarguesian and, by Theorem 5.25, A is not a translation AH-plane. Observe that the gross structure of A is Desarguesian. Remark . By Theorem 7.7, the ring of trace preserving endomorphisms of the translation group of the AH-plane A discussed above is an H-ring, and hence A is 'projectively Desarguesian' (see Definition 10.16). Remarks . J. VÂ» T . Lorimer was the first to state portions of Proposition 9.10: see A. 24. We discuss Lorimer 's and Lorimer and Lane's recent results in Discussions A.25-A.27, and we give a brief history of this research in subsection A. 28.

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10. HJELMSLEV STRUCTURES In this section we show that, for any AH-ring S, we can extend the AH-plane A generated by S to a lined Hjelmslev structure generated by S. The automorphisms of a Hjelmslev structure are (P,g ^-transitive for every point P and every line g. We use this property to show that the automorphisms of a Desarguesian AH-plane are (T ,g)-transitive for every direction r and every line g. We show that the Hjelmslev structure generated by S is a PH-plane (projective plane) if and only if S is an H-ring (division ring). We construct a ring S which is an AHring but not an H-ring, and show that the Desarguesian AH-plane generated by S is not projectively Desarguesian and cannot be extended to any Desarguesian PH-plane. We show that any automorphism (any ( P ,g)-automorphism) of a Desarguesian AH-plane can be extended to an automorphism (a (P,g )-automorphism) of the associated lined Hjelmslev structure. We characterize the automorphisms of a Desarguesian AH-plane and the automorphisms of a Hjelmslev structure. 10.1 Proposition . Every division ring is an H-ring. Moreover, if R is an AH-ring, and if N = N is the set of zero divisors ii R R, then the quotient ring R/N is a division ring. /) 150

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151 10.2 Construction of H:R Â— Â» S. Let S be an AH-ring. Let N be the set of two-sided zero divisors in S. If x ,x ,x e S, define r(x 1 ,x 2 ,x 3 ) to be the set {(a^a ,a )\ 3 r e S\N, a. = rx . V j , j = 1,2,3]. If y 1 ,Y 2 ,y 3 Â£ S, define (y-py^y^s to be the set l(b lt b 2 ,b 3 ) i 3 s e S\N, b = (y )s Vj, j . 1,2,3]. Let ft = Cr( X;L ,x 2 ,x 3 ) \ 3 j } x. ^ N] and let oj = ICy^y^y^s I 3 j ^ y.. ^ n3. Define incidence by rCx^x^Xg) I (y^y ,y )s <\$=p X l y l + X 2 Y 2 + X 3 y 3 = Â°* Incidence can be seen to be well-defined. We usually write rCx-^x^x^ as rx.; and (y^y^yjs as y s. Let H(S) = CP,oj,I). If <*>:S Â— * S' is a raorphism in R, define H(u)):H(S) Â— *K(S') by H(w)(rx.) = r(wx.) and H(w)(y.s) = (uov )s. Remark . We will show in Proposition 10.13 that S is an H-ring if and only if K(s) is a projective Hjelir.slev plane. 10.3 Definition . Let S be an AH-ring. We call H(S) constructed above the Hjelmslev structure of S. 10.4 Proposition . The map H:R Â— Â» s defined above is a functor from the category of AH-rings to the category of incidence structures. Proof . Observe that if w is a morphism in R, H(uj) preserves incidence. // 10.5 Proposition . Let S be an AH-ring with maximal ideal N. Let H(S) be the Hjelmslev structure of S. Then points rx, and

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152 ra are equal if and only if there is an element k t S\N such that x. = ka. for j = 1,2,3. If c,m Â£ S\N, and if rx . is a J j i point with x . = m for some j, then if we let a, = (cm )x for J k k k = 1,2,3, then rx. = ra . and a. = c. The symmetric statements hold for lines. // 10.6 Definitions . Let S be an AH-ring with maximal ideal N and Hjelmslev structure H(S). Let D = S/N. Then D is a division ring and there is a map v:H(S) Â— Â»Â• H(D) induced by the map taking s Â€ S to (s + N) fe D. We say P is near Q in H(S) and write P A Q whenever vP = vQ; and g is near h, g & h , whenever vq = uh. We say P is near g, P Â£. g, and g is near P, g A P, whenever vP I \)g. 10.7 Lemma . Let S be an AH-ring with Hjelmslev structure H(S). Any two points rx.,ra. of H(S) are joined by at least one line. They are joined by exactly one line if and only if rx . is not near ra . . The symmetric statements hold for two lines y.s.b.s whenever S is an H-ring. Proof . Assume that rx . = r(l,x_,x_). The points rx.,ra. are i 2 3 ii joined by a line m.s if and only if m = -(x p m + x_m ) and C 2 m 2 + C 3 m 3 = Â° where c 2 = a 2 ~ a l X 2 and C 3 = a 3 ~ a l x 3* since there is an h & S such* that either c_h = cor c h = cÂ„, the second equation can be solved for one of m ,m in terms of the other, which can be set equal to one. Thus the two equations always have a solution with m or m equal to one. If either

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153 one of c ,c is invertible, then m.s is unique: if for example cÂ„ 4 N, then m_ = -c_ c.m., and m.s is uniquely determined. 2 ' 2 23 3 l -i j However, if both cÂ„,c. are in N, and if, say, m.s = (m,,l,m_)s 23 ' J l 1 ' 3 and rx.,ra. I m.s, then there is a k / such that cÂ„k = 0, and ill' 3 hence Cm,' x_k,l,m_ + k) is not in m.s, but corresponds to a 1 3 ' 3 l ' line throuqh rx.,ra., and hence the solution m.s is not uniaue. 3 i* i' i The case where c.,c. Â€ N, (m_ ,m_ ,1) Â€ m.s is similar. The 2 3 12 l property that at least one of c.,c is not in N is equivalent to rx . being not near ra . . The cases with x. eN are similar. The li 1 corresponding statements hold for lines y.s,b.s whenever S is an H-ring since the antiisomorphic image of an HÂ— ring is an H-ring. // Â• Remark . The initial part of the above argument is adapted from Klingenberg [(1955), page 1083. 10.8 Proposition . Let D be an AH-ring. Then D is a division ring if and only if the Hjelmslev structure H(D) is a projective plane. Proof . ( =^ ) Assume D is a division ring. Two points are distinct in H(D) if and only if they correspond to distinct subspaces of _(D.. _). Let rx.,ra. be distinct points; then they Dlx3 11 2 are not near. Hence, by Lemma 10.2 they are joined by a unique line. Symmetrically, -any two distinct lines meet in a unique point. The points r (1,0,0), r( 0,1,0], r( 0,0,1) and r( 1,1,1) have the property that any three of them span _(D -J ; hence no one dimensional subspace of linear functions is orthogonal to

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154 any three of the four points; that is, no line intersects any three of these four points: see Jacobscn [(1953), page 55]. / ( 40 Assume H(D) is a projective plane. By Lemma 10.3, if P = r(l,0,0) and Q = r(l,n,0) for n e N, then P ~ Q. But P Q implies P = Q; hence N = \0\ and D is a division ring. // 10.9 Definition. Let S be an AK-ring. Let h be a line of H(S), and let G (K(s))h) be the incidence structure obtained from His) by removing all the points near h but not on h and all the lines near h but not equal to h. If h = (1,0, 0)s, we call (G h) (H(S),h)Â» the lined affine Hjelmslev structure of S, and denote it by G ' S* 10.10 Definition. If A is an AH-plane; if S is an AH-ring and if /* : S g (A) ~* (H(S),h) is a lined incidence structure embedding such that the image of G(A) is G then we say that A can be extended to (H(S),^ g<)0 ) ( through h ) , or that A can be extended to H(S) (throuqh u.). / Remark. If K(s) is a PH-plane, this definition of 'extended' is equivalent to our earlier one. ' 10.11 Proposition . Let S be an AH-ring with maximal ideal N; let h = (1,0, 0)s be a line in H(5); let A s be the AH-plane generated by S. Then A s can be extended to H(S) through X:G (A ) Â— * (H(S),h) ~g b ~ where X is defined as follows: K(x,y) = r(l,x,y), X[m,d] = (d,m,-l)s, X[u,v]' = (v,-l,u)s, X(m) = r(0,l,m), X(u)' = r(0,u,l) and XtgJ = (1,0, 0)s. Also, the cross structure of A S can be extended to H(S/N) through a map \Â» which is induced by

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155 X. The map X induces an isoraor; hism \' :G"(A ) Â— * (G h) ~g S (K(S),h)' ' and hence (G (H(s) h) Â» h ' is a generalized AH-plane, and the nearness end neighbor relations coincide in G . S Proof. Let us first shew that if k = [u,v]' with u Â£ S\N then k = Lu ,-vu J and that k is well defined. Observe that (v,0) is on both [u,v]' and [u~ ,-vu** ] and that they have the sane slope; hence they are equal. Also, X[u,v]' = (v,-l,u)s = (-vu ,u" ,-l)s = \[u~ ,-vu" ]. If P = ( m ) Â„ ( U )', then u = m" 1 and X(m) = \(u)'. Hence X is well defined. One can easily see that X is one-to-one on affine points, on affine lines of the form [n,d], on affine lines of the form [u,vj' where u Â€ N, on directions of the form (m), on directions of the form (u) ' where u e N and on the line g . One can also see that the sets which are images of each of these six classes of points or lines are pairwise disjoint. Thus X is injective. In H(S), a point P is near h ^> v P I vh; thus r( X;L ,x 2 ,x 3 ) Ah <=> xÂ± Â£ N. A point P = rCx^x^x ) is on h if and only if X;L = 0. Thus, X takes the generalized points of A onto the points of G . -> CrtvS),h) In H(S), a line k is near h if and only if vk = vh. Denote the element x + N in S/N by x*. Then, a line y.s is near the line h = (1,0, 0)s if and only if (y Â• ,y Â• ,y Â• ) s * (1*,0* ,0*)s* ; hence if and only if y2>y 3 e N. Thus, X takes the generalized lines of A s onto the lines of G (H(S) h) . Hence X takes G(A ) onto G. . , , (H(S) ,h) Observe that (x,y) I [m,b] <^ y=xm+b <=>b+xm-y=

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156 <^> \(x,y) I XU.b]. Also, (x,y) I [u,v]' <^ \(x,y) I A.[u,v]Â»; (m) Ig <^> X(m) iXg; and (u) ' I g O X(u) ' I X(g). Also, no point (x,y) is on g^ and no point r(l,x,y) is on h. Thus, X preserves and reflects incidence, and hence X is a lined incidence structure embedding of G (a ) into (K(S),h). ~g S ~ The remainder of the proposition is easily shown. // 10.12 Proposition . Let H(S) be the Hjelnnslev structure generated by an AH-ring S with maximal ideal N. Let D = S/N and let V f H(<) where t;S Â«Â—Â» D is defined by * s = s + N. Let h,k be lines; then h Â£ k if and only if lb. P\ k\ = 1. If h / k, then there is a point P which is on h, but not k. If P is a point and g a line of H(S) such that VP I vg in H(D), then there is a line g' and a point P' such that P & F'; g Â£> g ' and P' I g; P I gÂ». Hence there are at least three pairwise non-near points on each line (lines through each point). Proof . Let K be any point of H(S); then, K is not near one of the lines (l,0,0)s, (0,1, 0)s and (0,0, l)s of H(S): let g be one Of these three lines such that K is not near g . By Propositions 3.17, 3.27 and 10.11 and symmetry, there is a lined generalized AH-plane embedding u :G (A J Â— * (H(S).gÂ„) such that / K ~g S ~ ^K /* K (g w ) * gÂ„, and such that P Q *& y^.,(?) t. u. K CQ); n ~ m <Â±> ** (n) & u (m). Let G 'denote the generalized AH-plane (G (H(S), gif )' g K K Assume I h D k \ * 1 in H(5). Let 3 = h ft k. Then lh !\ k\ = 1 in G ; hence, h \$. k.

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157 Assume \>R I vg in H(d-). Then, by Proposition 2.31, 3 R* (in G ) such that R' A r and R < j g> and 3 g . ( in G j such that R g ' A g and Rig'. Assume h,k are lines of H(S), h ^ k. Let UK I vh,vk. By the results of the preceding paragraph, there is a point Q on h, Q Â£ 9 K > K Â« Thus, Q is not on k in H(S). Assume h,k are lines of H(S), hÂ£ k. Let B be a point such that vB = vh A vk. If M I h,k in H(S), then M A 3; hence, M is in G , and, since h 41 k in G_, In ft kl = 1 in G , H(S). The remainder or the proposition is now easily shown. // 10.13 Proposition . L.?t S be an AH-ring. Then S is an H-ring if and only if the Hjelmslev structure R(S) is a PK-plane. If S is an H-ring, the gross .structure of K(s) is isomorphic to H(S/N) under the map induced by the morphism taking s Â£ S to s + N. If S is an H-ring, thpn PAP' <=\$> P~P'; PAg <=> P ~ g and gAg' <Â£=?> g g Â• in [((s) . Proof. Assure S is an H-ring. By Lemma 10.7, any two points of H(S) are joined by at least one line and any two lines of H(S) intersect in at least one point. Also by Lemma 10.7, two points (lines) are neighbor if and only if they are near. Hence, the map v:H(S) Â— H(S/N) induced by the AH-ring homomorphism taking 5 6 S to (s + N) Â€ S/N. identifies points (lines) if and only if they are neighbor. Since S/N is a division ring, H(S/N) is a projective plane. Thus, H(S) is a FH-plane and H(s/N) is isomorphic to the gross structure of H(S). Observe that in H(S)'

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158 PAF' & P ~ PÂ»; g a gÂ« g ~ g Â• . Since P A g implies that there is a point PÂ» I g such that F A?', we have that F A g <^> P ~ g. Assume K(5) is a PH-plane. Let n,rn e N. Trie lines (-n,m,-l)s and (0,0, -Da intersect. Let rx . be a point on both. Observe that x (-1) = implies x = 0. Thus, x. (-n) + x^m = j 1.2 where at least one of X Â± ,x 2 is in S\N. If x, Â£ S\N, then n = km for k = x x " x 2 . If x 2 Â€ S\N, then m = kn for k = ^^V Thus, S is an H-ring. // Remarks . The concluding paragraph of the above argument is adapted from Klingenberg [(1955), page 107, the argument for S26]. Klingenberg claims to have shown [in (1955), page 103, S28l that if R is an H-ring, then H(R) is a PH-plane. There is an error in his argument where he says (lines 13 and 14), "Dann lost aber mit (u ,l,u )s auch das davon verschiedene Tripel (u^jl + c,u )s die Gleichungen (14), also Po ?'." Let K be a field and let R be the H-ring K[x]/(X 2 ). Then N, = ?kX.l k Â£ k] R and N R = \o]. In such an H-ring this argument fails as follows: let c,u 1 ,u 3 Â€ N Rt c /. 0; then (u^l.u^s = (u.^1 + c,u 3 )s since (u 1 ,l + c,uj(l c) = (u 1 ,l,u 3 ). 10.15 Example of an AH-rinq which is not an H-ring . Skornjakov [(1964), page 87, proof of Theorem 15] has constructed rings (R,+,-) in the following manner. Let K be a field which has an

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159 isomorphism
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160 10.19 Definition . Let H(S) be the Hjelmslev structure of* an AH-ring S. We say an automorphism
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161 10.21 Definition , Let S be an AH-ring ana let Z be a non-singular matrix in S^ %3 ' We define K(S) to be the map which takes a point rx. to the point rx.Z and which takes a line v s i l J i to (Z (y.s) ) , where t is the transpose operator. 10.22 Proposition . Let S be an AH-ring with Hjelmslev structure H(S). If Z is an invertible element of S , then cr :H(s) Â— * H(S) defined above is an incidence structure automorphism which preserves and reflects the nearness relation. Also.
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162 10.24 Proposition . Let S be an AH-ring with maximal ideal N. The automorphisms {o-^ \ q e s\N} of the Hjelmslev structure H(S), where Z q 10 1 ( 'O/'O"^) -transitive. , are ( '0, "0 "'0)-automorphisms which are Proof. Any line k through '0 has the form (0,c,d)s for some c,d 6 S where not both c and d are in N. Observe that each of the given matrices Z is non-singular. Then
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163 -1 some z,c & S, Z = Z (Z ) , where each of Z^ and ZÂ„ is of the form Z = v , are ( *"0, (O,l,u)s)-automorphisms which are 10 1 v 1 + uv ("'O,(0,l,u)s)-transitive. Here, '0 I (0,1, u)s; (0,1, u)s A'0'"C, and every line through '0 near 'C'C is equal to (0,1, n)s for some n 6 N. Proof . Let g be a line through *0 and near 'C'C. Then a = (0,1, n)s for some n Â€ N, since 'C'C = (0,1, 0)s; '0 = r(l,0,0). Assume u 6 N. It is obvious that '0 I (0,1, u)s and that (0,1, u)s A '0'"0. Any line through "'0 can be written in the form k = (e,f,0)s. Observe that each of the matrices Z with v e S is non-singular. Also
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164 Q v = r(-f,l,0). Then cr (Q ) = Q and
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165 to show that for any given full 4-vertex there is a matrix C such that or, takes this full 4-vertex tc ( ' C , "0, ,M ,E) . Let (Q-,Q ,Q ,Q ) be a full 4-vertex. Form a matrix Z by letting the Â±" row of Z be an element of C. for i = 1,2,3. Then Z is non-singular since its image in (S/N) is. Observe that Z _^ takes Q 1 to '0, Q to "0 and Q to '"0. Let (x l' x 2Â» x 3 ) Â€
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166 (G (H(R)
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167 R such that K is isomorphic to H(R) is unique up to isomorphism. // 10.31 Proposition . Let A be an AH-plane. Then A can be extended to a Desarguesian PK-plane if and only if A is protectively Desarguesian. If A can be extended to a Desarguesian PH-plane H isomorphic to H(R) (where R is an H-ring), then the ring E (A) of trace preserving endomorphisms of the translation group of A is isomorphic to R. Proof . Assume A is protectively Desarguesian. Let R = E (A). Then R is an K-ring, and A is isomorphic to the AH-plane generated by R. By Proposition 10.11, A can be extended to H(R), which is a Desarguesian PK-plane. Assume A can be extended to a Desarguesian PH-plane H isomorphic to H(R) (where R is an H-ring). Assume 9:H Â— *H(R) is an isomorphism. By Proposition 10.29 (G, , . ,,q) is ( H ( R ) , g } ' ^ isomorphic to A for every line g of H(R). Thus, A is isomorphic to A . Hence by Proposition 9.10 A is projective Desarguesian, and R is isomorphic to E (A). // W * 10.32 Corollary . There exist Desarguesian AH-planes none of which can be extended to a Desarguesian PH-plane. Proof . The AH-plane generated by any AH-ring which is not an H-ring has this property. // 10.33 Proposition . Let A be an AH-plane. The following are equivalent.

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168 1) A is a translation AH-plane and there is a line g of A such that, whenever r and t' are translations which have g as a trace, there is a trace preserving endomorphism of the translation group, S , such that either Sr = t' or Sy = T. 2) A is protectively Desarguesian. 3) A is isomorphic to the AH-plane generated by an H-ring. 4) A is a translation AH-plane, and, whenever T and t' are translations which have some trace g in common, there is a trace preserving endomorphism of the translation group, I , such that either St = r' or W = t. Proof . 1) =* 2). Assume that A is a translation AH-plane and that g is a line of A satisfying condition 1). Let K be a coordinatization of A such that g = g . Let (Q,+,x, Â•) be the quasiring of (A,K), and let s Â€ Q. By Theorem 7.7, there is a q t Kern Q such that either T, t , . or T (q-l,q.O) (s,0)' Â° r (q-s,q.O) ' ^(1 0)* If q ' 1 = S ' then s e Kern Q * If q ' s = 1 Â» then s ( ". Assume qÂ«s = 1. Then q 4 N. Observe that if a,b fe Q, and if qa = qb, then q(a b) = 0; so that, a = b. Thus, q(sx.) = (qs)x = (qs)-x = q(s-x) implies sx = s-x. Also, q(s(x-y)) = (qs)(x-y) = ((qs)x). y = (q(sx))-y = q((sx)-y) implies s(x-y) = (sx).y. And q(s(x + m)) = (qs) (x + m) = (qs)x + tqs)m = q(sx) + q(sm) = q(sx + sm) implies s(x + m) = sx + sm. Similar arguments show that s \ Kern Q. Hence Q is a kernel quasiring and by Proposition 9.5, (Q,+,x) is an AH-ring. If m,n Â€ N, then by Theorem 7.7 there is a q eKern Q such that T r, Â„, or r , . = T . Henrfa fo j. *} (qm,0) (n,0) (qn,0) (m,0) ence ' lÂ«Â»+Â»*>

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169 is an H-ring, and, since (G,+,X) is isomorphic to E (A) by W Proposition 9.10, A is protectively Desarguesian. / 2) =\$ 3). Obvious by Proposition 9.10. / 3) =^4). Let 0:A Â— * A be an isomorphism from A to the AH-plane generated by some H-ring R and identify the points (lines) of A with their images in A . Let K be the usual K R coordinatization of A n . Assume that T. lN and X, , are K I a ., b ) ( c , d ) translations which each have some line g as a trace. Assume g = [m,e]. Then gÂ« = [m,o] is a trace and thus (0 + a,0 + b) I [m,o] and b = am. Similarly, d = cm. There is an s eR such that sa = c or sc = a. Assume sa = c. Then A(s) defined by X(s) T (x,y) = T (sx sy) is a trace Preserving endomorphism of th translation group of A by Theorem 7.7. Observe that X( s ) takes T (a b) to T (c d)" The other case s are similar, and hence condition 4) holds in A. / 4) =S> 1). Obvious. // 10.34 Lemma . Let A be an AH-plane; let P and g be a generalized point and a generalized line of A and let Q,Q' be affine points of A; Q + P,g. Then there is at most one ( P,g )-automorphism * of A which takes Q to Q 1 . If K is a (P.g)-automorphism then at least one of the following three conditions holds: a) g = g ; b) P I g^; c) Â« = 1 A . Proof . Assume
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170 Assume g = g^. If P is an affine point, Proposition 8.2 can be used to show that ^ = 1 and hence * = p, . If P is on g , Proposition 5.17 can be used to show this. Assume that g is an affine line and that P I g , P f TT(g). Let r = P. Let f be a direction, t H-T\{g) f V, The line L(Q,2) meets g and hence is fixed by q. If S I L(Q,Â£), then S = L(Q,Â£) [\ L(S,D and hence vj_S = S. If R is a point of A, then R = L(R,P) A L(R,g). Observe that L(R,g) meets L(G,2) in a single point and hence Â«^L(R,g) = L(R,g). Thus, Â»^R = R for every point R of A. Hence ^ fixes all lines and
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171 10.35 Proposition . Let R,S' be AH-rings and let 6:H(S) Â—* H(R) be an incidence structure isomorphism. Then P A Q in H(S) c^> 6P A 6Q in H(R); g Ah in H(S) <^> 6g a Sh in H(R) and P 6 g in H(S) Â«=> 6?^fig in n(R.). Proof. By Lemma 10.7, P a Q in H(S) if end only if P is joined to Q by more than one line. Hence P A Q in H(S) <=> Â©P A. &Q in H(R). We use Proposition 10.12 frequently in what follows. Assume 0g Â£ 6h. Then there is a unique point Bp on both. Hence there is a unique point P on both g,h. Thus, g A" h. Thus g A h implies 6g A 0h. If P A, g, then there is a point Q I g such that P A Q. Thus 6P a SC and 6Q I Gg. Hence Sp A 9g. The proposition follows by symmetry. // 10.36 Definition . If H is an incidence structure; if S is an AH-ring and if 0:H Â— H(S) is an incidence structure isomorphism, then we say that H is a Hjelr.slev structure . We say P is near Q in H and write P A Q whenever 8P a Sq. We define g A h, P A g, g A P similarly. An incidence structure homomorphism between 4 Hjelmslev structures which preserves the nearness relation is called a Hjelmslev structure homomorphism . We denote the category of lined Hje l mslev structu res by (H ) . If H is a S g Hjelmslev structure, v/e call the incidence structure obtained by considering equivalence classes of near points and near lines the gross structure H* of H. We call k:H Â— *Â• H* defined by KP = (Q 1 Q A ?1 and so on, the neighbor map of H. We extend our notation (G. , , ,,g) in the obvious way to (G.,, >,g). IHis ; ,g ; J (H,g)

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172 10.37 Proposition . The nearness relation in a Hjelmslev structure is well-defined. Proof. This is immediate by Proposition 10.35. // Remark . If H is isomorphic to both H(S) and H(R), then, by Proposition 10.29, S is isomorphic to R. 10.38 Proposition . Let H be a PK-plane (Hjelmslev structure) and let P and g be a point and a line of H. If Q,Q' are points, Q,Q' / P,g (Q,Q' # P,g), then there is at most one (P,g)-automorphism of H which takes Q to Q'. Proof. Assume Â«,p are both (P,g )-automorphisms taking Q to Q'. -1 Let
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173 determined by its action on the affine points of (K,g). Two affine points (affine lines) are near in H if and only if their images are near in H'. Moreover,
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174 lined incidence structure isomorphisms n:A Â— Â» G , n':A' Â— G induced by tr and o"Â„,6' preserve and reflect A and ~, the neighbor relation and the nearness relation coincide in each of A, A'. Thus, since a preserves the nearness relation and since A' a ' Â°a is a 3 eneralized AH-plane homomorphism. Since
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175 Therefore ') is the inverse of H(^) and Â« is the inverse of k. Thus, oC^ is a generalized AH-plane isomorphism. // 10.42 Proposition . Let S be an AH-ring and let Z be a nonsingular matrix in S . Then the following are equivalent.

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176 1) o" z induces a generalized AH-plane automorphism * of G . Â£ s 2)
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177 10.45 Proposition . Let H,H'Â» be Hjelmslev structures; let S,S' be AH-rings and let 8:H Â— * H(S), 6':H* Â— * H(S') be incidence structure isomorphisms. A map Â£ b S US A ll^ G S'^ and let K S' K S ' be the obvious coordinatizaticns, then IT induces a coordinatization homomorphism tt':(A,K ) Â— > (A' K ). Then R(A,K s > = S, R(A',K ) = S' and we let ^ . R(rr'). Then tt takes r(l,x,y) to r ' (1 ,4>x,^y ) . Observe that tt induces a full lined Hjelmslev structure homomorphism tt :(H(S),h) Â— *Â• (H(S') h') where h = (1,0,0) s and hÂ» = (1,0, 0)s'. Since HCf) takes h to h ' and agrees with tt on the affine points of (H(S),h), H(
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178 TT agree on the affine points of (H(S),h); hence K(^) = it by Proposition 10.40. Thus, H(Â«f) =
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179 are not all near a single line k' in H(S*)", a contradiction. Thus, cr is full. // 10.46 Definition . Let A be an AH-plane. We say (C,X,E) is a basis triple of A if the images of the three points 0,X,E are not collinear in the gross structure of A. 10.47 Definition . Let S be an AH-ring. If A is an AH-plane and if there is a generalized AH-plane isomorphism Q:G (A) Â— G . g s we say (S,9) is a Klingenberg coordinatization of A. Once such a coordinatizaticn has been given, we usually identify a point P (line g) with its image G? (Qg). 10.48 Proposition . If A is a Desarguesian AH-plane, and if (0,X,E) is a basis triple of A, then A has a Klingenberg coordinatization (S,9:G (A) Â— *G e ) where S = EÂ„(A) is the rinq of trace preserving endomorphisms of the translation group of A such that 9C = r(l,0,0); 9x = r(l,l,0); 0E = r(l,l,l) and if S fE w (A); if T (ajb) (r(l,0,0)) = r(l,a,b) for all a,b in E W
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ISO since K = R(A,K ), there is a coordinatization K_ such that E W (A) = R(A,K E ); = C,0); X = (1,0); E = (1,1) and such that T (M) l ' T (l,l)Then V(S) * l and W ( r ,v) " T (V,Sv)Define 6: S n (A) ~~* G rai b ^ S ^ f y) = r(l,x,y). Then (E Â„(A>,&) is a coordinatization with the desired properties. // 10.49 Definition . If H is a Hjelmslev structure; if S is an AK-ring, and if 6:H Â— * '^ s ^ is an incidence structure isomorphism, we say (3,8) is a Klingenberg cocrdinatization of H. Once such a coordinatization has been given, we usually identify a point P (line g) with its image BP (Gg). 10.50 Proposition . Let A be a Desarguesian AK-plane. Then the automorphisms of A are (P,g)-, (T.g,,)and (P jC^J-transitive for every direction P, every line g and every point P. If (SÂ»6:G (A) Â— * G ) is a Klingenberg coordinatization of A, and if ~g i o( is a central axial automorphism of A, then there is a nonsingular matrix Z in S (with first column (v,C,0) for some v Â€ S\N) such that * is defined (in terms of the Klingenberg coordinatization) by <*(rx.) = rx 2 . l i Proof . Let A be a Desarauesian AH-olane and let 8:G (A) Â— * G ' ~g S be the morphism of a Klingenberg coordinatization of A. In H(S) the automorphisms of the form
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181 automorphisms of G are ( P,g) -transitive whenever any one of the following three sets of conditions holds: 1) P I h, g is affine; 2) P I h, g = h; 3) P is affine, g = h. Since 9:G (A) Â— G is ~g s an isomorphism, the automorphisms of A are (r,g)-, (T,g )and (P,g M )transitive. Let (S,0:G (A) Â— G ) be any Klingenberg coordinatization g o of A, and let <* be a central axial automorphism of A. Then
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182 to either 0Â£ or OE". Let g' e EE" and let I be a (TT(g ) ,k)-automorphism taking E" to E: such an automorphism exists by Proposition 10.50. Then Sr'(0' ) = O and St'(E') = E. Let X 1 " = St'(X'). Since I is an automorphism, O , E and X"* are not colin the gross structure. Let h fc XX'" and let * be a (TT(h),OE)automorphism which takes X 1 " to X. Then Â«h' takes O'.X'.E' to 0,X,E. // 10.52 Definitions . Let \:(5,g) Â— Â» (5,g) and X':(S',g') Â— * (S' ,g Â• ) be lined incidence structure embeddings. If et:(S,g) Â— *Â• (S',g') andcr:(S,g) Â— * (S',gÂ«) are lined incidence structure homomorphisms such that
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183 v e S* and y^ is defined by 0'G (u) = ( is. If u:A Â— > A' is an AH-plane homomorphism defined by 0'G UO = (cr^ ,H(vJ )+ 9 for some non-singular matrix 2', and some AH-ring homomorphism A" are respectively defined by 6'G U) = (
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184 X" to X' and E" to E Â• . Thus, if we coordfnatize A by K in such a way that (x,y) = r(l,x,y) and make a similar coordinatization of A', then e< = yS S t^. induces a coordinatization homomorphism *' = ((A,K ), (A',K ),f ). Let R(Â«Â«) = Â«f. Then since R(A,K ) = S, R(A*,K ) = S* by Proposition 10.11, we have that "f : S Â— S' is an AH-ring homomorphism, and that tf is defined by (r(l,x,y)) = H(f )r(l,x,y) . Thus, if we define A(f):A-Â»A' by A() is an isomorphism if and only if ^f is, we have that v* is an isomorphism if and only if *f is. If uj is a Kjelmslev structure homomorphism we here denote the appropriate associated lined Hjelmslev structure homomorphism by us : we only treat homomorphisms u> f or which this association is possible. Since the automorphisms of H(S') of the form 0" for some non-singular matrix Z are ( P,g)-transitive for every P,g by Proposition 10.29; since there is a unique ( P,g )-automorphism of A' which takes Q ^P,g to Â«jQ -f. P,g by Lemma 10.34, a/id since any ( X' 6'P, X' 0'g) -automorphism of (H(S'),h') (where V is the lined inclusion map) induces a ( P,g )-automorphism of A ' , any central axial automorphism r^ of A* can be extended to a (P,g)automorphism (for some P,g) of the form cr (where 2 is a nonsingular matrix) through \'Q' and X* 6' . Thus, t ' , S' and p' can be extended to
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185 through X9 and \'Â©Â» (where X is the lined inclusion map). Kence p. :A Â— * A can be extended to a: (H(S) ,h) Â— * (H(S') ,h') where cr = # # # a Â°" 2 ,(r z Â°" 2 H(Â«f) through X6 and X*&'. This extension ) )+ :G Â— Â» G is ~g / z'~ z'~ ' s s 1 a lined restriction of *:A Â— * V is 3*3 / * defined by Q'G U) = () ) + &, where we are here using G to g ' z ~ ~g indicate a construction in terms of G (A) and G (A') rather than ~g -g the functor G . We wish to show that p. is a non-degenerate AH-plane homomorphism or equivalently that G (>*) is a non-g / degenerate generalized AH-plane homomorphism. Since cr must take h' to h', the first column of Z' is (v,0,0) fc for some v e S' by Proposition 10.42. Since 6, 9' are generalized AH-plane homomorphisms, we wish to show that (^-.Ht^)) induces a

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186 generalized AH-plane homomorphism from G to G (this is immediate by Proposition 10.40 once we have shown that (er H(:A' Â— * A" are defined by 6'G (jO = -g r (oz ,HCf)) + e and 9"G g (v) = (
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187 and conditions (AH4) and (0A3) of the definitions of AH-plane and affine plane, A has a Klingenberg ccordinatization (S,0) for some AH-ring S. Assume that A has a Klingenberg coordinatization (S,9) for some AH-ring S. If 9:G^ -+ (H(S ) , (1 , C , ) s) is the lined inclusion map, then A can be extended to H(S) through AÂ©. Assume that A can be extended to (H,g) through to and that &:H -* H(S) is an isomorphism. Let Â»|:H(S) Â— * H(S) be an isomorphism such that)|.e(g) = (1,0, 0)s: one could construct n by letting (G',G") be a full 2-vertex whose points are on&g and extending (G',G") to a full 4-vertex and letting ^ = ofor an appropriate non-singular matrix Z. Then S:G (A) Â— > G induced ' -~g S v. #*# Â• by n io is an isomorphism, and, since E.(A(G J) is isomorphic to S by Propositions 9.10 and 10.11, E (A) is isomorphic to s. // 10.55 Proposition. Every Desarguesian PH-plane is a Hjelmsl? structure. Also, any full lined Hjelmslev structure hcmomorphism reflects nearness. Proof . The first statement is immediate by the definitions. The second statement follows from Theorem '2.63, and Proposition 10.13. // 10.56 Proposition . Let A be a Desarguesian AH-plane. Let (S,8) be a Klingenberg coordinatization of A. Assume A can be extended to a Hjelmslev structure H through u>:G (A) Â— * (H a)~g Â»'

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188 then, there is an isomorphism ^: (H,g) Â— * (H(s) , (1 ,0,0)s) such that |Â«P = 9P for every point P of G (A) and ^wk = <3k for every line k of G (A). In fact, n Â« = \ B where Xrx . = rx . V rx . in G (A) g < i l i -vq Proof. Since H is a Hjelmslev structure, there is an incidence structure isomorphism \$:H Â— * H(S') for some AH-ring SÂ». Let Â©Â• = & . There is an automorphism j^p. Similarly, Gk = huik for every line k of G (A). // g 10.57 Proposition . ' If j*:A Â— * A' is a non-degenerate AH-plane homomorphism between Desarguesian AH-planes; if A can be extended to a Hjelmslev structure H through w:G (A) Â— Â» (H,g) and if A' can be extended to a Hjelmslev structure H' through uo' :G (A') Â— * (hÂ» g') then there is a unique extension cr: (H,g) -* (HÂ»,gÂ») of yu through w and ui 1 , and cr is full; moreover, ois an isomorphism if and only if u is.

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189 Proof. This follows from Propositions 10.53 and 10.56. // 10.58 Proposition . Let A be a Desarguesian AH-plane, and let w: 2 (A) Â— " < H ig) fa e an extension of A to a Hjelmslev structure. The map Tt:(t<*l.e< is a non-degenerate endomorphism of A],o) Â— y (toI ois a full endomorphism of (H,g)3,Â°) defined by t\( and to, and if cr is an automorphism, then * is also. Then -uU) =
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191 G (A). Then 6 is a generalized AH-plane homomorphism, since each of 9^ and (Â«r 2 ) is a generalized AH-plane homomorphism. Hence (S,8) is a Klingenberg coordinatization of A. // 10.61 Lemma . Let S be a ring. The non-singular matrices in S 3x3 whose fi rst columns have 2 = 1 and Z = for j = 2,3, form a group under composition. // 10.62 Definition . Let S be an AH-ring. We denote the set of non-singular 3x3 matrices over S whose first columns are (1,0, 0) t * Remark . Lorimer[in (1971), page 155, Theorem (5.5.2) and ( 1973), page 15, Theorem 5.1l] states a theorem for a Desarguesian AH-plane A with a fixed coordinate frame lo,x,Y} which corresponds to what Artin [(1957), page 83, Theorem 2.26] calls the fundamental theorem of projective geometry. Lorimer's theorem concerns the automorphisms of A which fix the origin (Aut A) q , a map from (Aut A) q into a set of semilinear transformations and a surjective group homomorphism from (Aut A) to o (Aut E W (A)) whose kernel is (G. L. A) . The fundamental theorem of projective geometry deals with bijective homomorphisms. Lorimer's theorem together with Theorem 2.26 in Artin (1957)3 and our Proposition *10. 53 inspired the following theorem. 10.63 Theorem. There is a functor Y:(D) n Â— R from the category of Desarguesian AH-planes with non-degenerate AH-plane homomorph-

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192 isms (D) to the category of AH-rings R such that, if ^:A Â— * A* is a non-degenerate AH-plane homomorphism between Desarguesian AH-planes A, A', and if (5,6) and (S',9') ace canonical coordinatizations of A and A 1 respectively, then ^ is defined by /*(r-(l,x,y) ) = r' Cl,y(^*)x,Y(u)y)Z' for some non-singular matrix Z' in ( S') such that the first column of Z" is (1,0, O) 1 . if ^ is defined by y*(r(l f x,y) ) = r Â• (l,Â«?x f S' and some non-singular matrix Z, then there is an e fc S'\W such that U? is defined by
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193 from G s to G g , , and that Z' Â« ^ . By Proposition 10.40, ^Z-^ = ^Z'-H 5 ' Then, for some a,b,c Â£ S', (1,0, 0)Z = (a,0,0)ZÂ«; (1,1,0)2 = (b,b,0)Z' and (1,1,1)2 = (c,c,c)ZÂ». Since Z ' Â€ *^S' and since the fir st column of Z is (v,0,0) for some v 6S', a = b = c = v and v -1 Z = Z'. Thus, since r(l,^x,C) (vZM = r(l,^x,0)2Â« for every x 6 S, we have that f v ^x) = r^x for every x Â€ S; hence p^ = rj. Define Y(S,6',u) to be f, and let Z^,^. ZÂ». We wish to show that Y is independent of the coordinatizations. Let (S,6 ) and (S',8,,) be our choosen fixed A A canonical coordinatizations of A, A': see Definition 10.59. Construct Y( 6 , 9 ,u.) as before. Recall that 0(P) = 6 (P)z A A J A 9 (where Z g is a non-singular matrix in 2 ) for every point P of G g (A). Similarly, Â©Â»(PÂ«) = Â© A ,(PÂ«)Z &I where Z & , Â£ J g| . Thus, Q'(^P) = (H(Y(6, 6',^))(rx i ))Z = 9 ,(/*P)2 . Thus, & A ,(^P) = (H(Y( e , e ., / .)) ( rx.))Z 9>6l ^Z G r 1 (H(Y( 0A ,0 A( ^))(rx i ))2 OA ^ A( ^. Hence since Z^^-," 1 * ^,, Y(G,9*,^) = X^Â»Â» e Â» l Â»/'Ji and (using canonical coordinatizations only) Y is independent of the coordinatization. If yu e [a,A'1 ^ (3)" define Y(^) to be Y( 6 , 9 a , ,^) . If A is a Desarguesian AHplane, define Y(A) = E (A) , the ring of trace preserving endomorphisms of the translation group of A. Obviously, Y(l ) = Let j*zk Â— *Â• A' , n):A' Â— * A" be morphisrns in (D) n . Then if P is a point of A with 8.(P) = rx . , then & Â„(vi*P) = (H(Y(v^))(rx.))Z H(Y(\>vO)(rx, ))Z n Â„ = 9 v(9 ) -1 (up) -

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194 (H(Y(v)) ((H(Y(vJ) (rx.))Z ))Z ~~r x G A ,e A ,, r e A ,,d A Â„,v (H(Y(v))H(Y(^)Hrx.))Z = (H(Y(v)Y(ii) ) (rx. ) )Z where Z = ((Y(tf)Z )Z. . . Since Z,Z^ n Â€ ^ , , , they are equal by our earlier argument. Hence Y(vlJ = Y(v)Y(u). Thus, Y is a functor. One can easily show that Y is full and representative by using Propositions 9.10, 10.11 and 9.11. We wish to show that Y is not faithful. Let A be a Desarguesian AH-plane. Define /^:A Â— *Â• A by 0\.O*P> = (6 (P))Z where Z 10 1 10 Then Z 6 3y(a) and Y(^) = ly(A)* But /*" ^ 1 A 5 hence I is not faithful. If ^:A Â— A 1 is a morphism in (D) ; if (S,&), (S',0') are canonical coordinatizations of A, A', and if y^ is defined by ^(r(l,x,y)) = r ' (l,fx,M*y)Z for some AH-ring homomorphism
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195 of Z, 0' , in terms of 2*: see Definition 10.59. Let Â«. be the identity map on A. Then in terms of the coordinatizations, eyP = (9P)Z" for some Z" Â£ 2 ( ., and for all affine points P. W Also, GP = G A (P)Z and 9'P = Â© (P)Z'; so that (P)Z' = (6 A (P))ZZ". Thus, 9 (P) = (e^(P))ZZ"(Z')~ 1 . Observe that ZZ"(Z')~ and Z" each take (1,0,0) to (1,0,0), (1,1,0) to (1,1,0) and (1,1,1) to (1,1,1) since each is in ^ . ,. Hence ic, A A; -1 w Z" = I = ZZ"(Z') where I is the identity matrix. Thus, Z = Z', G . &'. // 10.65 Example . We now give an example of a non-degenerate affine plane homomorphism which is not an epimorphism. Let F be a field and let K be a finite separable extension field. Let 9 be a non-trivial automorphism of K which leaves F fixed. Then R'(F) and R'(K) are Desarguesian affine planes and R'(X):R'(F) Â— > R'(K) (where X:F Â— K is the inclusion map) is a coordinatization homomorphism as is R'(0):R'(K) Â— > RÂ«(K). Observe that R'(l ) 4 R'(G), but that R'(9)R'(X) = R'(lÂ„ )RÂ»(X). Thus, RÂ«(X) is not an epimorphism in the category of Desarguesian affine planes. Since R'(X) takes (0,X,E) of R'(F) to (0,X,E) of R*(K), R'(X) is non-degenerate. 10.66 Theorem . There are reciprocal equivalences D:((H ) ) f -* (b) n and D':(D) n Â— ((H) ) f where (D) n is the i> g ~ S g category of Desarguesian AH-planes with non-degenerate AH-plane homomorphisms and ((H) ) is the category of lined Hjelmslev i> g structures with full lined Hjelmslev structure homomorphisms.

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196 Proof. Observe that (D) n and ((H) ) f are cateaories by S g * Corollary 2.64 and Theorem 2.63. For each Desarguesian AH-plane A, let (S,& ) where S = A E v/ A ^ be our chocsen fixed coordinatization of A (see Definition 10.59), and define D 1 (A) to be (H(S) , (1,0,0) s) . If^:AÂ— *AÂ» is a morphism in (D) , let S = E, ,(A) and S' = E (A'), and let D Â• GuJ w w i be the unique extension of u. through X G :G (A) Â— * D'(A) and S A * g ~ ^S ,6 A' : Sg (A,) Â— *D'(A') where X s and X are the lined inclusion maps. For each lined Hjelmslev structure (H,g), let D(H,g) = ~l| (G (H,g)' g} * If ^' AH ^ ) -* CH',gÂ«) is a morphism in ((H) ) f , let D(ui):D(H,g) Â— Â»D(H',gÂ«) be defined by restricting f to the affine points and lines of (H,g). Observe that D and D' take objects (morphisms) into the desired categories, and that D and D* are functors. Define S:1 A Â— Â» DD' by S A = A^(fr )j then I is an isomorphism since 9 A is. Let f.A Â— * A 1 be a morphism in (D) n ; let S = E W (A) and let S' = \, (A '>Then, X S ,6 A ,G (^) = D'(^)X 9 ; hence ~ll (e A' 5 g ( ^ )) = ^'fy^VV' and thus VV^ = DD Â• (ja) A < 6 ) . Thus, ^> AI/ h = 22.'^/ A ^Â° A Â» and ^ is a natural isomorphism. For each lined Hjelmslev structure (H,g), let 1(H q) : ^ H ' g ' 1 ~~ V ^-^ S 'Â» d> Â»Â°^s) be a lined incidence structure isomorphism such that, if A * D(H,g) and S = E (A), then ~ W -S 6 A = 1(H,g)\ Where X s'' G S ~^ (H',g') be a morphism in ((H) ) f . Let S g

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197 n = 1( H ,g)' *!' " 1(H',g')* A ' 2 (H <5>, A< = fiCH'.g'), S = E^A) and S' = E (A"). By the definition of D', we have that X qA.S < D( a g ~ ~~ b A o S inclusion maps. Then, since X 6, = hX snd XÂ„,Â£\, = n'X,,, v;e have that r^'(
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198 composition and has the natural identities. Let (H,g) be a lined FH-plane, and let k be an affine line of (H,g). Let P be an affine point on k, and let Q = k g. Define a map
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11. DESARGUESIAN PH-PLANES In this section we give several sets of necessary and sufficient conditions for a PH-plane to be Desarguesian. * 11.1 Remark . Klingenberg claims to have shown ["(1955), page 106, S 25 J that if H is a PH-plane, and if one of the AH-planes derived from H is 'Klingenberg Desarguesian', then H is Desarguesian. His argument breaks down however. If one has a lined uniform Desarguesian PH-plane (H,g), if Pig and if one removes all the points and lines neighbor to g and replaces the class of points neighbor to P (and the associated incidences) by a non-Desarguesian affine plane of the appropriate order, one can use Artmann's construction [(1969), pages 130-132"] (see Bacon [(to appear), Theorem 2.l3, Artmann [(1969), page 134, argument for Satz 2], Drake [(1968), page 203, Theorem 5.2]) to construct a PH-plane H' such that A(H',g') is equal to A(H,g): this is a counterexample to Klingenberg ' s assertion [(1955), page 107, lines 10-12] , "Mann erkennt nun unmittelbar ,. dass die vorstehenden Zuordnungen umkehrbar eindeutig sind und dass dabei die Inzidenzrelation durch (6) beschrieben wird." Thus, Klingenberg has failed to prove S 25 [(1955), page 106]. We could, for example, let H = H(Z [x]/(X )) and construct a (non-Desarguesian) 'derived' affine plane of order 25 in the 199

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200 manner indicated in Hughes' and Piper [(1973 ) , pages 202-205], In this section we prove (Proposition 11.4) that if H is a PH-plane and if there are three derived AH-planes which are Desarguesian and whose affine point sets cover H, then H is Desarguesian. 11.2 Lemma . Let H be a PH-plane. Let *g and "g be non-neighbor lines of H. Let the associated derived AH-planes 'A = A(H, 'g) and "A = A(H,"g) be Desarguesian. Then, 'A and "A are protectively Desarguesian and there is an H-ring R such that both 'A and "A are isomorphic to the AH-plane generated by R. If 'K is a coordinatization of 'A with AH-ring S and with 'g = "g, then there is a coordinatization "K of "A with AH-ring S and with "9 X = *g, "g = '9 x and "E = Â•Â£ such that Â»(l,y) = "(y,l) for every y in S. Then, S is an H-ring. If Â«K and "K are any coordinatizations of *A and "A such that 'g = "g, "g = 'g, y x "g v = 'g Â» "E = 'E, 'M = "M = S (the symbol sets are both equal y * to S), and '(l,y) = "(y,l) for every y in S, then '[m,d] = " [d,mV for all m,d in S and if P is a point of 'A such that P + Â»g an d P = '(x,y), then x I N and Â»(x,y) = " (x -1 y,x -:L ) . Proof . Assume *g and "g are non-neighbor lines of a PH-plane H such that 'A = A(H,'g) and "A = A(H,"g) are Desarguesian. Then there is a coordinatization 'K = ( 'g , 'g , *E ,R, '* : '0 'E Â— R) of 'A such that *q = "q. y Assume we have such a coordinatization 'K. Let "O = 'g H 'g and "E = 'E. Then one can easily see that there is a

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201 coqrdinatization "K = ( "g^ Â»g Â» E| R, "\$ : Â»0"E -^ R) of "A such that "9 X = 'gÂ» "g y = 'g x and ' (l,y) = "(y,l) for every y in R. Then CA,'K) generates an AH-ring 'S = (R, + ,X) and (*'A,"K) generates an AH-ring "S = (R,Â»,-). As usual, we write axb as ab. Assume we have coordinatizations 'K and "K. Let O = 'g Â»g and E = Â«E. Since P I OE is neighbor to 'g if and only if the 'y-coordinate of P is in ' N (and if and only if the "xcoordinate of P is in "N) , we have that n fc 'N <-* n 6 "N. Let N = 'N. If x 6 R\N, x has a multiplicative inverse x -1 in 'S and a multiplicative inverse x in "S. We wish to show that -1 -2 -2 x = x Here we are using x for notational convenience and -2 o do not mean that x is the multiplicative inverse of "x ; " in 2 fact, "x " is undefined in this context. By construction, 'tm,0l joins '(0,0) and *(l,m). Also Â•(m) = '[m.Ol A 'g = "L( Â» (m,l ) , Â» (0 ) ' ) r\Â»[0,0] = Â»(m f 0). .Since interchanging the roles of 'g and "g is the same as interchanging the roles of the xand y-axes in each of the coordinatizations; we have by a symmetric argument that "(m)' = '(0,m). Thus, Â•[m,bl = Â«(m) v Â«(0,b) = Â»(m,0) v "(b)= Â»[b,m]'. Since '[m,b] = "[b,m]' and since '(l,y) = "(y,l) for all m,b,y Â£ M, and since OE meets , [m,b'\ at '(l,m + b) and "[b,m]' at "(m Â• b,l), we have that m + b = m * b for all m,b in R. Let x Â£ R\N. Then, Â• (x,l) = '[x -1 ,Â©"! f\ '[0,l] = "[0,x -1 ]' A "UfOl* = "(x" 1 ^" 1 ). Using this, we have '(x,0) = C(x,l) v 0) f\ '[0,0l = ("(x -1 ^ -1 ) v 0) fl "[0,0]' = *'[0,x J (\ "[0,0]' = "(0, x ). Using the symmetric argument, we have that "(0,y) = '(y~ 2 ,0). Hence, '(x,0) = "(O^ -1 ) = -1 -2-15 '((x ) ,0). Thus, x = x for x 6 R\N.

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202 If x e R\N, then Â• (x,y) = '[0, x ]' f\ ' [0,y] = "[0,x ] A "[y,0]Â» = "( x ~ -yjx" 1 ). By the symmetric argument, "(x,y) = ' (y~ ,y~ x) . We next -wish to show that a>b = ab for all a,b e M. Let a e R\N. FroT. our relations above we have '(x f y) = "(x -y,x ) = '(x,x(x .y)). Thus, y = x(x~ .y). Letting x = a" 1 , y = b, we obtain b = a (a.b) and hence ab = aÂ»b when a <Â£ N. If a e N, then (a 1),1 j N and a-b = ( ( a 1) + 1 ) b = (a l)-b + b = (a l)b + lb = (a 1 + l)b = ab. Hence, a.b = ab for all b 6 R. Thus a.b = ab for all a,b in R. In order to prove that (R,+,X) is an K-ring, we need only show that given any n,n G R there is a k Â€ R such that kn = m or km = n. The lines 'g and '[m,-n] must intersect in (affine) points of 'A or "A since a point is on g only if it is in 'A or "A. If P Â£ '[0,0] A ' [m,-n] is in -A, then P = '(x,0) for some x, and hence = xm n; hence xm = n. If Q Â£ "l0 t 0")' H "[-n.m]' is in "A, then C = "(0,y) for some y and hence = (y)(-n) + m; hence yn = m. Thus, (R,+,x) is an H-ring. Our previous arguments showed that (R,+,x) = (R,Â»,-). // ' Remark . The argument that the AH-ring (R,+,X) is an H-ring was adapted from Klingenberg t(1955), page 107, the argument for S 26] 11.3 Definition . Let H be a PH-plane. We say Q = (P,P* ,P";s,s' ,s") is a triangle if P,P' I s; P',P*' I s ' and P",P I s". Vie say each of P,P',P" is a vertex of Q and that each of s,s',s" is a side of Q. We say Q is a full triangle if the

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203 images of its vertices and sides are all distinct in the gross structure of K. 11.4 Proposition . Let H be a PH-plane, let F be a full triangle with sides Â»g, "g and "'g, and let 'A = A(H,'g), "A = A(H,"g) and '"A = A(H,'"g) be the derived AH-planes. If Â»A, "A and '"A are each Desarguesian, then H is Desarguesian and the H-ring of H (which is unique up to isomorphism) is isomorphic to the rings of trace preserving endomorphisms of the translation groups of each of 'A, "A and *"A. Proof . Let F be a full triangle with sides 'g, "g and M, g. Let E be a point such that E Â•/'g/'g,'*^; it is easy to show that such a point exists by using the gross structure. Let 'K be a coordinatization of 'A with AH-ring R such that 'E = E, 'g = "g and 'g = "*g. By Lemma 11.2, R is an H-ring and there are coordinatizations "K and ,M K with H-rings R such that "E = 'E, "g v = 'g, "g v = 'g ; ,m e = -e, -g = Â» g , g = Â» g and such A j * x y x that if P ^ 'g,"g, P = *(x,y), then x 4 N and *(x,y) = "(x~ y,x~ ) and if Q f Â»g, "Â»g, Q = "(x,y), then x <Â£ N and "(x,y) = m (x" 1 y,x" 1 ). We wish to show that if P f ' M g,'g, P Â« '" (x,y); then x ^ N and Â•Â•Â• (x,y) = ' (x y,x~ ). We first show that the points on "OE which are in both the affine point set of "'A and the affine point set of 'A have the property that Â•Â•Â• (l,y) = '(y,l). Assume x,y Â£ N; then '"(x,y) j. 'g^'g.-g. Let '"(x,y) = Â»Â•Â• (a'-Va" 1 ) . Then a = y~ , b = y~ 1 x, and Â•Â•Â• (x,y) = "(y~ 1 ,y" 1 x) . Let

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204 "(y~ Â»Y~ x) = "(c~ ci,c~ ). Then c = x _1 y and d = x" 1 ; so that "(y~ ,y~ x) Â« '(x~ y,x" ). Thus, 'Â• (x,y) = '(x^y,*" 1 ), and for x = 1, ,M (l,y) = *(y,l) for y <Â£ N. Let n Â€ N. Let k = ,M [n,dl for some d Â€ r. i n "'A, "'(x,y) I '"[n,dl Â«=> y = xn + d. In 'A the line k meets "'g in the same point that the line h = '"[n,0] does. When a,b \$ N, "'(a,b) = Â• (a^b.a" 1 ) . Thus, when x,d Â£ N, '" (x,xn + d) Â« Â• (n + x~ d,x~ ); thus in 'A, k = '[d,nVÂ« The line h goes through '"(0,0), '"(l,n) and Â• (n,0). Thus, h = Â•C0,n3' and '(n,l) I h. Thus, h A "OE = "'(l,n) = '(n,l). By Lemma 11.2, if P ^ m g,'g and P = '"(x.y), then *"(x,y) = '(x y,x ), and the relationships between the three coordinatizations 'K, "K and ,M K can now be treated in a cyclically symmetric fashion. We wish to construct incidence structure embedding maps X: S Â— * H(R) where S in the incidence structure of 1 A such that if P,k are in A,^A, then H P = hp and hk = hk. Define 'X by 'X'(x,y) = r(l,x,y); "X by "X"(x,y) = r(y,l,x) and "'X by ,M X M, (x,y) = r(x,y,l) for all x,y in R. Let h be a line of H(R). Let A (H(R) denote the incidence structure obtained from H(R) by removing h and all the points and lines neighbor to h from K(R). By Proposition 10.11 and symmetry, 'X is an embedding of Â£ into H(R) and onto A (H(R),(l,0,0)s) 5 "*' Â° f " S onto A (H(R),(0,l,0)s) ; " ,X Â» Â° f '" S Â° nt Â° A (H(R),(0,0,1) S )Let X:H Â— * H(R) be defined by XF = XP whenever P is an affine point of A. We wish to show X is well-defined. Let P be a point in 'A and in "A. Assume P = *(x,y). Then P =

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205 "(x y,x ) and 'XP = "\P. The other cases follow by symmetry, and X is well defined on points. Let k be in 'A and in "A. Since 'A'Crr^d] = (d,rr,,-l)s; Â»X'tu,v"]' = (v,-l, u )s; "X"U,d] = (-l,d,m)s and "A"[u,v"]' = (u,v,-l)s, and since k = '[m,d] <^> k = "[djirQ', there are two cases. Assume k = '[n\,d]. Then 'X'C m Â»d] = "X"[d,m"l , Â« Assume k = 'Lu,v]', u 6 N, and k = "[m,d], m e N. Let P = k Â»g = k ft "g . Then P = '(v,0) = "(0,d): x ^y Â» Â» hence v Â£ N and d = v~ . Let Q = k P\ ' [l,ol = k Pi "[0,l"]'. Then Q = Â»(v(l u)~ 1 ,v(l u) -1 ) = "(l,m + v -1 ). Thus, Q = "(1,(1 u)v~ ); hence m = -uv~ . Observe that 'X'CujV ] 1 = "X"C-uv" ,v~ ]. Thus, X is well defined. Assume P I k in H. Since P is not neighbor to one of 'g, "g, ,M g; say g, both P and k are in A. But P I k in 1 A implies XP I \k in H(R); hence \ preserves incidence. Assume ^P I Xk in H(R). Since Xp is not neighbor to one of 'h = (1,0, 0)s, "h = (0,1, 0)s and m h = (0,0, l)s; say h, then both XP and Xk are in A (H(R) i h). But ^ P I ^ k in A (K(R) Si) im P lies P I k in H; hence X reflects incidence. Observe that any point (line) of H(R) can be written with at least one of the x.'s (y.'s) equal to 1 (-1); hence X is surjective on points and lines. Assume XP = \Q, P 4 Q. Then P, Q are not in the same A. Assume P is in *A; Q is in "A. Then P = '(x,y) where x 6 N. Thus, if XP = rx . , then x 6 N, a contradiction to Q being in "A. The other cases are similar by symmetry; hence XP = XQ implies P = Q. Moreover, since X preserves and reflects incidence, X is injective on lines. Hence X is an isomorphism and H is isomorphic to H(R) where R is the H-ring of ('A,'K). Thus, H is Desarguesian.

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206 The remainder of the preposition follows easily from Proposition 9.10 and Corollary 10.30. // 11.5 Definition . Let H be a PH-plane. We say H is Hjelmslev Desarguesian if each derived AH-plane A(H,g) is Hjelmslev Desarguesian. Remark . Obviously, 'Hjelmslev Desarguesian' can be defined directly in terms of geometric properties of the PH-plane itself. 11.6 Theorem . Let H be a PH-plane. The following are equivalent. 1) H is Desarguesian: this is an algebraic characterization. 2) The automorphisms of H are (P,g)-transitive for every point P and every line g: this is a characterization in terms of morphisms. 3) There is a full triangle Q in H such that the automorphisms of H are ( V,s )-transitive for each vertex V of Q and each side s of Q. 4) There is a full triangle Q in H such that each of the three derived AH-planes A(H, s), where s is a side of Q, is Desarguesian: this represents an algebraic characterization (see Proposition 9.10, 2) <& 4) <Â£> 5)). 5) Every AH-plane derived from H is projectively Desarguesian. 6) H is Hjelmslev Desarguesian: this is a geometric characterization.

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207 Remark . A full triangle is a triangle whose image in the gross structure is 'non-degenerate.' Proof . 1) =^ 2). If H is Desarguesian, the H is isomorphic to a Hjelmslev structure H(R) for some H-ring R. By Proposition 10.29, the automorphisms of H(R) are (P,g )-transitive for every point P and every line g. / 2) =^ 3). Obvious. / 3) =^4). Let Q be a full triangle such that the automorphisms of H are ( V, s )-transitive for each vertex V of Q and each side s of Q. Let s be a side of Q and let A = A(H,s) be the AHplane derived from (H,s). There is a point P in A which is a vertex of Q. Let h and k be the sides of G which contain P. The automorphisms of H which fix g induce automorphisms of A. Hence the automorphisms of A are (P,gÂ„Â«,)-, CTT(k) t gÂ„)and ttT(h) ,g^)transitive. Since h + k; TT(h) * TT(k). Hence by Proposition 9.10, A(H,s) is Desarguesian. / 4) =^1). Immediate by Proposition 11.4. / 5) -=^> 4). Obvious. / 1) =^> 5). Let g be a line of H. Since H is Desarguesian, there is an H-ring R and an isomorphism 9:H Â— Â» H(R). By Proposition 10.29, if g is a line of H, (G. . . ., g) is iso(H(R) , g) morphic to A , the AH-plane generated by R. Hence A(H,g) is isomorphic to A , and by Proposition 9.10, A(H,g) is protectively Desarguesian. / 5) =^6). Immediate by Proposition 9.10. / 6) -=5> 4). Immediate by Proposition 9.10. //

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12. PAPPIAN CONFIGURATIONS In this section we define 'Pappian configuration* and use this definition to give several sets of necessary and sufficient conditions for a coordinatized AH-plane to be Pappian and (T.g.JH-Desarguesian for every direction T. We use the results of the last section to give sets of necessary and sufficient conditions for a PH-plane to be Pappian and Hjelmslev Desarguesian. 12.1 Definitions . Let A be an AH-plane. If Â£1,1 are directions in A, we say an ordered set of points and lines

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209 * Figure 12.1 . Let H be a PK-plane, and let g,g',g" be the sides of a full triangle in H. If pÂ» i s a point on g", we say that H is (gtg 1 > P 1 'Â» p "Â»g">P_appian whenever A(H,q") is (g,g',P ',P")~ Pappian. We say H is Pappian if it is (g,g',P ',P",g")Pappian for all g,g ' ,gÂ»,P ;L ' ,P" such that g,g',gÂ» are the three sides of a full triangle; P" 4> g,g'; P" I g"; P ' Â± g,g"; P 1 I g'. Let J be a full triangle in a PH-plane H. We say H is Pappian for the full triangle J if H is (g,g',P ' ,P",g")and (g'ig.P^P-'.g'O-Pappian for all ? 1 Â• I g'; P Â• 4. g,gÂ»; for all P x I g; P 1 ^ g',gÂ» and for some P" I g"; PÂ» + g, g Â« whenever

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211 AH-plane with prequa siring .(M, + ,*,-). Let b Â£ M\N; a,c Â£ M, then the following are equivalent. 1) C is ((a),(c)lC0,o] ',[0,o],(-b,0),(l))-Pappian. 2) (ba)c = (bc)a. Remark , If b = 1 , then 2) implie s ac = ca. Â• Remark . Our proof of 1) ^ 2) below is an adaptation of an argument of Klingenberg ' s [(1954), page 399, s 5.10] obtained by letting g Â« [l,l~l; P Â» (0,a) j g Â« g ; gÂ« . g . if our thuslyx a y x J coordinatized AH-plane Â€ is axially regular, then P Â• = (-a,0): a * a P a+b = (0,a + b); P ab = (0 ' ba) 5 h = [a, a] and so on. Proof . 1) :zÂ£ 2). Observe that (0,0) = [0,0]Â» (\ [0,o]; (-b,0) + (0,0); (1) -/. g ,g and (a),(c) -h g . Assume C is y x r y ((a), (c)l g Â»9 X Â» (-b,0),(l) )-Pappian. By Proposition 12.2, there is a unique ((a),(c)\g ,g , (-b,0 ) , ( 1 ) )-Pappian configuration F. y x By construction, P Â± = (0,(ba)c) and Jc = [a,(bc)a]. Consequently, (0,(ba)c) I [a,(bc)al, and hence (ba)c = (bc)a. / 2) =} 1). Assume (ba)c = (bc)a and that F is a ( (a) , (c)\ g Â»9 X Â» (-b,0) , (1) )-Pappian configuration. By Proposition 12.2 and construction, P^ = (0,(ba)c) and k = Ta,(bc)a]. Thus, P Â± = (0,(bc)a) and hence P I k . // 12.4 Corollary . Let C be an axially regular coordinatized AHplane with prequasiring (M,+,x,Â«), and let u e. M\N. The following are equivalent.

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212 1) C is < 2). Since u ^ N, (u)' = (m) for some m & M, and um = 1. By Proposition 12.3, urn = mu if and only if C is ((u),(u)Mg ,g , (-1,0) , (1 ) )-Pappian. / Y X 2) ^3). Assume there is an m eM such that um = mu = 1. Then 3) follows by (VW9s). / 3) =* 2). Let x = 1. // 12.5 Proposition . Let (M, + ,x,0 be a prequasiring, and let (M,x) be abelian. Then (M, +,*,Â•) is a quasiring, and, if a Â£ M\N, then there is a unique element a~ of M\N such that (xÂ«a)a = (xa ). a = (x.a )a = (xa).a" = x for every x fr M (including x = 1). Proof . Since (M,x) is abelian, (M,+,x) satisfies both distributive laws; hence by Lemma 9.7, (M,+,x,-) is a quasiring. If a Â€ M\N, there is a unique element a" such that a 'a = a'O + 1. Thus, a'a = aa* = 1 and hence a' Â£ N. By (VW9s), (x-a*)a = (xa)-a' = x = (x-a)a' = (xa')-a. // 12.6 Lemma . Let (M,+,x,-) be a prequasiring. The following are equivalent. 1) (ba)c = (bc)a for all a,c 6 M and all b e MSN. 2) (M,X) is associative and abelian.

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213 Proof. 1) =\$> 2). Assume (ba)c = (bc)a for all a,c Â£ M and all b Â€ M\N. Let b a 1; then ac = ca for all a,c e M, and hence (M,x) is abelian. Since (M,x) is abelian, both distributive laws are satisfied in (M,+,X), and by Lemma 9.7, (M,+,X,Â«) is a quasiring. Let a,c e K; n c N. Then 1,(-1 + n) Â€ M\N; so that (na)c z ((1 + (-1 + n))a)c = (a + (-1 + n)a)c = ac + ((-1 + n)a)c = ca + ((-1 + n)c)a = (c + (-1 + n)c)a = (nc)a. Thus, (ba)c = (bc)a for all a,b,c Â£ M. Hence if a,b,c e M, then a(bc) = (bc)a = (ba)c = (ab)c, and (M,x) is associative. / 2) =5> 1). Assume (M,x) is associative and abelian. Then (ba)c = Mac) = Mca) = (bc)a for all a,b,c in M. // 12.7 Proposition . Let C be an axially regular coordinatized AH-plane with prequasiring (M,+,X,Â«). The following are equivalent. 1) C is (g ,g , (z ,0 ) , ( 1 ) )-Pappian for every z e MSN. y x 2) (M,x) is associative and abelian. 3) (M,x) is associative and abelian; (M,+,x,.) is a quasiring and aq = a-q for every a in M and every q Â£ M\N. Proof . 1) =^ 2). Assume C is (g ,g , (z,0) , ( 1 ) )-Pappian for y * every z e M\N. By Proposition 12.3 and Lemma 12.6, (M,x) is associative and abelian. / 2) => 3). Assume (M,X) is associative and abelian. By Lemma 9.7, (M,+,X,Â«) is a quasiring. Let q 6 M\N. Then there is an m such that qm = 1. But qm = mq and hence by Corollary

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214 12.4, there is a q 1 such that (x-q)q' = (xq')-q = (x.q')q = (xq)-q' = x for all x Â£ M. Observe that if a Â£ K, then (aq)q' = a(qq') = a. Thus, ((aq)q'j-q = aq = a-q for all a e M. / 3) =^ 1 ) . Assume (K,x) is associative and abelian. Then by Lemma 12.6 and Proposition 12.3, C is ((a),(c)lg ,g ,(z,0),(l))_ y x Pappian for every z e M. // 12.8 Proposition . Let (A*,KÂ«) be a coordinatized translation affine planeLet (!, + ,*, Â•) be the quasiring of (AÂ»,K*). The following are equivalent. 1) (A',KÂ») is (g ,g ,(z,0),(l))-Pappian for every z / 0. y x 2) (M,x) is associative and abelian. 3) (M,x) is associative and abelian and (M,x) = (M,-). 4) A* is Desarguesian and Pappian. 5) H Â»(A*) is Desarguesian and Pappian. Proof . 1) =Â£ 2). Immediate by Proposition 12.7. / 2) =\$> 3). Immediate by Proposition 12.7 and aO = a-0. / 3) -=5> 4). Since (M,x) = (M,Â«) is associate and since both distributive laws hold, (M, + ,X,-) is a kernel quasiring and by Proposition 9.10, A* is Desarguesian, and (K,+,x) is isomorphic to E (AÂ»). Let F be a (Cl,il g,g ' ,P ' ,T )-Pappian configuration in A*. Let K be a coordinatization of A* with g = g; g = g' and = y x P Q . Then R(A*,K) = (S,+,x) is isomorphic to E (AÂ«) by Proposition 9.10, and hence has abelian multiplication. Observe that (S,+,*,X) = XÂ»(R(A*,K)) = Q(A*,K) is the quasiring of

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215 (A*,K), and hence that (yx)z = (yz)x for all x,y,z 6 S. For some a,c Â£ S, Jl= (a) and Â£ ( c ) since il,l 4g , and for some bÂ€ S\M, P = (-b,0). Then P = (0,(ba)c) = (0,(bc)a) and k = [a,(bc)a]; so that P I k . Thus, A* is a Pappian Desarguesian affine plane. / 4) => 5 ) . Assume A* is a Pappian Desarguesian affine plane. By Corollary 9.13 and Proposition 9.10, E (A*) is a w division ring. Let (H.g*) = H *(A*). Then, since the ~g * extension of an affine plane to a projective plane is unique up to isomorphism, there is an isomorphism 0:H Â— + K(S) where S = E (AÂ») by Propositions 9.10, 10.11 and 10.8; hence H is Desarguesian. Let g" be a line in H. By Proposition 10.31, A' = A(H,g") is Desarguesian and S 1 = E(A') is isomorphic to S. Let K ' be a W coordinatization of A'; then by rroposition 9.10, R(A*,K') is isomorphic to S*. Hence ( S Â• , + * , x' , x* ) = X'RtA'.K 1 ) = Q(A',K') has both multiplications associative and abelian. By 2) =Â£ 4) (shown above), A' is Pappian. Thus H is Pappian. / 5) => 1) . Obvious. // 12.9 Proposition . Let C = (A,K) be a coordinatized AH-plane. The following are equivalent. 1) C is ((Ojg^)and ( (0 ) ' ,g M )-H-Desarguesian and is ( 9 Â»9 Â» (z,0) , (1) )and (g ,g , (0, z) , ( 1 ) )-Pappian for all z e M\N. a y y x 2) C is axially regular and V(C) is a prequasiring both of whose multiplications are associative and abelian. 3) C is Desarguesian and the AH-ring R(C) has abelian multiplication (hence C is protectively Desarguesian).

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216 4) A is a Pappian AK-plane which is (P , g.J-H-Desarguesian for every direction T. 5) A is a Pappian translation AH-plane. * 12.10 Remark . The equivalence 3) <=> 4) above is very closely related to two propositions of Klingenberg ' s [(1954); pages 401, 405: S 5.16, S 6.6]. As remarked earlier (following Definition 12.1), our definition of 'Pappian' for derived AH-planes is not the same as Klingenberg 's. Also, our definition of ' (r ,gÂ„)-H-Desarguesian for every direction V is not the same as Klingenberg ' s definition of 'dem affinen kleinen Satz von Desargues S' for derived AH-planes. Even our definition of Â• (r,g >() )-3-Desarguesian for every direction V differs (for derived AH-planes) from Klingenberg 's 'dem affinen kleinen Satz von Desargues I': we do not require P iP . (See Theorem 4.21 and Definition 4.10 for the relationship between (r,g w )-Hand (t^g^-S-Desarguesian. ) Proof . 1) => 2). Immediate by Theorem 4.21 and Proposition 12.7 and the xy-dual of Proposition 12.7. / 2) ^ 3). Assume C is axially regular and that both multiplications of V(C) = (M,+,x,.) are associative and abelian. By Proposition 12.7, aq = a-q for every a fe M; q e M\N. Observe that since (M,x) and (M,-) are both abelian, all four distributive laws hold. Let a 6 M; n Â€ N. Then 1,(-1 + n) ^ N and an = a(l + (-1 + n)) = a + a(-l + n) = a-1 + a. (-1 + n) = aÂ»(l + (-1 + n)) = a-n. Thus, (M,*) = (M,-). By Lemma 9.7,

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217 V(Q) is a quasiring, and hence V(C) is a kernel quasiring with abelian multiplications. Since an AH-ring with abelian multiplication is an H-ring, R(A,K) is an K-ring, and by Proposition 9.10, (A,K) is protectively Desarguesian. / 3) => 4). Let F be any Pappian configuration in A. Let K' be a coordinatization of A such that q = c; q = q ' : (1) = V. y "x ' ' where g^'.r 1 are the g,g',H of F. Then Q(A,K) is isomorphic to Â£'(E (A)) by Proposition 9.10. Hence using Lemma 12.6 and Proposition 12.3, one can easily show that P I k . Thus, A is Pappian. By Theorem 4.21, A is (r,g (lo )-H-Desarguesian for every direction P. / 4) =^ 5). Immediate by Theorem 5.25. / 5) =^> 1). Immediate by Theorem 5.25. // 12.11 Example . The coordinatized AH-plane (A ,K ) constructed in Example 9.14 is not axially regular, but it is (g ,g , (z,0),(l))y * Pappian for every z 6. S\N. Observe that the gross structure of # * A is a Desarguesian Pappian affine plane. 12.12 Proposition . Let H be a PH-plane. The following are equivalent. 1) H is Pappian and Desarguesian. 2) There is a full triangle J such that H is Pappian for the full triangle J and such that the automorphisms of H are (V,s)transitive for every vertex-side pair (V,s) such that Vis. 3) There is a full triangleJ with sides s,s*,s" such that each of A(H,s), A(H,s'), A(H,s") is a Pappian translation AHplane.

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218 4) H is Desarguesian and there exists a line g of H such that A(H,g) is Pappian. 5) There is an H-ring R with abelian multiplication such that H is isomorphic to the Hjelmslev structure H(R) generated by R. 6) H is Desarguesian and every AH-ring S such that H is isomorphic to the Hjelmslev structure H(S) generated by S has abelian multiplication. 7) H is Pappian and Hjelmslev Desarguesian. Proof . 1) => 2). Immediate by Theorem 11.6. / 2) =^ 3). Assume that H is Pappian for a full triangle J and that the automorphisms of H are ( V, s )-transitive for all incident vertex-side pairs. Let s" be a side of J. Since H is Pappian for J, there is a point P" on s" such that P" /. s,s' and such that A = A(H,s") is ( s, s Â• , P ' , P")and (s ' , s, P, P")-Pappian for all P' I s' and Pis such that P' + s; P p s'. Let K be a coordinatization of A such that g = s: q = s 1 and (1) = P". x y Let V" = s" r\ s < a nd V = s s". The (V",s")and ( V,s")-automorphisms of H induce ((0) Â• ,gÂ„,)and ( ( 0) ,g ao )-transitive automorphisms of A. Hence (A,K) is axially regular. Hence by Proposition 12.9, A is Pappian and Desarguesian. / 3) =* 4). By Proposition 12.9, a Pappian translation AHplane is Desarguesian; hence by Proposition 11.4, H is Desarguesian. / 4) =f^ 5). Immediate by Propositions 12.9, 9.10 and 10.31. / 5) ^> 6). Immediate by Corollary 10.30. /

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219 6) =\$ 1). Let h be a line of H and let A = A(H,h). ByProposition 10.31, A is Desarauesian and E (A) has abelian W multiplication. If F is any Pappian configuration, choose a coordinatization K such that g = g; g = g Â• ; (1) = T, where the y ** g,g',r are the g,g*,P of F. Then by Propositions 9.10 and 12.3, P l I k 13* Thus > A is Pappian. Since h was arbitrary, K is a Pappian Desarguesian PH-plane. / 1) <=^ 7). Immediate by Theorem 11.6. // 12.13 Proposition . Let C be a cocrdinatized translation AHplane with quasiring (M,+,*,-). The following are equivalent. 1) C is (L0, 1"]',[1, 01,(0, C),(0))-Pappian. 2) (M,X) is abelian. Proof . 1) ^ 2). Assume C is a translation AH-plane which is (l0,ll',[l,0], (0,0) , (0))-Pappian. Let n, x be elements of M. Let F be the ( (n) , (x) \ L0, 1] ' , [l ,0], (0,0 ) , (0 ) )-Pappian configuration; then P 1 = (l, x + n nx) is on k 13 = [n,x xn] . Thus, x+n-nx=n+xxn, and hence nx = xn. Therefore, (M,x) is abelian. / 2) =# 1). Let F be the Ul,f. \[0,ll ' , [1,0] , (0,0 ) , (0) )Pappian configuration. Since -fl-,5. 4[0,1]', there are n,x Â£ M such that A . (n); Â£ = (x). Since (M, x) is abelian, P = (l,x + n nx) is on k = fn,x xn]. Hence A is
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220 AH-plane with prequasiring (M, +,*,-). The following are equivalent. 1) C is (l0,0l,[0,0]',(0,l),(-l) M-Pappian. 2) (M, Â• ) is abelian. * 12.15 Remark . The argument for 1) => 2) below is an adaptation of an argument given by Klingenberg [(1954), page 399, S 5.9] obtained by letting g = Lo,Cf); g' = [0,0]'; P = (a,0); P a Â« = (0,(-a)(-D); P 1 Â» = (0,1); gÂ± = [-l,l]'; h a = [-a, a]' and so on. Here, P ^ = (-( (-b) (-1 ) ) Â• (-a) ,0 ) , and one can show that if 6 q is a derived (g,g ' .Pj, Â• ,TUg ) )-Pappian translation AH-plane, then P . = ((-a)(-b).O). ab ' Proof . 1) =Â£ 2). Assume C is (g ,g , (0 ,1 ) , (-1 ) Â• )-Pappian. Let x y -a,-b Â£ M. Let F be the ( (-a) ', (-b) '\ g ,g , (0,1) , (-1) ' )-Pappian x y configuration. Then P = (-( (-a) (-1 ) ) (-b) ,0 ) and k = C-a,-((-b)(-l))(-a)l *. Thus, ( (-a ) (-1 ) ) Â• (-b) = ( (-b) (-1 ) ) (-a ) . Letting b = -1, we have (-a)(-l) = a for all a e M. Thus, a-(-b) = b-(-a) and hence (-a)'-(-b) = (-b)-(-a) for all -a,-b Â£ M. Therefore, (M,Â«) is abelian. / 2) =5> 1). Assume (M, Â• ) is abelian. Let s 6 M. Then Then since (s(-D)-(-l) = s, we have that (-l).(s(-D) = s. Hence -s(-l) = s, and s(-l) = -s. If F is a (-TL, Z.I g ,g , (0, 1 ) , (-1 ) ' )-Pappian configuration, then A = (-a)', x y 1 = (-b) ' for some a,b Â£ M. Then, since (-a)-(-b) = (-b).(-a), -((-a)(-l))-(-b) = -((-b)(-D). (-a); so that P I k . Thus, A is (g ,g ,(0,1), (-1) -)-Pappian. // x y

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221 * 12.16 Example . The AH-plane A(H,g) of Remark 11.1 is a Pappian Desarguesian AH-plane which can be extended to a non-Desarguesian PH-plane: H'. Hence H' is a counterexample to Klingenberg ' s S 5.17 [(1954), page 40l].

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APPENDIX A. RESTRICTED BITERNARY RINGS In this section we prove a theorem somewhat similar to a theorem which is stated by V. K. Cyganova [(1957), page 51, Theorem 18*1 : we prove that there are reciprocal equivalences between the category of restricted biternary rings and the category of coordinatized AH-planes. We also construct a functor which is an isomorphism from the category of biternary rings to the category of restricted biternary rings. A.l Definitions . Let M be a set with distinguished elements 3 and 1, with a ternary operation T:M Â— M defined on M. Let N = {n 6 Ml 3 k fe H, k ^ 0, 3 T(k,n,0) = 0"5 . Let T":.M* NxM Â—* M be a partial ternary operation defined on M. Define a relation Â«on M by a ~ b (read 'a is neighbor to b') if and only if every x which satisfies the equation a = T(x,l,b) is an element of N. The negation of a ~ b is written a 4b and is read 'a is not neighbor to b.' The ordered triple (M,T,T") is said to be a restricted biternary ring if the following sixteen conditions are satisfied. (Rl) = (Bl); that is, we let condition (Rl) be the same as condition (31) of the definition of biternary ring. (R2) = (B2). (R3) = (33). 222

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223 (R4) = (B4). (R5) = (E5). (R6) = (B6). (R7) = (37) with T' replaced by T". (R8) = (88) with T* replaced by T". (R9) If a ~ a', b ^ bÂ», the system ^T"(b,u,v) = a, T"(b',u,v) a 1 ) is uniquely solvable for the pair u,v (where u 6 N) ; if a -A a ' , the system does not have a solution. (RIO) If u,u' 6 N, and if a,b,v,v' e M are such' that a = T"(b,u,v) and a = T" (b,u * , v ' ) , then v ~ v' and there exists at least one other pair a',b' such that a* = T"(b',u,v) and a' = T"(b',u , ,v'). (Rll) T"(b,u,z) = a is uniquely solvable for z for any a,b from M, and any u from N. (R12) T"(y,0,d) = d for any y,d from M. (R13) = (E10). (R14). T"(b,u,v) ~ v for every b,v from K, and every u from N. (R15) T"(l,u,0) = u for any u from N. (R16) T"(0,u,v) = v for any u from N and any v from M. Each element of N is said to be a right zero divisor . We call the elements of M symbols , and we say that is the zero and 1 is the one of (M,T,T"). Â» A.2 Remark . Cyganova [in (1967), page 61, Definition 4"} defines an 'H-ternar' to be a set M with two distinguished elements 0,1 and twooperations T,T" which satisfy properties which imply all the properties of a restricted biternary ring except possibly

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224 (R13), (R15), (R15). Her definition of N (and hence ~) is unclear. She requires that T* be a ternary operation. Her arguments to show that T* is a ternary field fail to show any of the three required uniqueness properties. If (R15) and (R16) were not required in the definition of restricted biternary ring, then our proof that if R is a restricted biternary ring, then ZBC (R) = R, would not go through (see Proposition A. 18,) but our proof that if R is a restricted biternary ring, than C (R) is a coordinatized AK-plane, would (see Proposition A. 17.) A. 3 Proposition . Let (R,T,T") be a restricted biternary ring and let u fc R. Then u *> if and only if u 6 N. Proof . Replace (33) by (R3) and ( 35 ) by (R5) in the proof of Proposition 3.5. // A. 4 Pro no sit ion . If (R,T,T") is a restricted biternary ring, then 1 40. Proof . By (R13) 1* / 0*; hence 1 Â•/< 0. // A. 5 Proposition . In a restricted biternary ring (R,T,T"), if a = T(n,l,b) for some n Â€ N, then a ~ b. Conversely, if a ~ b, then there is a unique n Â£ N such that a = T(n,l,b). Proof. Replace (BIO) by (R13); (B5) by (R5); and (32) by (R2) in the proof of Proposition 3.7. //

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225 A, 6 Definitions . Let (R,T,T") and (K,S,S") be restricted biternary rings. A concrete mcrphism oc:R Â— * M is a restricted biternary ring homo-orphism if <*(N ) Q N ; *(C) = 0; Â«(1) = 1; ) by oo'n = m for all svmbols m in D. The map "F Z *:F* Â— * Z* is an isomorphism; that is, Z * is an equivalence ~"F ~r which produces a one-to-one correspondence 'between the objects of F* and Z*. Define F *:Z* Â— Â» F* to be the obvious inverse functor. // A. 8 Construction of C^:Z Â— > C. Let R = (K,T,T") be a restricted biternary ring. We construct an incidence structure S = CP,"J,I) in the following way. We let *P= M x i-i, and we let fl] = fjÂ„,^ Â°\mÂ„ where oj and Of are defined as follows: 0J = M* M*\$0l and Â°J_Â„ = NxM*^ll. For convenience (m,b t 0) is written [m,bl and (u,v,l) is written [u,v]'. We define the incidence relation by

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226 (x,y) I [m t bl Â£=> y = T(x,m-,b) and by (x,y) I [u,v]' <^> x = T"(y,u,v). We say (m,b,s) II (n,c,t) <^ s = t, m = n. Let A = (S R ,I1). Define K R to be ( [0,0] , [0,0l ' , (1 ,1 ) ,K,%) where 5 R : UÂ»0J Â— * M is defined by ? (rn,m) = m for all points (m,m) on [l,ol. We define C_(R) to be (A_,K_ ) . If u:(K,T,T") Â— * (R,S,S") is a restricted biternary ring homomorphism, define CÂ„(Â«Â»i}:C (M,T,T") Â— Â» C (R,S,S") by ** Z " Z * z C 2 (uj)( x ,y) = (wx,wy), C (w)tm,b*l = [ion,*obl and C (w)[u,v]' = [uju,u)v1 ! for every point (x,y) and every line Â£m,b], [u,v~]'. A. 9 Proposition . If R* is a restricted biternary field, then C^(R*) constructed above is a coordinatized affine plane. Proof. If R* = (K*,T*,T*") is a restricted biternary field, then (M*,T*) is a ternary field and it is easily seen that C^(R*) is the coordinatized affine plane commonly constructed over the ternary field (M*,TÂ»): see Hall [(1959), top of page 356]. // A. 10 Construction of Z:5 Â— * Z. Let B = (M,T,T*) be a biternary ring and let N be the set of right zero divisors of B. Define T":KxNxMÂ— >M by T"(y,u,v) = T'(y,u,v). Let Z(B) = (M,T,T"). If io = (B,B',f M ) is a biternary ring homomorphism, let Z(uÂ») = (Z(B) ,Z(B') ,f ). A.ll Proposition . The map Z:3 Â— * Z defined above is a functor from the category of biternary rings to the category of

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227 restricted biternary rings. If B is a biternary field, then Z(B) is a restricted biternary field. Proof . Assume 3 is a biternary field. Obviously N_ = N . . and B Z ( 3 ) (a ~ b in 3 <=? a ~ b in Z(3).) Conditions (Rl) through (R8) obviously hold in Z(B) = (M,T,T"). Skipping (R9) and (RIO) for the moment, we see that (RID follows from (34)'; (R12) follows from (B?)'; (R13) follows from (BIO); (R14) holds since (T(b,u,v))* = T' * (b* ,0* ,v* ) =v* when u Â€ N by (310)' and Proposition A. 3; (R15) follows from (33)' and (R16) fellows from (32)'. Now look at (R9). If a* a', b -p b Â• , the system Â£T"(b,u,v) = a, T"(b',u,v) = a'"} is uniquely solvable for u,v by (36)'. If u,v is a solution, then by (R14), proved above, a v, a' ~ v'; so that a ~ a'. Hence if a
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228 A. 12 Corollary , If (K,T,T') is a biternary field, each of (M,T) and (K,T') is a ternary field. // A. 13 Definition . Define a functor Z*:B* Â— * ZÂ» by Z*(B) = Z(B), Z*(w) = Z(uj) . A. 14 Corollary . The functor Z^*:F* Â— * ZÂ» is coual to the functor * Alf Z'E'C^*, where B* and C * are defined in Corollary 3. 23 and Definition 3.29 respectively, and Z Â• is defined in Definition A. 7. // * A.15 Remark . Cyganova [(1967), page 61, Theorem 18] states that if A is an AH-plane, then what we call ZB(A) is an H-ternar. Lorimer [(1971), pages 162-188] states that if A is an AH-plane, then what we call ZB(A) satisfies many conditions among which are: (M,+) is a loop; a ~ implies a Â€ N; (M/-,T*) is a ternary field. * A.16 Remark . Cyganova [(1967), page 61, Theorem 18] states that if R = (M,T,T") is an H-ternar then C r7 (R) is an AH-plane. Her argument to show this contains a number of omissions; for example, she fails to show that (a,b) is affinely neighbor to (a*,b') if and only if a ~ a ' , b ~> b Â• , and she does not show the corresponding lemma for lines; although these lemmas are used in the argument she gives. (These lemmas correspond to Lemma 2 and Lemma 4 in the proof of the following proposition.)

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229 A. 17 Proposition . The map C :Z Â— Â» C defined in Construction A. 8 * z_i is a functor from the category of restricted biternary rings to the category of ccordinatized AH-planes. Also, C (Z*) Q C*. Proof . The proof of this proposition is so much like the proof of Proposition 3.23 that we will indicate only where the two proofs differ. Replace ( Bl ) by (Rl); (32) by (R2); (32)' by (R12); (B3) by (R3); (B4) by (R4); (35) by (35); (36) by (R6); (B7) by (R7); (B8) by (R3); and (BIO) by (R13) in the arguments given for Proposition 3.23. Let R = (M,T,T") be a restricted biternary ring. Replace B by R and T 1 by T" in these arguments. (Lemma 1) (Case 3) Assume P ^ Q, a ~ a ' , b /Â• b ' . By (R9), there is a solution [u,vl' (with u e N) to the system {a = T"(b,u,v), a' = T"(b',u,v)l. / (Lemma 2) ( => ) (Case 2) Assume P,Q I g,h, g 4 h and <3Â» h * Â°} T Â„The system t'T"(b,u,v) = a, T"(b*,u,v) = a*3 is solvable but not uniquely solvable for the pair u,v; hence by (R9) , a ~ a', b ~ b'. / (Lemma 3) ( <= ) (Case 2) Assume g,h Â£ Â°] TU ' Let g = [u,v] Â• ; " h = [u'jvO 1 . By (RIO) , v v' . If G I g where G = (c,d), then c ~ v by (R14). Let G' = (T" (d,u ' , v ' ) ,d) ; then G' I h, and T"(d,u',v') v' by (R14). Observe that G ' G by Lemma 2. By a similar argument if W * is an arbitrary point -of h, there is a point W on g such that W D W. Thus g D h, a contradiction. This case does not occur. / (Case 3) Assume g = [m,e"l, h = tu,vl'. By (R8) \ g C\ h\ = 1. /

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230 (=}) (Case 1) (Second Part) Let g = h = Lu,vl'. Then (T"(0,u,v) ,0) and ( T" ( 1 ,u, v) , 1 ) are on g, and since C / 1 by (R13), this case does not occur. / (Case 3) Assume g,h & oj g 4, h. By (RIO), Ig ft hi ^ 1; hence this case does not occur. / (Lemma 4) (Case 2) Assume g = [u,v]'. (<=) Assume h = [u',v'l' with u u', v ~ v'. Let (a,b) be a point on one of the lines, say g. Then u ~ by Proposition A. 3, and by (R14) a ~ v. By (R8) the line l0,b] intersects h in a unique point; by (R2) this point is of the form (a',b ). By (R14), a* v'. Hence a ~ a' by (Rl). Thus (a,b) (a',b). Since (a,b) was an arbitrarily chooser, point on the arbitrarily choosen line g, g h. / (=5>) Assume g h. By the early part of the argument for (=\$>) in Case 1; h = [u',v'3'. Let t = (a,b) be any point on g. Let Q, Q P, C = (a',b'), be a point on h. If P I g, such a point C exists since g h. By Lemma 2, a a*, b ~ b'. By (R14), b ~ v, b' ~ v 1 . Hence v ~ v'. Since u,u' Â€ ;.", u,u' ~ by Proposition A. 3; so that u' u' by (Ri). / (Lemma 6) (Case 1) Let P = (x,y) and g = [m,b]. Then by (R4) there is a unique symbol z such that y = T(x,m,z). Hence [m,zl is the unique line parallel to Cm,b] which contains P. (Case 2) Let P = (x,y) and g = tu,v"]'. Then by (RID there is a unique symbol z such that x = T"(y,u,z). Hence Lu,zl' is the unique line parallel to fu,vl' which contains P. / The remainder of the proof that C (R) is a coordinatized AH-plane follows as we have indicated, and the remainder of the

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231 proof that C :Z Â— Â•* C is a functor closely parallels the remainder of the proof of Proposition 3.23. // A. 18 Proposition . The functor (22)C 2 is the identity functor en Z. Proof . This can easily be seen by using the various constructions. Observe that (R15) implies (l,u) I [u,c"]' and that (R16) implies (v,0) I [u,v"]'. // A. 19 Construction of f : 1~ Â—Â• C (ZB) . It can be seen by looking at the usual identification of points ar.d lines with their representations, that there is an isomorphism f between a coordinated AH-plane C and C
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232 A. 22 Proposition . The functors C Z:3 Â— " Z and C:B naturally isomorphic. // A. 23 Proposition . The functors Z, 3, C , Z* , B*, and Z Â• (where Â£ 2 *:Z* Â» c* is defined in the obvious way) are equivalences. The functors Z, Z* and F z *Z':E* Â— * FÂ« are isomorphisms. Moreover, (EC )2 = 1 & . Thus, going around the triangle with vertices Z, 3, C takes one back to the same object (morphism) if one starts at either Z or 3; similarly for the triangle with vertices Z*, B*, C*. Proof. By Proposition A. IS, Z(BC ) = 1~. It will suffice to show that (BC )Z = 1^. *** "^ ~ *"B Let B = (M,T,T') be a biternary ring. Observe that B' = Â§? 2 2^ B ^ = (QfS,S*) is a biternary ring and that Q = M; S = T; N Q = N M and s ' ( yÂ» u ' v) = T'(y,u,v) when u e N t y,v Â€ M. V/e wish to show that Â£* = T'. Let u Â£ (FAN ). By (B6) there is a M unique m,d Â€ M such that T(u,m,d) = 1 and T(0,m,d) = in B and B'. Since 1 f 0, m f and by (32) d = 0. For v in M, there is an e in M such that = T(v,m,e) in B end B'. For y Â€Â• M, there is an x in K such that T(x,m,e) = T(x,0,y) = y. Since = T(v,m,e) and v = T'(0,u,v) = S'(0,u,v); T(x,m,e) = y = S(x,m,e) implies that x = T'(y,u,v) = S'(y,u,v) in ( B9 ) . Thus since u,v,y are arbitrary elements of (M\N },M and M respectively,S' = T'. Thus BCZ(B) = B. Since BC Z(uo) =
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233 Â•Remark. The discussions which follow are confined to results which closely resemble or which are closely connected to material presented in this paper. A.24 Discussion of Lorimer [(1971)] . In Theorem (5.1.1), page 99, Lorimer states that if A is a translation AH-plane, then A is Desarguesian if and only if the endomorphisms of A are strongly (P.g^-transitive for every point P: this is similar to our Proposition 9.10, 1)<Â£>3). Beginning on page 126, Lorimer indicates the construction of an AH-plane A(K) from an AH-ring K. In Theorems (5.3.1) and (5.3.7), pages 134, 142, he states that A(H) is Desarguesian: this is similar to Proposition 9.10, 7) ^5> 1). In Theorem (6.2.6), page 202, Lorimer states that if A is Desarguesian, then (in our notation) (M,+,*) is an AH-ring: this resembles part of 9.10, 2) ^ 4). Lorimer states that the ternary operation T satisfies a number of properties some of which correspond to properties which a biternary ring is defined to have. On page 181, Lorimer defines the map which we call Z:M Â— -> M. On pages 188, 190, Lorimer defines a geometric property which he calls D.. In Theorem (6.2.4), pages 190, 191, he states that if A is an AH-plane; if the set of translations of A form a group, and if there is a line of A with three pairwise non-neighbor points on it, then the following are equivalent: 1) For every coordinatization, the first ternary operation T is linear. 2) Property D holds. 3) A is a translation AH-plane.

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234 In view of our Theorem 4.21, this theorem resembles our Theorem 5.25: property D Â± resembles ( r, g#o )-3-Desarguesian for all P except that (in our notation) P is required to be not neighbor to P . In Theorem (5.2.6), page 193, Lorimer states that if A is a translation AH-plane, then (in our notation) a + b = a * b: this resembles part of Theorem 5.25, 1) =\$> 4). Â• A. 25 Discussion of Lorimer and Lane [(1973)1 . Lorimer and Lane let H denote the ring of trace preserving endomorphisms of the translation group of an AH-plane H : we let A = X . In Corollary 4.4.6, page 22, they state that if A is a translation AH-plane, then H is a local ring: in view of our Theorem 7.7, this is essentially our Proposition 7.3 (see A. 28). In Theorem 5.3, page 35, Lorimer and Lane state that if A is a translation AHplane, then the following are equivalent: Â• 1) A is Desarguesian. 2) The endomorphisms of A are strongly (P,g ) -transitive for every point P. 3) There is a point P such that the endomorphisms of A are strongly (F,g M )-transitive. This resembles portions of Proposition 9.10, 1) 4=> 2) 3) (see A. 28 and A. 24). In Theorem 6.9, page 50, Lorimer and Lane state that if A is Desarguesian, then H is a local ring with the properties that every zero divisor is a two-sided zero divisor and if a,b Â£ H, then a Â€ bH or b e aH: this resembles a portion of Proposition 9.10, 2) => 6 ) . In Theorem 8.4, page 69, Lorimer and Lane state the following (we use our notation): if

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235 S is an AH-ring, if A = A(S) is the AH-plane they construct from S, and if W is the translation group of A, then the following are equivalent: 1) If r,T' e '.v, and if the traces of r are traces of T * , then there is a S eE., 7 (A) such that Jr = r'. If r, T Â• fc w, and if T,T' are ( (0) jg^-translations, then every trace of t is a trace of T' or every trace of r' is a trace of T. 2) S is an H-rinc. 3) A can be embedded in a Desarguesian PH-plane. This resembles portions of Propositions 10.31 and 10.33 (see A. 28). * A.26 Discussion of Lorimer [(1973)a"]. Lorimer begins (page 9, Theorem 2.9) by quoting Cyganova's Theorem IS (see our Remarks A. 2 and A. 16); although he calls an 'H-ternar' a 'biternary ring' (and gives a definition slightly different from Cyganova's: he explicitly defines N and ~ in terms of the ternary operations, but omits the 'only if in (S4)). In Theorem 2.14 (page 12) he states various properties of a 'biternary ring'. In Theorem 2.15 (page 13) he states that the 'biternary ring' B of the AH-plane generated by a 'biternary ring' B is isomorphic to B. One constructs a counterexample to Lorimer *s Theorem 2.15 as follows. Let (A S? K S ) be the cocrdinatized Desarguesian AH-plane generated by the AH-ring (S,+,x) = p Z 3 LXJ/(X ), and coordinatize all the lines and points as usual. Let [0 f m]Â» = [o,m]Â«, [x,m]" = [2X,m]Â» and [2X,m]Â» = [x,m]' for

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236 every m Â« S. Define (S,T,TÂ») by (x,v) I [m,b] ^ y = T(x,m,b) and ( x ,y) I [u,v> <^> TÂ»(y,u,v) = x. Observe that (S,T,T") is not a restricted biternary ring since condition (R15) is not satisfied, but that (S,U,UÂ») = Z3(A s ,K s ) is. Both B = and B' = are Lorimer 'biternary rings' assuming (of course) we have added the requirement that (a ,b ) 4 (a ,b ) to Lorimer 's condition (p ? ). Observe that B and B' are non-isomorphic Lorimer 'biternary rings' and that B is isomorphic to BÂ«. This is a counterexample to Theorem 2.15. The requirement that l ,b l J ' (a 2' h 2 ) corres PÂ°nds to an error in Cyganova [(1967), page 54, Characteristic s], where she says (page 54), "ECJTH 5bJ g,h SbUTH pa3HHX THnOB , TO KaÂ«flaH H3 CHCTeM "I Itloy od = x u-x,. v = y. u-x . v = y. 6hIJia 6bl pa3eiUHMa OTHOCHTeJIbHO U,V COOTBeTCTBeHHO m,d ^to npoTHBopetiHJio 6bi CBOftcTBy 7." Obviously, Characteristic 7 (which corresponds to our (B8)) is not contradicted if x n = x~, y = y . In Comment 2.16 (page 14) Lorimer discusses modifications of some arguments given by Klingenberg [(1954)] which Lorimer uses in his argument for the validity of Theorem 3.5 (page 16). In Theorem 3.5 Lorimer gives two sets of conditions which he states each characterize translation AH-planes; one is algebraic and one geometric; both are different from any of the characterizations given in our Theorem 5.25. (The geometric characteriza-

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237 tion is omewhat similar to Theorem 5.25, 1) 8) of our Theorem 5.25. (Cur corresponding results were obtained in the spring of 1972.) In Theorem 4.5 (page 24) Lorimar states that an AH-plane is Desarguesian if and only if for at least one coordinatization the ternary operation T is linear and (M,+,a) is an AH-ring. Cur Example 9.14 is a counterexample to this assertion. Lorimer states the theorem correctly in his summary at the beginning of the paper: there he requires T = T'. (Our corresponding results were obtained in the spring of 1972.) In a portion of his argument for the validity of his Theorem 5.1 (page 25) (which concerns a Pappian configuration property), Lorimer relies on some arguments given by Klingenberg in [(1954)]. However, Lorimer fails to specifically connect (or differentiate between) his algebraic system and Klingenberg 's. See our Remark 12.15. Theorem 5.1 is very similar to our Theorem 12.9, 3) <=>4). (Our" Pappian configuration property is slightly different from Klingenberg ' s (which is the one Lorimer uses in Theorem 5.1): see our Remark 12.10. Our Theorem 12.9, 3) <=? 4), was obtained in the spring of 1972. In his final section, Lorimer states an interesting theorem (Theorem 5.3, page 26) concerning a special Pappian configuration property. Â» A.27 Discussion of Lorimer [(1973)b]. Lorimer defines AH-plane

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238 homomorphism (he does not require that the neighbor relation be preserved), and states some theorems concerning AH-plane and affine plane isomorphisms (Theorems 2.9, 3.5 and 4.5; pages 5, 8, 10), concerning the automorphisms (Aut A) which fix the oriain o ' of a coordinatized Desarguesian AH-plane (A,K) (Theorem 5.7, page 14), and concerning a map from (Aut A) to a set of o semi-linear transformations and a relationship betv/een (Aut A) o and Aut H where H is the ring of trace preserving endomorphisms of the translation group of A (Theorem 5.11, page 15). See our Propositions 2.55, 2.63, and Theorem 10.63. In Remark 4.3 (page 10), Lorimer states that if A and A* are AH-planes and if f:A Â— A' is a surjective incidence structure homomorphism, then ^ induces an incidence structure isomorphism tf*:A* Â— AÂ»*. He uses this in his argument to show (Lemma 4.4 on the same page) that ^ preserves and reflects the neighbor relation. However, his argument fails to show that ^ * is well defined; that is, that f preserves the neighbor relation. He uses Lemma 4.4 frequently in the arguments which follow it. See our Proposition 2.33 and Corollary 2.65. In his argument for Theorem 3.5 (page 8), Lorimer uses Corbas' Teorema [(1965), page 375] without commenting on its proof. Corbas' argument breaks down where he says, "Se ^(P) = (R) Â£ c?(?) (un punto siffatto dev'esistere sens' altro perche altrimenti tutti i punto di -rr avrebbero la stessa immagine). Ma ailora le rette distinte e non-parallele FR e QR avrebbero la stessa immagine <-?(R)M>(P) " What if R,F,Q I g for some line g? This argument can easily be repaired: see our Theorem 2.6 3.

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239 * A.28 A brief history of this research . The following announcement appeared in the April, 1973, .Notices of the American Mathematical Society , pages A33 7, A3 33: 73T-D10 PHYRNE YOUENS BACON, University of Florida, Gainesville, Florida 32601. Ccordinatized Hjelmslev planes . Preliminary report. Two categories of algebraic structures are defined: biternary rings and quasi-rings; the definition of (group) congruence is generalized to cuasi-congruence; and it is shown that there exist equivalences between the following pairs of categories: biternary rings and coordinatized affine Hjelmslev planes; quasi-rings and coordinatized translation H-planes; quasicongruences and translation H-planes with a base point; AH-rings and coordinatized Desarguesian AH-planes. The definition of (P,g)-transitive for a projective plane with a base line is generalized to the AH-plane case. Necessary and sufficient conditions geometric that an AH-plane have endomorphisms which are (TTjg^) -transitive are given. Necessary and sufficient algebraic conditions that a coordinatized AH-plane with y-axis g have endomorphisms which are ((0,0)^^)((TTjg^)-, ((0),g )and ((Â«o),g )-) transitive are given. If the endomorphisms of an AH-plane are (TT, g 00 )-transitive for three nonneighbor directions TT, then A is a translation H-plane. Results analogous to Andre's about kernels are shown. (P,g)-transitive is defined for the projective H-plane case. It is shown that a PH-plane H is isomorphic to Â€ (R) for some H-ring R if and only if the P endomorphisms of H satisfy certain transitivity conditions. (Received February 13, 1973.)

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240 In the spring and summer of 1972, I proved the following results: a theorem similar to Cyganova's Theorem IS (pre-restricted biternary rings); Theorem 4.21 ( ( F , g^) -HDesarguesian) ; an early version of Proposition 5.7; Corollaries 5.17, 5.16, 5.19 and 5.20 ( ( (0) , g^-automrophisms, ((0),g )transitive, axially regular); Theorem 5.25, 1) <Â£> 4) <^> 6) <^> 7) (translation. AH-plane Â£\$> axially regular and (?:, + ) abelian); Proposition 7.3 (local ring), the last sentence of Theorem 7.7 in terms of the operations + and so on directly from the ternary operations (kernels of quasirings); Proposition S.2, 1) &? 2) (left modular for s); Proposition 8.11, 1) O 2) ( ( (0) , [0,0] ' )-normal for s); Proposition 8.16 ( ( (0 ) ' , [0,0] Â• )normal for s); Proposition 9.10, 1) <Â£> 2) ((P,g w )and two (f 1 Â»gÂ„)-transitivities) ; Proposition 10.31 (extension of a protectively Desarguesian AH-plane); Proposition 10.33 (protectively Desarguesian and H-rings); the first sentence of Proposition 10.36 ( (n,g) -transitivity) ; Lemma 11.2 (two derived Desarguesian AH-planes); Proposition 11.4 (three derived Desarguesian AHplanes); Theorem 11.6, 1) <Â±? 2) <Â£=> 3) C=> 4) <=> 5) (Desarguesian PH-plane and full triangle); Proposition 12.13 (an early version of a partially Pappian result); Proposition 12.9, 3) <^> 4) (Pappian and commutative H-ring ) ; and Proposition 12.12, 1) <^> 6). I then succumbed to the temptation to define an algebraic structure with two full ternary operations, and in the process of doing this got coordinatization homomorphisms. In attempting to relate my results to Lorimer's I proved Theorem 2.63 and 10.63.

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241 Remark . The following information was obtained by talking to Drake, Kleinfeld, Bruck and Brauer, and by research in the Rice University Library. David A. Drake got his degree at Syracuse and his advisor and his advisor's advisor and so on were as follows (the schools are listed in parentheses): Erwin Kleinfeld (Wisconsin), R. H. Bruck (Harvard), Richard Brauer (Berlin?), Issai Schur (Berlin?), Ferdinand Georg Frobenius (Friedrich-Wilhelm-Universitat Berlin), Karl Weierstrass (Konig Theologische und Philosophische Akademie, Kunster), Christof Gudermann (?). Gudermann worked on power series expansion of functions. In 1840, Weierstrass presented a paper to the Examination Committee at Kunster titled "Uber die Entwicklung der Modular-Functionen" (on expanding elliptic functions in terms of power series). Frobenius' dissertation was written in 1870 in Latin and was titled, "De functionum analyticarum unius variablis per series infinitas repraesentatione. " Issai Schur is the 'Schur' of ' Schur 's Lemma' (If M is an irreducible R-module then the commuting ring of R en M is a division ring.). Brauer has worked in groups, Bruck in finite projective planes, Kleinfeld in finite projective Hjelmslev planes and Drake in finite affine Hjelmslev planes. The following proposition does not fit conveniently in any of the sections or appendices since it depends on results from Section 6 and Appendix A.

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242 A. 29 Proposition . The category of quasifields 8 Â• is eauivalent Â— p to the category of biquasif ields Q* which in turn is equivalent to the category of coordinatized translation affine planes C * T and to the category of regular biternary fields B Â•, where each of these categories is defined in the obvious way. Proof. The only equivalence that does not follow easily from a corresponding equivalence for AH-planes and so on, is that involving the category of quasifields Q *. Observe that if C is a coordinatized translation affine plane and if (C, + ,x,.) i s the biquasifield of C , then (Q,+,x) is a quasifield, and that if (Q,+,x) is a quasifield, then the coordinatized affine plane generated by the restricted biternary field naturally associated with (Q,+,x) is a translation affine plane [see Hall (1959), page 362 J . We know that Z' is equivalent to C*, and it is routine to show that the appropriate subcategory of Z* is eauivalent to Q *, C *. // F ' T

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APPENDIX 3. QUASICONGRUENCES The purpose of this section is to extend some results of Luneburg's [(1962), pages 274, 276 and 277, Satze 4.1, 4.3 and 4.5]. In this section we show that the category of pointed translation AH-planes is equivalent to the category of quasicongruences, and v/e characterized non-degenerate AK-plane homomorphisms between translation AH-planes. B.l Definition . Let (0), + ) be a group with identity and let D be a set of proper subgroups of 0]; the subgroups in D are said to be the components of ((0J, + ),D). V/e denote the set of elements of Oj which are in more than one component of D by N , and say that ((<\$, + ), D) is a preconqruence (on C^ ) if each component contains an element not in N . We usually denote a precongruence ((0], + ),D) by D or by (0J,D) . B.2 Definition . Let C(0]*, + ),K*) be a precongruence, let a* be abelian, and let 0*. be the identity of Q\* . We say (0\Â»,K*) is a congruence if it satisfies the following conditions. (KÂ»0) Any two distinct components of K* have only the element 0* in common. (K*l) Every element of (Jj *\\0*] is in exactly one component of 243

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244 (K*2) If g* and k* are distinct components K*, then 0\* is equal to the direct sun, g* Â© k*, of g* and k*. B.3 Definitions. Let ((CJ t + ) t S) be a precongruence, and let be the identity of OJ. We say (((\$, + ), s) is a seniconaruence (.on 0]) if and only if S satisfies the following conditions: (Kl) The components of S form, a cover of 0); that is, US = OJ. (K2) There is a surj active group homomorphism * from 0] to a group 0)* where y satisfies the following conditions: (K2a) K* = \tg \ g e si is a congruence on 0J* . (K2b) \(g + G) H (p + P) 1 Â« 1 if and only if U(g + G)(1 |(p + F)| a l: here Iwl denotes the number of elements in the set V. 7 . (K2c) Let N g = {c Â£ 0] \ G is an element of at least two components of s]; then N is equal to the set of elements of OJ which map to 0* (the identity of 0)*) under the map r. Thus, N S is the kernel of the group homomorphism *. We call such a map *:S Â— > K* a projection rap for S. A semicongruence (C0J, + ),K) such that (0j, + ) is abelian is called a quasicon.Gruer.ee . B.4 Definition. Let K and K' be quasicongruences on (0], + ) and Â«JJ', + ') respectively. Let w:K -* K be a concrete rcorphism such that ((0J, + ) , (0]* , + ') ,f w ) is a group homomorphism such that for each g e K there is a g' 6 K' with w(g) Q g< as sets, such that if gÂ£ K there is an element Peg suc h that sP <Â£ M f , suc h that w(N K ) c n^^ and such that if r , is a pro j ecticn map for K ,

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245 then there are two components of K whose images under y*u) are not contained in a single component of (K')Â»; then, if all these conditions hold for oo:K Â— *Â• K', we say that u is a quasicon gruence homomorphism. B.5 Proposition . Any quasicongruence K such that N = {0} is a congruence; moreover, any congruence is a quasicongruence. Proof. Assume ( 0] , K ) is a quasicongruence such that N. = ^0"). K Let k:0)Â— Â»0]Â» be a projection map for K. Since N__ = \ 0} is the K kernel of / , y is an isomorphism which induces a one-to-one correspondence between the components of K and those of K*. Since K* = {* g \ g 6 Kl is a congruence on 0]*, K is a congruence on 0]. Assume (0],K) is a congruence. Since UK = 0], condition (Kl) is satisfied. Let t = 1 be the identity on (01, + ). One can easily see that (K2) is satisfied for this Y. Since (0], + ) is abelian, K is a quasicongruence. // B.6 Definition . We denote the ordered pair whose first element is the class of quasicongruences and whose second element is the class of quasicongruence homomorphisms by K; similarly, we denote the pair whose first element is the class of congruences and whose second element is the class of quasicongruence homomorphisms between congruences by K*. B.7 Proposition . If the natural composition of two quasicon-

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246 gruence homomorphisms is always a quasicongruence homomorphism, then K and K* are categories. // B.8 Definitions . Let A be an AH-plane. Let P be a point of A. Then we say that (A,: : ) is a pointed AH-plane (v/ith base point P). If (A,F) and (A',F') are pointed AH-planes and if us:(A,F) Â— * (A',P*) is a non-degenerate AH-plane homomorphism which takes P to PÂ», then we say ). If (0},K) is a quasicongruence, we define J(K) to be (J (K),0), and J rr (K) to be J Q (K). If ui:K Â— * K ' is a quasicongruence

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247 homomorphism, we let J(
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248 for some G* e g*, H* Â£ h*, G* + G* = H* + H*; thus, g* + G* and h* + H* meet. Hence any two lines which fail to meet are parallel. 'Thus, (CA4) is satisfied in A*. If ?* is a point and g*+ G" is a line, then g* + F* is the unique line through P* parallel to g* + G*. Thus, (CA2) is satisfied in A*. Since each element of K* is a proper subgroup, and since UK* = 0]*, there are at least two non-zero elements G*,H* which are not in the same component of K*. Since every line through 0* is a component of K*, the points 0*, G* and H* are not collinear, and (0A3) is satisfied. Hence A* is an affine plane. / (=?) Assume A* = J (K*) is an affine plane. If P* is a point of A*, then ?*6.g*+0* for some line containing the identity 0Â« of OJ*. Hence P* 6 U^.g*l g* e K'') , and (K*l) holds in K*. Assume two components g*,vÂ» have a point P* ^ 0* in common. Then the lines g* + 0* and v* + 0* have two distinct points in common; hence g* = v* and thus (K*0) holds in KÂ». Let PÂ» Â£ 0]*, and let v*,g* & K* such that v* d g*. Then V* + P* 4 g" + P*; so that g* + 0* is not parallel to v* + F*. Hence there is a point G* on both. Thus G* = V* + P* for some V* fc v*. Hence P* e (v* + g*), and by (K*0), which we proved above, P* Â€ v* Â© g*. Thus, (K*3) is satisfied and K* is a congruence. Let Q* Â€ q*\t0*J and define A* by a,
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249 line parallel to the line 0*Q* fixed; hence
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250 It is easily seen that * induces a surjective incidence structure homomorphisn X:J Q (K) j q (k-). Let Q I (g + G),(p + p). Then g + G = p + P <=>g + Q s p + Q <^> g = p . Let P,Q be points. Then \P = XQ <^> (e Q) e M <=\$> "3 distinct components' h,k such that (F Q) Â£ h,k <^> 3 distinct components h,k such that P,Q I (h + Q) , (k + Q) <^> P ~ q <=> (P Q) 0. Let g be a component of K; then by (K2a), Yg is a component of K*. By Proposition B.1C, J Q (KÂ«) is an affine plane. Let g + G be a line, and let PI X(g + g) . Then \(g + G) = Kg + TG = Xg + XG; so that (?Â• XG) I Xg. Thus, by (K2a), there is a point 2 I g such that *Z Â« PÂ« Xg. Hence \(Z + G) . P* and (Z + G) I (g + G). Thus, two lines (g + G),(p + P) are af finely neighbor if and only if X(g + G) = Mp + P). Let Q I (g + G) , (p + P). Then Kg + G) (p + P)l=Â„l if and only if Wig + G) C\ *( P + P)\ = 1; hence if and only if (g + G) + (p + p); thus, (AK2) holds in A. Assume Kg + G) H (p + P)\ =0. Then l\(g + G) Mp + P)l I 1; so that X(g + G) 11 X(p + P). Hence (AH4) holds in A, and A is an AH-plane. Observe that if 5S Â« rK Â— K" is the map induced by *, then <' is a quasicongruence homomorphism and J ?f (x') = X is an AHplane homomorphism. Also, J (KÂ») is isomorphic to (J (K))Â». Let q be a line and let Q,0 I q. Define T :A Â— * A by ' f Q (G) = G + Q, r Q (p + P) = p + p + q. since r Q preserves the parallel and incidence relations; fixes the directions; preserves the neighbor relation on points, and is a bisection on points,

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251 by Propositions 2.55 and 2.56, r is an automorphism. Also, T y q is a (TT(q) .g^) -automorphism. In fact, the set W = {r \ Q I q] is (TT(q),g w ) -transitive. By Theorem 5.25, A is a translation AH-plane. / ( =^) Assume J n (K) is an AH-plane. Let N = ^_P \ p ~ o]: this is equivalent to our previous definition of K . Let K K: ^ (K) ~* A " be the nei g h bor map. Let G fe 0]; then G,0 6(g + 0) for some g in K and hence UK = Oj and (Kl) holds in K. Observe that P ~ Q 3 distinct components h,k such that 0,(Q P) I h,k <^ (Q P) Â€ N K . Let G e 0); let G,0 fc g where g Â€ K, and define c-^J^K) -^ J Q (K) by (Q P) e N <=> ((Q + G) (P + G)} fi m c^ Â°" (P)
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252 the kernel of 1. Let KÂ» = \Ug,l) 1 g Â£ K"J. If (*g,.L) is an element of KÂ«, observe that Xg = (*? \ P Â£ g"^ and, since g is a subgroup of 0] , * g i s a subgroup of Op. Also, by Proposition 2.31 P* I g* <=\$ 3 Q such that Q ~ P; Q Â£ g. Or P* I g* <^> P* t *g. Also, P* I (g + G)* 4=5 3 Q such that Q 6 (g + G). Thus, P* I (g + G)Â« 4=> P* 4. *(g + G) <Â£\$> P* e (*g _i_G*). Hence J (KÂ») is an affine plane isomorphic to A* and K* is a congruence. Thus, (K2a) is satisfied. Observe that l(g + G)fi(p + P)| =1 <Â£> (g + G) -ft (p + p) <=> K(g + G) tt K(p + P) <=> li(g + G) ft *(p + P)t = 1. Hence (K2) holds and K is a quasiccngruence. // B.12 Proposition . Let C = (A,U) be a coordinstized AH-plane such that the automorphisms of A are (!0),gjand ((0)',g )-transitive. Let V = (M, +,*,-) be the associated prequasiring. Then if we let SiC) = ((M, +)*(H, +),t(s,+) Q (M,+) *(M, +) 1 3 m * M * s = Â£(x,xm) 1 x Â€ !-'.] or 9 u fe M 4 s = ^(y-u,y) i y e k}"? ) , then S(C) is a seirdcongruer.ee on (K,+)x(H,+) and A is isomorphic to J^S(C)) under the obvious map. If A is a translation AH-plane, then S(C) is a quasicongruence. Proof . By left distributivity in V, the elements (s,+) in the definition of S(C) are subgroups of (K,+) x(K,+) . If g is a line in A not quasiparallel to g , then there exist m,b such that (x,y) I g if and only if (x,y) = (x,xm) + (0,b), and xy-dually if g is not quasiparallel to g . Hence the set of points on any line g is a right coset of some element of S(C) . In fact, it is

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253 obvious that A is isomorphic to Jjs(C)). One can use Luneburq's proof of his Satz 4.1 [(1962), page 2741 to show that S(.C) is a semicongruence: Luneburg does not assume the underlying group is abelian as we did in Proposition B.ll. If A is a translation AK-plane, then (M,+) is abelian; hence S(C) is a quasicongruence. // Remark . The asterisk preceeding "B.12 Proposition" indicates that this proposition is included primarily to relate our results to Luneburg 's: this proposition is not used in what follows except in an asterisked remark. B.13 Proposition . If A is an AH-plane and if t is a translation on A, then f is uniquely determined by its action on any point P of A. Proof . Let g be a line containing both P and tP. Then g is a trace of Tand hence r is a (TT(g) , g^)-endomorphism. By Proposition 4.7 (3), the action of T is uniquely determined by its action on P. // B.14 Construction of K:T Â— Â» K, of t and of W . .Let T = (A,P) be a pointed translation AH-plane. Let (W,o) be the translation group of A. If Q is a point of A, define T to be the translaQ tion which takes P to Q. If g is a line through P, define V! to' 9 be the set of all translations r such that Q I g; hence W is Q g the set of all translations with direction 1T(g) . The map

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254 g IÂ—* W is well defined by Proposition B.13 and is one-to-one by Proposition 2.34. Let K(T) = ( C.V, Â°) , t( W ,Â») I P I q}). If w:T Â— Â»Â• T 1 is a mcrphism in T , define K(w):K(T) Â— ^K(T') to be the map which takes f to t for every point Q in T. B.15 Proposition [Luneburg (1962), page 275, Satz 4.3 and its proof]. Let T = (A,P) be a pointed translation AH-plane. The raap /"T :T Â— * ~^ T ^ defined by letting **(C) be the translation which takes P to Q is a pointed AH-plane isomorphism. We combine the proof of this proposition with the proof of the following theorem. B.16 Theorem . The maps K:T p Â— * K and J:K Â— T p are functors and are reciprocal equivalences. If uj:T Â— * T' is a morphism in T , then r uQ w ^5 = wr (R) for all points Q,R in T. Proof . If K is a quasicongruence , Proposition B.ll shows that J(K) is a pointed translation AH-plane. If ^:Y. Â— * K' is a quasicongruence homomorphism, and if g e K, thenl^P \ p Â£ g^ is contained in a unique element of K' which we denote wg; hence if g + G is a right coset of an element of K, then ^(g + G) is contained in a unique right coset of an element of K': wg +' Â«g. Thus J(<-o) is an incidence structure homomorphism which takes base point to base point and preserves the parallel relation. Since uj(N ) C N , J(oo) preserves the neighbor relation on points. Since each image line contains at least two non-neighbor

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255 image points, and since J do.) preserves the neighbor relation on points, J(H = G + Q 3 G I g and KIk<=Â£K = G + Q' 3 G I g. Let T (Q) = Q'. Then H I h TÂ„ (H) = T (G + Q) 3 G I g <3> T_(H) = r(T (P)) = b b b b G + Q T G + (S + Q) (P) = T G (T S (T Q (F))) V Q,) = G + Q? 9 G Ig ' ThUS ' T (h) = k and h II k. Conversely, assume h \\ k. Let h = g + Q;

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256 let Q' Ik and let T (Q) = Q'. Then T (h) = k; so that since f s s s is an automorphism, k = {? I P = r (G + Q) 3 Q I gf = g + Q Â« . Thus, two lines are parallel if and only if they are right cosets of the same line g through P. It is new easily seen that T is isomorphic to j'( (ffj, + ) , \ (g, + ) I P I q\ ) and hence that the map /* T :T Â— J(K(T)) defined by^u. (Q) = T for every point Q is a pointed AH-plane isomorphism. Since J(K(T}) is a pointed AH-plane, J (K(T)) is an AH-plane, and K(T) is a quasicongruence by Proposition B.ll. Assume that w':(A,P) Â— Â» (A',:-') is a pointed translation AH-plane homomorphism. Let u = (A,A',f ). V/e wish tc show that K(w' ) is a quasicongruence homomorphism. Let W & K(A,P) and let T_ e W . Then F,C I g and K(Â«j') is non-degenerate and wP = pÂ«, there is a point Q I g such that wC HP ' by Proposition 2. Â£9; thus K(Q by Theorem 2.63. Hence P'-jQ' ^ P'uiQ. Then if V is a projection map for K(A',P'), then 8'K(w')W D , = X'V/ ^ PQ P ujQ *' W PV,o' = * ,K(rt,)w poÂ« by deposition 3.11. If r & n then T Â£ V/ ,W ,; g Â£ g'. Thus, Q ~ P, and ^C P Â• ; so that (K(w' )r ) Â€ M , , , . , , . It remains to show that K(wÂ») is a Q K ( A ' , t" ) group homomorphism; that is, that K(w')(r or ) = ~ G R (K(to' ) t ) Â»(K(ui' ) r ) . Since i-o is non-degenerate, we can pick a coordinatization K of A such that C = P and such that the" point u>Â£, and the lines ujg and tÂ»ig are the unit point and the x y xand y-axes of some coordinatization K' of A*: this is possible because w induces an AH-plane embedding u>*:A* Â— Â•> A'*

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257 where A* and A** are the gross structures of A and A 1 respectively by Theorem 2.53. Thus, *jR " (K(-i , )T G )Â«(K(w>')r ). Hence K(wi* ) is a group homomorphism and a quasicongruence homomorphism. If K is a category, it is then easily seen that K:T Â— Â» K is a functor. Let T = (A,P) and T' = (A 1 ,!' ) be pointed translation AHplanes. Let ^^, y* be the isomorphisms indicated earlier. We wish to show that a*:1^ Â— Â» JK is a natural isomorphism. Let is non-degenerate, W Â« is also, and hence by Proposition 2.68, ^i" 1 = J(Â£(u3))Â»a , and if K is a category, m> is a natural isomorphism. Assume ((0], + ),3) is a quasicongruence. Then J(S) is a translation AH-plane with translation group (V/,o) = (\r Â£ End J(S)I Q 6 0] and T Q (G) = G + Q for all G e 0)1,Â°). Observe that K(J(S) ) is a quasicongruence on (W,o) and if 1" Â»r Â€ W, then for all Q fc OJ, (lyr^Q = Tyo. + V)=Q + V + U= T V + (J (Q), and since (W,o) is abelian, TV r = T Thus r:(1*, + ) Â— * ( W Â•') i s a v U v + U group isomorphism. Recall that W fe K(J(S)) <=> 3q Â€ S. W' = g Â— g It I Q 6 gl. Thus, the map X :K(J(S)) --> s defined by A (r ) = Q is a group isomorphism and a quasicongruence isomorphism. We wish to show that if K is a category, then X:KJ Â— > 1~ is a

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258 natural isomorphism. Let ^:S Â—* S' be a quasicongruence homomorphism. If t^ fc K(J(s)), then X (K(J(o)))Q and u^r^ = Q; thus, X s ,K(J(uj)) = wA,,, and if K is a category, then X:KJ Â— * 1is a natural isomorphism. We now wish to show that the composition of two quasicongruence homomorphisms is a quasicongruence homomorphism. Let t<:K * K' and jj:K' Â— * K" be quasicongruence homomorphisms. We have that Â« = X K ,K(JU))X K ~ and f = X K Â„K( J(p ) ) A ~ 1 . Thus, since K(J(a)J(e<)) is a quasicongruence homomorphism, K"~^[ i *~^^K = P* is also Â» and K is a category. Hence, K:T p Â— * K and J:K Â— T p are functors and reciprocal equivalences. // ' Remark . The operation + defined on the points of the pointed translation AH-plane (A,P) in the proof of the above theorem is the same as that defined by the coordinatewise T-addition in (A,U) where U is any coordinatization such that = P: this can be seen by locking at Theorem 5.25. B.17 Corollary (compare Andre [( 1S54) , pages 162-165, \$ 2*]) . The functors K*:T* p Â— * KÂ» and J* :K* Â— Â• f* defined as restrictions of K and J respectively are reciprocal equivalences between the category of pointed translation affine planes and the category of congruences. Proof. This is immediate by Proposition B.10. //

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259 B.18 Proposition. Let (0],S) and (0]',S-) be quasicongruences. If w:J (S) Â— * J (S 1 ) is a non-degenerate translation AK-plane homomorphism which takes to Q, then ^ can be written as ^ = t J # ( S'; hence wis defined by ) which is defined by pR * ' = T ~ ui 1 Q Then
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260 isomorphism defined by (Q) = f where T is the translation which takes P to C, then we say (K(T),9 ) is a quasieongruence coord inatizat ion of A, and we usually identify the points of A with their images under 8 tn . For every translation AH-plane A, we pick a point P which we call the canoni cal base ooint of A. We A Â— Â— ^ Â— ___^_ Â— r call (K(A,P ) , 9, , .) the canonical auasiconqruence ~" A IA,P J * A coord inatizat ion of A. Since we usually denote the operation in the underlying group of a quasieongruence additively, we will abuse our notation and let (W,o) (C.v f o),K(A,F )) Â„ ({Â», + ), K(A,P )). ~ A ~ A abuse our notation and let (W,o) ('.Â•/, + ) and 1 =0. Then, A B.20 Proposition . There is a functor K :(T) n Â— * K from the category of translation AH-planes with non-degenerate AH-plane homomorphisms to the category of quasicongruences such that if u:A Â— * A' is a morphism in (T) ; and if (K,0) and (K',9*) are the canonical quasieongruence coordinatizations of A and A' respectively, then uo is defined (in terms of algebraic properties of the coordinatizations) by wQ = (K (oo)q) + loO; here to is an isomorphism if and only if K (<*>) is. Moreover, if A, A' are objects in (T) with canonical quasieongruence coordinatizations (K,6) and (K',0') respectively, if RÂ» is a point of A', and if *^:K Â— * K' is a quasieongruence homomorphism, then u.:A Â— Â» A* defined by j*Q = M>G + R' is a morphism in (T) n and Â»*0 = R', K T Q = (K (w)Q) + uO would read: <Â»)Q = r~ ((K t (Â«*>)t ) Â»r p ), where 6 and &Â• are both denoted by T. A

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261 Proof. Let A be an object in (T) n , and let K (A) = K(A,P ). ~T ~ A Let to: A Â— Â» A ' be a non-degenerate translation AH-plane homomorphism and let (K.,6) and (K',Â©') be the canonical quasicongruence coordinatizations of A and A' respectively. Let 6 = ^ r w0 ^ uj; then A takes to 0. Let 6' = ((A,P A ),(AÂ»,F A ,),f p ), and let p" = K(a'). Then J^(ft # ) agrees with a under the cocrdinatization identification since /* T .P' = 2 ( P >/*T implies & Tl(3 = J # (p # )Â© T where T = (A,P a ); TÂ« = (A.,P A ,). By Proposition B.ll, if r is the translation on A' which K takes P to R' t then T can be expressed in terms of the algebraic properties associated with the canonical quasicongruence coordinatization of A* by T (Q') = QÂ« + R'. Thus, in terms of the coordinatizations, to is defined by *>Q = J (^5 )Q + ujO = ^, lr Q + ujO. Denote p = K(ja') by K (w) , and observe that K (w) is a quasicongruence homomorphism and that Â£ T is now well defined. Also observe that since K and J are reciprocal equivalences, & is an isomorphism if and only if K T (w) is; hence, that "0 is an isomorphism if and only if K do) is. We wish to show that K :(T) n Â— * K is a functor. By our construction, K (1 ) = 1 . . for every object A. Assume Â«:A Â— * A', a:A" Â— Â» A" are morphisms in (T) . Then kQ = K T (ot)Q + *0 and f> R = Y^i^R + p0; hence, p (*Q) = K T (|J)(K T (Â«)Q + Â«0) + p0 = K T (p)(K T (Â«)Q) + K (p)(Â«0) + pO. Also, fi<*,(Q) = K T (^ot)Q + p*0. Letting Q = for the moment, we get pUO) = K' T (^)(c<0) +p0 = ^>Â«0. Thus, K (fl)(K (Â«)Q) =

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262 K T (p*)Q for all points Q of A; hence, K ~/*5 thus, K(jjM = \$ and K_,(^; = jf. //

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BIBLIOGRAPHY Andre, Johannes (1954) . liber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Mathematische Zeitschrift 60, 156-186. Artin, E. (1957) "Geometric Algebra." Interscience Publishers, Inc., New York. Artmann, Benno (1969) Hjelmslev-Ebenen mit verfeinerten Nachbarschaf tsrelationen. Mathematische Zeitschrift 112 , 163-180. (1970) Uber die Einbettung uniformer affiner HjelmslevEbenen. Abhandlung aus dem Mathematischen Seminar der Universitat Hamburg 34, 127-134. Bacon, Phyrne Youens (1971) On Hjelmslev Planes with Small Invariants (Master's Thesis). University of Florida, Gainesville, Florida. (1972) Strongly n-uniform and level n Hjelmslev planes. Mathematische Zeitschrift 127 , 1-9. (to appear) On the extension of projectively uniform affine Hjelmslev planes. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg. 263

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264 Corbas, Vassili (1965) La non esistenza di omomorf ismi propri fra piani affini. Rendiconti di Matematica e dell sue Applicazioni 24, 373-376. Cyganova, V. K. (UeraHOBa, B. K.) (1967) The H-ternar of an affine Hjelmslev plane. (Russian) (H-TepHap eJIbMCJieBOBOft a\$6HHHOK nJlOCKOCTH) CMOJieHCK. ToeIlefl. Hhct . y^en . 3an. 18, 44-6 9. Dembowski, P. (1968) "Finite Geometries." Springer-Verlag New York Inc., New York. Drake, David A. (1968) Projective extensions of uniform affine Hjelmslev planes. Mathematische Zeitschrift 105 , 196-207. (to appear) Existence of parallelisms and projective extensions for strongly n-uniform near affine Hjelmslev planes. Geometeriae Dedicata. Hall, Marshall, Jr. (1959) "The Theory of Groups." The Macmillan Company, New York. Hughes, Daniel R. , and Piper, Fred C. (1973) "Projective Planes." Springer-Verlag, New York. Jacobson, Nathan (1953) "Lectures in Abstract Algebra. Volume II Â— Linear Algebra." D. van Nostrand Company, Inc., Princeton, New Jersey.

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265 Klingenberg, Wilhelm (1954) Projektive und affine Ebenen mit Nachbarelementen. Mathematische Zeitschrift 60, 384-406. (1955) Desarguessche Ebenen mit Nachbarelementen. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg 2Â£, 97-111. Lorimer, Joseph Wilson (1971) Hjelmslev Planes and Topological Hjelmslev Planes (Ph.D. thesis). McMaster University, Hamilton, Ontario, Canada. (1973)a Coordinate Theorems for Affine Hjelmslev Planes. Mathematical Report No. 63. McMaster University, Hamilton, Ontario, Canada. (1973)b Morphisms and the Fundamental Theorem of Affine Hjelmslev Planes. Mathematical Report No. 64. McMaster University, Hamilton, Ontario, Canada. Lorimer, J. W., and Lane, H. D. (1973) Desarguesian Affine Hjelmslev Planes. Mathematical Report No. 55. McMaster University, Hamilton, Ontario, Canada. Liineburg, Heinz (1962) Affine Hjelmslev-Ebenen mit transitiver Translationsgruppe. Mathematische Zeitschrift _79, 260-288. Mitchell, Barry (1965) "Theory of Categories." Academic Press, New York and London.

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266 Pickert, Gunter (1955) "Projektive Ebenen." Springer-Verlag, Berlin. Sandler, Reuben (1964) Lectures on Projective Planes (Notes). Institute for Defense Analyses. Skornjakov, L. A. (Ckophhkob, Jl. A.) (1964) ' Rings chain-like from the left. (Russian) (UenHbie cneBa KOJibua) . "IlaMHTH H. r. ^e6oTapeBa. 1894-1947". Ka3aHb, Ka3aHCKHfl yH-T, 74-88.

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BIOGRAPHICAL SKETCH Phyrne Youens Bacon was born January 2, 1936, in Holbrook, Arizona. Her parents were Cynthia Tanner Youens and Willis George Youens, Sr., M. D. In May, 1953, she was graduated from St. John's High School in Houston, Texas. In Kay, 1959, she received the degree of Bachelor of Arts with a major in Physics from Rice Institute, Houston, Texas. From 1959 until 1961 she worked for Sperry Utah Engineering Laboratory. She enrolled in the Graduate School of the University of Tennessee in September, 1961, and in December, 1962, she received the degree of Master of Science with a major in Physics. She held a University of Tennessee Non-service Fellowship for the acedemic year 1962-1963. She was admitted to candidacy for a Ph. D. with a major in Physics in September, 1963. She held a NASA Traineeship from 1963 until 1966. She was admitted to candidacy for a Ph. D. with a major in Mathematics in September, 1965. In June, 1967, she enrolled in the Graduate School of the University of Florida. She worked as a graduate assistant in the Department of Mathematics in the academic year 1967-1963. In June, 1971, she received the degree of Master of Arts with a major in Mathematics. Phyrne Youens Bacon is married to Philip Bacon, and is the mother of two daughters, Jennie Webb Marquess and Cynthia Marquess. . 267

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, a< dissertation for the degree of Doctor of Philosophy. David A. Drake, Chairman Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as dissertation for the degree of Doctor of Philosophy. jÂ° JJR 0/ /Z Ernest E. Shult Professor of Mathematics 1 certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 4vk 7? ' ^v s Mark P. Hale Assistant Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. iVvj Â•-Kermit M. Sigmon Assistant Professor of Mathematics

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 7//^ <7 N /> Mark L. Teply Associate Professor Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. s rju Billy S. Tnomas Assistant Professor of Physics This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. June, 1974 Dean, Graduate School

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