UFDC Home |   Help  |   RSS
 Title: Coordinatized Hjelmslev planes
CITATION PDF VIEWER THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
 Material Information Title: Coordinatized Hjelmslev planes Alternate Title: Hjelmslev planes Physical Description: xxxiii, 267 leaves. : ; 28 cm. Language: English Creator: Bacon, Phyrne Youens, 1936- Publication Date: 1974 Copyright Date: 1974
 Subjects Subject: Geometry, Projective   ( lcsh )Mathematics thesis Ph. DDissertations, Academic -- Mathematics -- UF 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 Bibliographic ID: UF00098680 Volume ID: VID00001 Source Institution: University of Florida Holding Location: University of Florida Rights Management: All rights reserved by the source institution and holding location. Resource Identifier: alephbibnum - 000580698oclc - 14074879notis - ADA8803

coordinatizedhje00baco ( PDF )

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,