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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.

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

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

contradiction.

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.

(* (0) = 0*, K C (1) = *i, where 0* is the zero and I1

the one of (M*,T').

(Bll) Observe that since we have made no special require-

ments on C, the xy-duals of conditions (B0) through (B10) also

hold in B(C).

Thus we have shown that B(C) is a biternary ring.

Assume that w:C C' is a morphism in C, and that B(C) =

(N,T,T'), B(C') = (Q,S,S'). We wish to show that B(w):B(C) B(lC')
-I1
defined by B(C)m = V'("(C (m))) is a biternary ring homomorphism.

Define w':M -- Q by B(w)m = w'm. Then it is easily seen that

w(x,y) = (w'x,w'y), w[m,d] = [w'm,w'd] and wlu,v]' = ['u,t'vl'.

Thus since preserves incidence u'(T(x,n,e)) = S(w'x,w m,w'e)

and w'(T'(y,u,v)) = S'(w'y,w'u,w'v). Since (n,n) ~ (0,0) implies

(w'n,w'n) ~ (0,0), W'NM N N Observe that w'(0) = 0, w'(l) =

1. Hence B(w) is a morphism in B.

Obviously B(1 ) = 1 (C). If < :C1 a C2 and :C2 C3 are

morphisms in C, then if m 6 .1 B(rx)m = C3(K(l-l(m))) =

3( 0[ 2- f2 s(( (m)))) = B(p)(B(C)(m)), so that E(pc) = B(p)B(o)
and B:C --' B is a functor.

If C is a coordinatized affine plane., recall that N B(C)

{o] and hence observe that B(C) is a biternary field. //

3.18 Construction of C:; C and A:0)- ( I-x {( ),< )>')/=. Given

a biternary ring B = (C,T,T') we construct an incidence structure

SB = (P,c,I) and a parallel relation II in the following way. Let
A = M M and let cT = M M` xtO0 and T,' = M FIl. For

convenience we denote an element (m,d,0) of 0T by [m,d] and an

element (u,v,l) of qT, by [u,v]'. We define

6 : T -T M
S(g) = u>', if g = [u,v]'. We define incidence by (x,y) I Cm,d]
y = T(x,m,d) and by (x,y) I [u,v]' 4= x = T'(y,u,v). We

identify and (u>' 44> T(u,m,0) = 1 = T'(m,u,0). If

= (u)', and if there is a point (a,b) on both [m,dl and Cu,v]',

then by (B9) and (B9)', ((x,y) I [m,d] <-> (x,y) I [u,v]'), and

we identify [m,d] and [u,vl'. Denote the set of lines by &J where

(0 = (oT U T\ )/=. Define A: 0-- (M, A J >,<( )')/= in the obvious
way. We define g 11 h if and only if a(g) = &(h). We denote the

incidence structure and the parallel relation thus defined by

AB. We let B: [1,01 -- M be defined by SB(a,a) = a. Let K =

([0,0],[0,0]',(1,1),M, 5). We define C(B) to be (AB,KB).
Given a biternary ring homomorphism m:B -- B' we define

C(w):C(B) -- C(B') by C())(a,b) = (wa,wb), C(w)[m,d] = [Cm,wd]

and C(w)[u,v]' = [u,wv1]' for all points (a,b), lines [m,dl,

[u,v]' in AB. Observe that C(a) is well defined.

3.19 Lemma. Let (B,T,T') be a biternary ring. Then the

following conditions hold in the construction given above.

(a) If m,d a ', m N ;, then there is a unique line [u,v3' of

OT' such that [m,d = [u,v]'. If m 6 N, there is no such line
lu,v]'.

(b) If u,v 6 M, u N, then there is a unique line [m,d] of

9T such that [u,vl' = [m,d]. If u e N, there is no such line
[n,d].

(c) The function L: 0--- (N:xs >,x )' )/= constructed above is

well defined.

Proof. (a) Assume g = nm,d], m 4 N. Then (0,d) I g. There is

a unique u & N such that T(u,m,0) = T(u,0,l) since 0 m. Hence

T'(m,u,0) = 1, and there is a unique v such that (0,d) I Cu,v]'.

Hence [m,d] = (u,v]'. If [m,d] = [w,zl', then T(w,m,0) = 1 and

u = w. Also since 0 = T'(d,u,z), z = v. Thus [u,v]' is unique.

Assume g = [m,d], m 6 N. Then, since m ~ 0, 0 7 1, there

does not exist an element u C M such that T(u,m,0) = 1, and

hence there is no [u,v'' 6 qT' such that [m,d] = [u,v3'.

(b) This is the dual of case (a).

(c) If [m,d] = [u,v]', then &[m,d] = (m>) = (u = u,v';

hence L is well defined. //

3.20 Definition. If B = (M,T,T') is a biternary ring and if B' =

(M,T',T) is the dual of B, then (x,y) I Cm,d] in C(B) 4=

(y,x) I [m,dj' in C(B'), and (x,y) I [u,v]' in C(B) 4=

(y,x) I [u,v] in C(B'). The structure C(B') is said to be the

structure dual to C(B).

60
3.21 Prorosition. Let B be a biternary ri-ng and let B' be the

dual of B. The map 9:C(3) --C(B') defined by O(x,y) = (y,x),

[lm,d] = [m,d)' and 6[u,v]' = Eu,vl is an incidence structure

homomorphism which preserves and reflects the parallel relation. //

3.22 Proaosition. If 3* is a biternary field, then C(B*) con-

structed above is a coordinatized affine plane.

Proof. If B* = (7M,T*,T*') is a biternary field, then (M*,T*)

is a ternary field and it is easily seen that C(B*) is the

coordinatized affine plane commonly constructed over the ternary

field (M*,T*): see [Hall (1959), top of page 3563. //

3.23 Proposition. The map C:B -* C indicated above is a functor

from the category of biternary rings to the category of coordi-

natized affine Hjelmslev planes.

oof. Let B = (M,T,T') be a biternary ring. The structure AB

instructed above is an incidence structure SB with a relation

)arallel' defined on the lines of SB. We wish to show that AB

s an AH-plane. We start by proving a number of lemmas. To

void confusion, throughout the remainder of this proof we will

Ise the symbol to indicate the neighbor relation in B, and

:he symbol C to indicate the relation 'affinely neighbor' in AB:

see the definition of AH-plane, Definition 2.21.

(Lemma 1) Whenever P,Q are points, there is a line g such

61

(Proof) Let P = (a,b), Q = (a',b'). There are four cases.

(Case 1) Assume P = Q = (a,b). Then by (B2),

(a,b) I O[,bl.

(Case 2) Assume FP 0, a + a'. The system T(a,m,d) = b,

T(a',m,d) = b' has a unique solution m,d by (B6), and hence

P,Q I [m,d].

(Case 3) Assume P g Q, b b'. This is Case 2 in the

dual structure; hence there is a line Cu,v]' joining P,C.

(Case 4) Assume P Q, a a', b ~ b'. By (B7) there is

at least one line joining the two points. /

(Lemma 1) (a,b) is affinely neighbor to (a',b') 4=q a a',

b b'.

(Proof) Let P = (a,b), Q = (a',b'). If P = Q, then a ~ a',

b b' by (Bl) and P is affinely neighbor to itself since (a,b)

is on both CO,b] and [0,a' by (B2) and (B2)'. Thus, we may

assume P 1 Q.

( ) Assume P is affinely neighbor to Q. Then at least two

distinct lines g,h join P and Q. By (BS) if P,Q I g,h; then

g,h 6 T or g,h C 0 .,

(Case 1) Assume P,Q I g,h; g I h, and g,h & o Then the

system T(a,m,d) = b, T(a',m,d) = b' has two solutions for Cm,d].

Thus, by (B6) av a' and b b'.

(Case 2) Assume ?,Q I g,h; g / h, and g,h & 0T,. This

reduces to Case 1 in the dual structure.

(c-) Assume a ~ a', b b', P 1 Q. Then, by (B7) there

are at .least two lines joining P and Q; so that P [ Q. /

(Lemma 3) Let P I g,h. Then gg A hi = 1 if and only if

g 0 h.

(Proof) (4 ) Assume g 0 h, P I Q,h. We wish to show

(g hi = 1. Let P = (a,b).

(Case 1) Assume g = [m,e], h = [m',e'] and that Q I g,h;

Q x P. Let Q = (a',b'). By Ler a 2, a ~ a', b b'. Hence by

(B7), m ~ m', e ~ e'. Let (c,d) be a point on one of the lines;

say g. Then d = T(c,m,e).

Let d' = T(c,m',e').

By (E10),

d' d. Thus by Lemma 2, (c,d) O

a contradiction. Hence Ig 0 h\ =

(Case 2) Assume g,h & C ,.

structure, so Ig A hi = 1.

(Case 3) Assume g [m,e],

may assume u 6 t;, since otherwise

by writing h in the form rm',e'].

( ) Assume Ig I h\ = 1, P

(Case 1) Assume g = h. If

(0,e),(l,T(1,m,e)) I c, and since

(c',d'). By symmetry, g O h,

1.

This is Case 1 in the dual

h = Cu,vi'. By Lenma 3.19,

this can be reduced to Case

Since u 6 N by (38) Ig g

we

1

h\

I g,h. We wish to show g P h.

g = mC,e] then

0 i 1 by (B10) this case

doesn't occur, and dually if g = Lu,v '.

(Case 2) Assume g = [m,e1, h = Cm',e'], g 6 h. By (B5)

the uniqueness of the solution x = a to the equation T(x,m,e) =

T(x,mn,e') implies m i m'. Since by (B10) 0 1, and since

- is an equivalence relation there is a symbol c such that c + a.

Let d = T(c,m,e). Assume g 0 h. Then there is a (c',d') I h

such that c c', d d' by Lemma 2 and the definition of

'affinely neighbor'. Hence the equation T*(x',m*,e) =

T*(x*,m'*,e'*) has two solutions: x* = a*, x' = c', a contra-

diction. Thus g 0 h.

63

(Case 3) Assume g,h a T'' g h. This is Case 2 in the

dual structure, hence g ( h.

(Case 4) If none of the preceding cases occurs, by

Lemma 3.19 we may assume g = [m,e h = [u,v1' where u E N.

There is a symbol w such that w v. Assume g 0 h. Let z =

T(w,m,e); Q = (w,z). There is a point R = (w',z') on h such

that R O Q. By Lemma 2, w ~ w'. Since u E N, u ~ 0, and since

w' = T'(z',u,v) by (B10), w' = v'; so that w' ~ v. Thus, since

is an equivalence relation on B; w ~ v, a contradiction.

Hence g 0 h. /

(Lemma 4) If g = [m,e]; then g h if and only if h =

[m',e'3 and m ~ m', e e'. If g = Cu,v]'; then g O h if and

only if h = [u',v']' and u -u', v v'.

(Proof) (Case 1) Assume g = [m,e].

(=) Assume h = [m',e'], where m m', e e e'. Let (a,b)

be a point on one of the lines: say g. Let b' = T(a,m',e'). By

(B10), b ~ b'. Hence (a,b') I h and (a,b') O (a,b) by Lemma 2.

By symmetry, g D h.

( ) Assume g D h. If g = h we are done. Assume g P h.

By Lemma 3, g E1 h implies Ig n hi 1; hence by (ES), h / [u,vj',

u & N; hence by Lemma 3.19 (b), h = [m',e'lfor some m',e' in M.

If m + m', then by (B5), there is a unique point of inter-

section, a contradiction by Lemma 3. Hence m ~ m'. The

point (0,e) is on g. Let (u,f) I h such that (0,e) O (u,f).

Then, 0 u, e ~ f. By (B10) there is a unique z* such that

T*(0*,m*,z") = e*. Then e* = e'* = z* and e e'. Thus, m ~ m',

e ~ e'.

(Case 2) Observe Case 2 reduces to Case 1 in the dual

structure. /

(Lemma 5) The relation 'affinely neighbor' is an equiva-

lence relation on the points and lines of A .

(Proof) The relation 'affinely neighbor' is an equivalence

relation on the points of AB and is a reflexive and symmetric

relation on the lines of AB by Lemmas 2 and 4. Assume g 0 h,

h D k. If g C gT' then h 6 )T; so that k c aT by Lemna 4. Hence

g k by Lemma 4 and (Bl). Similarly, if g 6 0T,, then h,k C qT'

and g 0 k. Thus 'affinely neighbor' is an equivalence relation

on the points and lines of A /

(Lemma 6) If P is a point; g, a line, then there is a

unique line h such that P I h, h II g.

(Proof) (Case 1) Let P = (x,y) and g = Cn,b3. Then, by

(84) there is a unique symbol z such that y = T(x,m,z). Hence

[m,z3 is the unique line of aT parallel to [m,b3 containing P.

Assu-e P I [u,v]' and [u,vl' n Inr,b]. Then (u>' =
that since P I [u,v]',nm,z1, and Eu,v]' = [m,z]. Thus, [ m,z is

the unique line through F parallel to [m,bJ.

(Case 2) This case reduces to Case 1 in the dual struc-

ture. /

(Lemma 7) There is a map :A -- A which is a surjective

incidence structure homomorphism such that g n hi= 0 in AB implies

qg II fh in AB, and such that (0P =Q -C F 0 Q) and ('g = Ph

4* gO h.)

(Proof) If a & K, denote the n-equivalence class of a in

B by 8*. Then define q(a,b) = (a*,b'), q[m,d] = m*,d* and

flu,v]' = [0o,v']' when u & N. Then, since the map v:B --> B

induced by is a biternary ring homomorphism, ? preserves the

incidence relation. The remainder of the lemma is immediate by

Lemmas 2, 4 and 5. /

Thus, AB is an AH-plane. Observe that [0,01 [0,01]; that

(1,1) Z [0,0],[0,0o'; and that SB is a bijection. Thus C(B)

(AB,([0,0,[O0,0',(l,1),M,M )) is a coordinatized AH-plane.

If w:B -- B' is a biternary ring homomorphism, then C(w)

defined earlier is an incidence structure homomorphism which

preserves the parallel relation. Recall that w(NB) c NB'

Hence, if a ~ b in B, the ua ~ wb in 3'. Thus, (a,b) D (a',b')

in C(B) implies (oa,eb) 0 (wa',wb') in C(B'), and C(w) preserves

the neighbor relation on points and similarly on lines. Since

c(w) takes gx gy and E to the x-axis, the y-axis and the unit

point of C(B'), C(w) is a coordinatized AH-plane hcomoorphism.

Since C(lg) = 1(B) and C(p) C(p)C(), C:B --C is a

functor.

If B" is a biternary field, then N, = tO3 and C(B') is a

coordinatized affine plane since the neighbor map of C(B*) is

essentially the identity map. //

3.24 Proposition. The functor BC is the identity functor on B.

Proof. Let (B,T,T') be a biternary ring, and let (Q,S,S') =

BC(B). By the constructions, Q = B, and y = S(x,m,d) in BC(B)

4c C (x,y) I Em,d] in C(B) 4> y = T(x,m,d) in B. Similarly,

x = S'(y,u,v) in BC(B) 4> (x,y) I [u,v]' in C(B) <, x =

T'(y,u,v) in B. Hence BC(B) = (B,T,T').

66

Let '3:3 -- B" be a morphism in 5. Then C(w)(a,a) = (Qa,wa)
-I
for every a 6 3, and E(g(())a = (w)-l a = g"(Wa,za) = wa.

Thus BC:B -- is the identity functor on 3. //

3.25 Construction of 1C --B CB. Let C = (A,(g ,g E,,,)) be a

coordinatized AH-plane. Then CB(C) = (AB(), ([0,0o,[0c,0,(1,1),

M,nB(C))) and it can easily be seen by locking at the usual
identification of points and lines of C with their representations

that there is an isomorphism fC:C -4 C(C) which takes a point

(x,y) of C to the point (x,y) of CB(C).

3.26 Proposition. The map o: : -- C2 indicated above is a

natural isomorphism.

Proof. Let w:C C' be a morphism in C. Recall that B(w)m =

{'(W( -im)). Hence, if P is a point of OE in C, (w)(?) =

t'(wP). Thus if ? I CE in C, CB(w)(p P) = CE(w)(P,P) =

(B(w)(gP),B(w)(sP)) = (r'(wP), '(P)) = C,(wP). Thus the

diagram:

C c
c ----- ()

C' ---C (C')

commutes with respect to the x-axis of C, the y-axis of C and

all the points of GE, and hence it commutes with respect to all

the lines which are parallel to either g or g and hence with

respect to all the points of C. The image of each line of C in

CB(C') intersects either both C0,0o and [0,1i or both [0,0]' and

[0,1]' in CB(C') in points which are non-n2ichbor images of

points in C. Thus, since the diagram commutes with respect to

all the points of C, it commutes with respect to all the lines of

C. Thus P:-l CB is a natural isomorphism. //

We have shown the following theorem:

3.27 Theorem. The functors B:C -- B and C:B C are reciprocal

equivalences where C is the category of coordinatized affine

Hjelmslev planes and B is the category of biternary rings. //

3.28 Corollary. The functors B*:* -- B* and C*:B* C*

defined as restrictions of B and C respectively are reciprocal

equivalences where B* is the category of biternary fields and C'

is the category of coordinatized affine planes. //

3.29 Definition. There are reciprocal equivalences F':C" F'

and C :F' -- '* (where F* is the catecorv of ternarv fields)
-F --- -----------------------__
defined by letting F*(C) be (M,T) if B*(C) = (I,T,T'); by letting

C F(F) be the usual coordinatized affine plane constructed over
-F
a ternary field F (see the proof of Proposition 3.22); by letting

F* take a morphism w:C -r C' to (F'(C),F*(C'),f .() ) and by

letting C take a morphism o = (F,F',f) to (C '(F),CF*(F'),f ,)

where f is defined by f ,(x,y) = (f x,fy), f ,tm,d] =[p(m,ad]

and f ,0,vo = [O,av]'.

The composition of B':C' -- B and C *:F* -- is a

functor B*C *:F* -- B* from the category of ternary fields to
the category of bitrnar fields. If (,T) is a ternary field,
the category of biternary fields. If (M*,T*) is a ternary field,

Full Text

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COORDINATIZED HJELKSLEV PLANES By PHYRME YOUENS BACON A DISSERTATION PRESENTED TO TEE GRADUATE COUNCIL OF THE UNIVERSITY C? FLORIDA IN PARTIAL FULFILLMENT OP THE REQUIREMENTS FOR THE DEGREE CF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1974

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Copyright (c) 1974 by Phyrne Youens Bacon

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To my husband, Philip Bacon, to mymother , Cynthia Tanner Youens, and to the memories of my father, Willis George Youens, Sr., M.D., and my maternal grandmother, Fhyrne Claiborne Tanner.

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The white people must think paper has some mysterious power to help them on in the world. The Indian needs no writings; words that are true sink deep into his heart where they remain; he never forgets them. Four Guns, Oglala Sioux (1891) quoted in "I Have Spoken" compiled by Virginia Irving Armstrong (1971), The Swallow Press Inc., Chicago, pages 130-131.

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ACK NOWLE DGEME NTS I would like to express my thanks and appreciation to my advisor, David A. Drake, for his superior example, for his contagious enthusiasm, and for his excellent suggestions. I would like to thank the other members of my committee, Ernest E. Shult, Kermit N. Sigmon, Mark L. Teply, Mark P. Hale, Jr., and Billy Thomas, for their comments and suggestions. I would also like to thank George E. Strecker for his comments. I would like to thank Benno Artmann, N. D. Lane, and Willian J. LeVeque for having indirectly provided copies of papers referenced in my bibliography, [cygancva (1967)], LLorimer (1971)] and [skornjakov (1964)] respectively, and I would like to thank Wladimiro Scheffer for his translation of Ccyganova (1967)]. I also wish to thank the many professors and fellow students who have helped make my graduate study an interesting and challenging experience. I would especially like to thank Richard D. Present, Vj'illiam M. Bugg, >,;. Edward Deeds, E. G. Harris, Harold C. Schweinler, Don D. Miller, David R. Hayes, John G. Moore, Dennison R. Brown, George E. Strecker, W. Edwin Clark, and Charles I. Babst.

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TABLE OF CONTENTS ACKNOWLE DGEMENTS LIST CF DEFINED TERMS LIST OF FUNCTORS LIST OF SPECIAL NAPS KEY TO CATEGORIES KEY TO SYMBOLS ABSTRACT Sections 1. INTRODUCTION 1 2. HJELMSLEV PLANES 7 3. BITER NARY RINGS 4. SEMITRANSLATIONS AND GEOMETRY 5. SEMITRANSLATIONS AND ALGEBRA 6. PREQUASIRINGS AND QUASIRINGS 7. KERNELS OF QUASIRINGS 116 8. OTHER CENTRAL AXIAL AUTOMORPHISMS 123 9. AH-RINGS 139 10. HJELHSLEV STRUCTURES 150 11. DESARGUESIAN PH-FLANES 199 12. PAPPIAN CONFIGURATIONS 208 Appendices A. RESTRICTED 3ITERNARY RINGS 222 vi 44 69 36 106

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TABLE OF 20NTENTS continued Appendices continued B. QUASICONGRUSNCES 243 BIBLIOGRAPHY oco 253 BIOGRAPHICAL SKETCH 267

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LIST OF DEFINED TERMS Term Subsection natural isomorphism, functor 2.1 function, graph, compose, identity function, underlying set, concrete morphism, underlying set function, natural composition, natural identity rr.orphism, identity, concrete category, honomorphisms , map, map 2.2 generated by, generated by, of, with 2.3 surjective, injective, bijective 2.4 equivalence, isomorphism 2.5' reciprocal equivalences 2.6 preserve, reflect 2.7 incidence structure, points, lines, incidence relation, is incident with, lies on, is a point of, is on, goes through, is a line through, join, collinear, copunctal 2.8 incidence structure homomorphism 2.9 the category of incidence structures 2.10 the incidence structure induced from S by ~, the induced incidence structure 2.12 projective plane, ordinary projective plane 2.14 parallel, affine plane, ordinary affine plane 2.15 viii

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LIST OF DEFINED TERMS continued Term Subsection neighbor, projectively neighbor, protectively neighbor, projective Hjelmslev plane, PH-plane 2.17 projective Hjelmslev plane homomorphism, the category of projective Hjelmslev planes, the category of projective planes 2.19 parallel, affineiy neighbor, affinely neighbor, affine Hjelmslev plane, AH-plane 2.21 affine Hjelmslev plane homomorphism, the category of affine Hjelmslev planes, the category of affine planes 2.23 the gross structure, the neighbor map, the gross structure, the neighbor map 2.25 . is neighbor to, is not neighbor to, is not parallel to 2.26 non-degenerate, degenerate 2.30 direction 2.35 quasiparallel, quasiparallel , quasiparallel 2.37 lined incidence structure, base line, affine points, affine line, lined incidence structure homomorphism, the category of lined PH-planes, the category of lined projective planes, the category of lined incidence structures 2.39 the generalized incidence structure, the lined generalized incidence structure, 2.40

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LIST OF DEFINED TERMS continued Term Subsection generalized point, generalized line, generalized incidence structure, line, point, incidence structure 2.40 neighbor, generalized neighbor relation, neighbor relation, neighbor, neighbor relation, generalized neighbor relation 2.43 derived from, derived from, derived from H by use of, derived, extended to, extended to, extensions of 2.51 embedding, incidence structure embedding of S into S' 2.52 generalized AH-plane, generalized AH-plane homomorphism, the category of generalized AH-planes 2.58 AH-plane embedding 2.62 ternary field, zero, one, symbols, ternary field homomorphism, the category of ternary fields 3.1 neighbor, not neighbor, biternary ring, right zero divisor 3.2 dual 3.3 symbols, zero, one, biternary field -3.4 biternary ring homomorphism 3.9 the category of biternary rings, the category of biternary fields 3.10

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LIST OF DEFINED TERMS continued ,?erm Subsection coordinatized AH-plane, coordinatization, symbols, y-coordinate , x-coordinate, representation, representation, representations, X-axis, y-axis, origin, unit point 3.11 xy-duals 3.12 coordinatized AH-plane homomorphism, coordinatization homomorphism, neighbor map 3.14 the category of coordinatized affine Hjelrr.slev planes, the category of coordinatized affine pl.ines 3.15 dual 3#20 the biternary field associated with 3.29 generated by, the AH-plane generated by a biternary ring 3.31 dilatation, trace, semitranslation, gemitranslation with direction P, translation 4.1 (P,g)-endomorphism, central axial endomorphism, center, axis, (P,g) -endomorphism, central axial endomorphism, center, axis 4.3 neighbor, neighbor endomorphisms 4.6 j-angle, vertices, sides, triangle, T-related, (^j^^-related, W .g^-j-Desarguesian, Cr r g w )-H-Desarguesian 4.10 the canonical expansion of a (j-l)-angle to a j-angle 4#12 (P,Â£) -mime tic 4.13

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LIST OF DEFINED TERMS continued T erm Subsection (P,g)-transitive 4.17 order, infinite order 4.20 T-addition, T-multiplication, T'-addition, T'-multiplication, linear, linear 5.1 (k)-regular for s, (k)-regular, (k) '-regular for s, (k ) '-regular , axially regular, regular 5.10 T-regular, regular, axially regular, regular in the direction T 5.13 the category of axially regular biternary rings, the category of axially regular coordinatized AH-planes 5.21 the category of coordinatized translation AH-planes, the category of regular biternary rings 5.26 quasifield, zero, one 6.2 addition, first multiplication, second multiplication, prequasiring, right zero divisor, zero, one 6.3 quasiring 6.5 dual 6.6 skew quasiring 6.7 prequasiring homomorphism 6.11 the category of prequasirings, the category of quasirings 6.12 generated by, 6.19 xii

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LIST CF DEFINED TERMS continued Term Subsection the AH-plane generated by a prequasiring 6.19 biquasif ield, the category of biquasif ields, the category of coordinatized translation affine Planes 6#20 kernel, the kernel of a quasiring 7.1 local ring 7 # 2 trace preserving, a trace preserving endomorphisrn of the translation group 7.5 left modular for s, left modular, strongly left modular, T'-v:eakly left modular for s S.l strongly (p, g w ) -transitive 8.6 P-related, (F,gÂ«)-related, (P.g^-H-Desarguesian, strongly (P.g^J-H-Desarguesian 8.8 ((0),C0,0]')-normal for s, ( (0 ) , [0 ,0] * )-normal , T'-weakly ( (0 ) , [0,0] Â• )-norxal for s, T-weakly ( (C ) , l0,0] Â• ) -normal for s 8.10 (<0>Â» t [0 f 0l')-normal for s, (*(0) Â• ,[0,0l Â• )-normal 3.15 affine Kjelmslev ring, AH-ring, Hjelmslev ring, H-ring 9<1 AK-ring homomorphism, the category of AH-rings 9.2 kernel quasiring, the category of kernel quasirings 9.3 Desarguesian, the category of coordinatized Desarguesian AH-planes 9.6 the AH-plane generated by an AH-ring, generated by g s xiii

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LIST OF DEFINED TERMS continued Term Subsection Hjelmslev Desarguesian, strongly Hjelmslev Desarguesian 9.9 the category of division rings 9.12 the category of coordinatized Desarguesian affine planes 9.13 the Hjelmslev structure of an AH-ring S 10.3 near, near, near, near 10.6 the lined affine Hjelmslev structure of S 10.9 extended to, extended to (HtSl^g^) through yu., extended to, extended to H(S) through yu. 10.10 projectively Desarguesian 10.16 Desarguesian 10.17 (P,g)-automorphism, ( P ,g)-transitive 10.19 full j-vertex 10.26 Hjelmslev structure, near, Hjelmslev structure homomorphism, gross structure, neighbor map, the category of lined Hjelmslev structures 10.36 full, a full lined Hjelmslev structure homomorphism 10.39 basis triple 10.46 Klingenberg ccordinatization -10.47 Klingenberg coordinatization 10.49 induces x through X and X 1 , is induced through X and X' by
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LIST OF DEFINED TERMS continued Term Subsection extension of Â« through X and V 10.52 the canonical basis triple, canonical cocrdinatization, basis triple 10.59 the category of Desarguesian AH-planes with non-degenerate AH-plane homomorphisms

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LIST OF DEFINED TERKS continued Terrr ' Subsection congruence 3 2 semicongruence, projection map, quasicongruence B.3 quasicongruence homomorphism ' B.4 pointed AH-plane, base point, pointed AH-plane homomorphism, the catecory of pointed translation AH-planes, the category of pointed translation affine planes E.8 parallel 3 9 quasicongruence coordinatization, the canonical base point, the canonical quasicongruence coordinatization B.19 the category of translation AH-planes with non-degenerate AH-plane homomorphisms B.20

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LIST CF FUNCTORS Funct

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LIST OF FUNCTORS continued Functor Subsection 6.18 6.18 6.20 6.20 7.6 7.6 9.4 9.4 9.11 9.11 9.16 9.16 10.2 10.63 D:((H ) ) f -* ( D ) n 10.66 Q

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LIST OF FUNCTORS continued Functor Subsection 2 F Â«:FÂ— Z* Am7 F * :Z* A. 7

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LIST OF SPECIAL MAPS Symbol In use Subsection f w (A,A',f M ) 2.2 K k:A Â— Â• A* 2.25 K K:H Â— * H* 2.25 TT TV(g) 2.35 g g 2.40 P P(TT) 2.40 S S (H,g) 2 47 T T(x,m,b) 3.2 TÂ« T'(y,u,v) 3.2 % y.OE Â— M 3.11 + a + b 5.1 X Â„ a%b = ab 5.1 a Â• b 5.1 a-b 5.1 Z Z(x,m,a) 5.4 2* Z'(y,u,b) ( 5.4 i rx. 10.2 i y Â± s 10.2 G G (H(S),h) 10 9 G G s 10.9 Kern Kern Q 7.1

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LIST OF SPSCIAL NAPS continued Symbol In use Subsection -f

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KEY TO CATEGORIES Equivalent categories are joined by [Â• Category Subsection S incidence structures 2.10 A s lined incidence structures 2.39 A rA* affine planes 2.23 H * lined projective planes 2.39 H* projective planes 2.19 A AH-planes 2.23 c A generalized AH-planes 2.58 H PH-planes 2.19 H lined PH-planes 2.39 9 A -C coordinatized AH-planes 3.15 a ~B biternary rings 3.10 A 2 restricted biternary rings A. 6 A "~C* coordinatized affine planes 3.15 ~B* biternary fields . 3.10 A -Z* restricted biternary fields A. 6 F* ternary fields 3.1 . xxii

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KEY TO CATEGORIES continued Category Subsection T< ~2 axially regular coordinatized AH-planes 5.21 ~ B 2 axially regular biternary rings 5.21 L " v prequasirings 6.12 -Cf coordinatized translation AH-planes 5.26 _B r regular biternary rings 5.26 U Q quasirings 6.12 'Â•"^T* coordinatized translation affine Planes . 6.20 ~ B r * regular biternary fields 5.30 -Q* biquasif ields 6.20 Â•-Qp* quasifields A. 29 -C n coordinatized Desarguesian AH-planes 9.6 kernel quasirings 9.3 *-R AH-rings 9.2 ~^ D * coordinatized Desarguesian affine planes 9.13 -Qj^* kernel biquasif ields ' 9.14 a R * division rings 9.12 H s Hjelmslev structures 10.36 ( T ) translation AH-planes with nondegenerate homomorphisnis B.20 a, L M left modules 7.5 K

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KEY TO CATEGORIES continued Category Subsection -(D) Desarguesian AH-planes with nondegenerate homomorphisms 10.63 L -((H ) ) lined Hjelmslev structures with S g full lined Kjelmslev structure 10.66 homomorphisms [~K quasicongruences B.6 T p pointed translation AH-planes B.8 |~-K* congruences B.6 T* p pointed translation affine planes B.8

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KEY TO SYMBOLS Symbol Subsection Â— neighbor 2.17 2.21 3.2 6.3 9.1 A.l ^ not neighbor (see above) W parallel 2.15 2.21 tt" not parallel (see above) \\ quasiparallel 2.37 Â•H" not quasiparallel (see above) ^ near 10.6 # not near (see above) *,
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KEY TO SYMBOLS continued Symbol Subsection TV(g) the direction containing the line g 2.35 Â°" semitranslation 4.1 T Â» T , T, ,. translation 4.1 7 4 B.14 w, w* , any small Greek letter homomorphism 2.2 zero 3.1 3.2 3.11 6.2 6.3 9.1 1 one 3.1 3.2 3.11 6.2 6.3 9.1 2.8 2.8 2.8 2.21 Â°1

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A v KEY TO SYMBOLS continued Symbol Subsection A * the gross structure of A 2.25 A* affine plane 2.15 (A,K), C, C* coordinatized AH-plane 3.11 A B the AH-plane generated by a biternary ring 3 3.18 3.31 the AH-plane generated by a prequasiring V 6.19 3.31 3.18 A the AH-plane generated by an AK-ring S 9.8 6.19 3.31 3.18 B, B', (M,T,T') biternary ring 3.2 C f C Â• , (A,K) coordinatized AH-plane 3.11 D set of semitranslations 4.22 Dp set of semitranslations with direction V" 1 Â• 4.22 E unit point 3.11 E W (A), E y the ring of trace preserving endomorphisrr.s of the translation group of A 7.4 7.6

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KEY TO SYMBOLS continued Symbol Subsection g, h, k line 2.8 9* the class of lines neighbor to g 2.17 2.21 2.43 g*Â» h *Â» k* line of an affine or projective plane 2.14 2.15 g set of directions 2.35 2.40 2.43 x-axis 3.11 y-axis 3.11 PH-plane Â• 2.17 the gross structure of H 2.25 projective plane 2.14 lined PH-plane 2.39 lined projective plane 2.39 incidence relation 2.8 coordinatization 3.11 quasicongruence Â• B.4 congruence B.2 direction containing [m,0] 5.9 a line not quasiparallel to g 3.11 side of a j-angle 4.10 the set of symbols 3.1 xxviii g x

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KEY TO SYMBOLS continued Symbol Subsection M Â» M" the set of symbols continued ( M , T , T Â• )

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KEY TO SYMBOLS continued Symbol Subsection Q* the class of points neighbor to Q continued 2.21 2.43 Q*> p * point of an affine or projective plane 2.14 2.15 Q, (M,+,x,-) quasiring 6.5 R, (R,+,x) H-ring, AH-ring 9.1 R, (M,T,T") restricted biternary ring A.l (Sig) lined incidence structure 2.39 S, R, (S,+,x) AH-ring 9.1 S, (OJ,S) semicongruence B.3 T > T' ternary operation 3.2 T " partial ternary operation A.l T Â» (A,P) pointed translation AH-plane B.B (u) ' direction containing [u,o]' 5.9 Lu,v]' line not quasiparallel to q 3.11 x V, (M,+,x,-) prequasiring 6.3 W the set of translations 4.22 (W,o) the group of translations4.22 5.24 7.4 x, a, c the x-coordinate of a point 3.11 YÂ» b Â» d the y-coordinate of a point 3.11

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Abstract of Dissertation Presented to the Graduate Council "of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COORDINATIZED HJELKSLEV PLANES By Phyrne Youens Bacon June, 1974 Chairman: David A. Drake Major Department: Mathematics A coordinatization may be thought of as an isomorphism between a geometric structure and a geometric structure which has been constructed from an algebraic structure. Affine Kjelmslev planes (AH-planes) are cocrdinatized by using biternary rings; translation AH-planes by using cuasirings; Desarguesian AH-planes by using AH-rings; and Desarguesian projective Hjelnslev planes (Desarguesian PH-planes) by using H-rings. An affine plane homomorphism crf:A Â— * A' is an incidence structure homomorphism which preserves the parallel relation. Iferf is non-degenerate (that is, it does not map all the points of A into points of a single line of A') then
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If attention is restricted to AH-plane homomorphisms which take x-axis to x-axis, y-axis to y-axis and unit point to unit point, then the following pairs of categories are equivalent: biternary rings and coordina tized AK-planes, quasirings and coordinatized translation AH-planes, AH-rings and coordinatized Desarguesian AH-planes. The category of quasicongruences is equivalent to the category of pointed translation AH-planes, and the category of Desarguesian AH-planes with non-degenerate AH-plane homomorphisms is equivalent to the category of lined Hjelmslev structures with full lined Hjelsmlev structure homomorphisms. Desarguesian PH-planes are Hjelmslev structures in which every two lines meet. The directions of the x-axis and the y-axis are denoted by (0) and (0)'. The translations of a coordinatized AH-plane C are (tO^tg*,)and ( (0) Â• ,g M )-transitive if and only if the biternary ring (K,T,T') of C satisfies the following conditions: 1) T and T' are linear. 2) The Tand T'-additions are equivalent: a + b = a Â• b for all a,b in K; that is, T(a,l,b) = T'(a,l,b) for all a,b in M. 3) (K,+) is a group. 4) xm + sm = (x + s)m and x-m + s-m = (x + s)Â« m for all x,s,m in M where the two multiplications are defined by ab = T(a,b,0) and a-b = T'(a,b,0) for all a,b in M. A coordinatized AH-plane whose biternary ring satisfies the conditions listed above is a translation AH-plane if and only if the addition + is abelian. Translation AH-planes, Desarguesian AH-planes, Pappian

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translation AK-plar.es, Desarguesian i : H-planes and Pappian Desarguesian PH-planes are each characterized geometrically, in terms of algebraic properties of their coordinatizations, and (for all except the Pappian planes) in terras of properties of their endomorphisms. Algebraic characterizations are given of those coordinatized AH-planes which have a semitranslation (in an affine plane a semitranslation is a translation) with direction (k) which moves the origin to (s,sk), and of those coordinat'ized AH-planes which have a ( (0,0 ) ,g 0O )-endomorphism which moves (1,1) to (s,s). There are similar results concerning ( (0 ) , [0,0] ' )and ( (0) Â• ,[0,C]' )-automorphisms where [0,0]' is the y-axis and (0)' is the direction of the y-axis. If H is a PH-plane; if s , s , s are the sides of a triangle whose image in the gross structure of H is non-degenerate, and if each of the three AH-planes A , A , A derived from H by use of one of the sides s , s , s is Desarguesian, then H is Desarguesian. There exists a Desarguesian AH-plane which cannot be derived from any Desarguesian ?H-plane.

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1. INTRODUCTION A coordinatizaticn may be thought of as an isomorphism between a geometric structure and a geometric structure which has been constructed from an algebraic structure. Klingenberg [(1955)1 began the solution of the coordinatizaticn problem for Hjelmslev planes by constructing a projective Hjelmslev plane from an H-ring, and he showed that this constructed FH-plane has a number of properties. Luneburg [(1962)3 defined an algebraic structure (which is here called a quasicongruencc ) and showed that any translation affine Hjelmslev plane (translation AH-plane) can be coordinatized (in the sense mentioned above) by using a quasicongruence. Lcrimer L( 1971)3 continued work on the coordinatization problem by constructing an affine Hjelmslev plane from an AH-ring and then giving various theorems relating these constructed planes to the class of Desarguesian AH-planes. Cyganova C(1967)l a i so did considerable work on the coordinatization problem: she undertook to define an algebraic structure which would have essentially the. same relation to affine Hjelmslev planes that ternary fields have to affine planes. Unfortunately, her arguments contain a number of serious omissions, not all of which I have been able to repair. Her algebraic system has a ternary opertion and a partial ternary operation. In Definition A.l, a similar (but different) 1

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algebraic system is defined which is called a restricted biternary ring. In Definition 3.2, an algebraic system with two (complete) ternary operations is defined: this system is called a biternary ring. An AH-plane homomorphism which takes x-axis to x-axis, y-axis to y-axis and unit point to unit point is called a coordinatization homomorphism (Definitions 3.11 and 3.14). In Theorem 3.2 7, it is shown that the category of biternary rings is equivalent to the category of coordinatized affine Hjelmslev planes (with coordinatization homomorphisms ) . This result is used (along with others) to show that the category of quusirings (these have two multiplications) is equivalent to the category of coordinatized translation AH-planes (Corollary 6. IS), and that the category of AH-rings is equivalent to the category of coordinatized Desarguesian AH-planes (Proposition 9.11). The quasiring equivalence is used to prove the existence of a module isomorphism (in fact a natural transformation between functors) which relates the kernel of a quasiring to the ring of trace preserving endomorphisms of the translation group of the associated AH-plane (Theorem 7.7). Even without considering possible algebraic connections, there is considerable interaction between geometric properties of a Hjelmslev plane and the existence of certain types of endomorphisms of the plane. In Theorem 4.21, a geometric characterization of those AH-planes whose automorphisms are (r f g^-transitive for any given direction V is given. In Proposition 8.9, a geometric characterization of those AH-planes

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3 whose automorphisms are (P,gÂ„,) -transitive for any given point ? is given. These results are used (together with some algebraic ones) to give geometric characterizations of translation AH-planes (Theorem 5.25), of Desarguesian AH-planes (Proposition 9.10) and of Desarguesian FH-planes (Theorem 11.5). An AH-plane homomorphism w:A Â— * A 1 is required to preserve the incidence, parallel and neighbor relations. An affine plane homomorphism oc:A Â— > A' (Â«( is an AH-plane homomorphism between affine planes) is said to be non-degenerate if it does not map all the points of A onto a single line of A'. A non-degenerate affine plane homomorphism is an incidence structure embedding and reflects both the incidence and parallel relations (Â°c g \\ reflects the quasiparallel and the neighbor relations (Corollary 2.65). In Theorem 5.11, an algebraic characterization is given of those coordinatized AH-planes which have a semitranslation with direction (k) which moves the origin to (s,sk) (in an affine plane a semitranslation is a translation); in Proposition 3.2, of those coordinatized AH-planes which have a ( (0 ; 0) .g^-endomorphism which moves the unit point to the point (s,s). Propositions 8.11 and 8.16 give similar results for ( (0 ) , to ,o] Â• )and ( (0) ' ,[0,0 3' )-automorphisms respectively: here ["0,01' is the y-axis and (0) and (0) 1 are the directions of the x-axis and the y-axis respectively.

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4 Theorem 5.29 shows that the translations of a coordinatized AH-plane C are 1(0), g M )and ( (0) Â» .g^) -transitive if and only if the biternary ring (K,T,T') of C satisfies the following conditions: 1) T and T' are linear. 2) The Tand T'-additions are equivalent: a + b = a * b for all a,b in M; that is, T(a,l,b) * T'(a,l,b) for all a,b in K. 3) (M,+) is a group. 4) xra + sm Â« (x + s)m and x.rn + s-m = (x + s)-m for all x,s,m in M where the two multiplications are defined by ab c T(a,b,0) ond aÂ»b = T'(a,b,0) for all a,b in M. Theorem 5.25 shows that a coordinatized AH-plane whose biternary ring satisfies the conditions listed above is a translation AH-plane if and only if the addition + is abelian. Klingenberg [( 1555)3 attempted to characterize in terms of their automorphisms those FH-planes which are isomorphic to some PH-plane constructed from an H-ring. His argument fails however. In Theorem 11.6, such a characterization is given. This theorem also indicates a geometric characterization of these t=H-planes (they are called Desarguesian PH-planes). Lorimer [(1971)3 generalizes part of what Artin [(1957)3 calls "the fundamental theorem of projective geometry" by proving some results relating the automorphisms of a Desarguesian AH-plane which fix the origin to a set of semilinear transformations. Theorem 10.63 shows that there is a functor Y from the category of Desarguesian AH-planes with non-degenerate AH-plane homomorphisms to the category of AK-rings; if ^:A -, ais a morphism

6 The development here depends only on well-known results from the theories of categories, sets, algebra and affine and projective planes. Paragraphs marked with asterisks are not part of this development and are intended to relate results here to other portions of the literature. Since some of the treatment of Desarguesian AH-planes given here resembles some of the recent work of J. W. Lorimer and X. D. Lane [(1973)1 and of J. W. Lorimer [(1973)a and Q973)b], there are discussions of their recent papers and a brief history of this research in Appendix A: A. 24 through A. 23. If it is desired, Appendix A (restricted biternary rings) can be read immediately following Section 3 (biternary rings) and Appendix B (quasicongruences ) can be read immediately following Section 5 ( semitranslaticns and algebra). Nothing in any of the sections depends on anything in the appendices.

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2. HJELMSLEV PLANES In this section we give a number of definitions (some of v.'hich differ slightly from the usual ones and some of which are entirely new), prove a few basic propositions, and discuss the relationship between affine and projective Hjelmslev planes. Proposition 2.31, Theorem 2.63, and Corollaries 2.64 and 2.65 concern non-degenerate homomorphisms. Proposition 2.56 concerns isomorphisms. 2.1 Definitions . We will assume that the reader is familiar with the elementary definitions of category theory. Those of the necessary category theory definitions which are not given here are stated by Mitchell [(1965), pages 1-5, 49, 52, 59"] . We will commit a common notational inconsistency by usually denoting a category with objects A and morphisms M by simply A; our names for the various categories will reflect this notation. We will use the term ' natural isomorphism ' in plane of Mitchell's term 'natural equivalence', and the term ' functor ' in place of Mitchell's term 'covariant functor'. 2.2 Definitions . A fu nction f from a set A to a set. B is an ordered triple (A,B,G ) where G is a subset of A*B such that, if a is an element of A, there is a unique element b of B such that (a,b) 6 G . G is called the graph of f. If (a,b) Â€ G , 7

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we write f (a) = b. Two functions f = (A.B.G,) and q = (C.D.G ) r g compose to give gf = (A,D,G ) if and only if the domain C of g is equal tc the codorr.ain B of f, and, if C = B, then G is gf defined by gf(a) = g(f(a)) for every a in A. We call the function (A,A,G ) defined by 1 (a) = a for every a in A the A A identity function on A. Let A be a class such that for each A 6 A, there is a set U(A), called the underlying set of A. An ordered triple to = (A,B,f^) is said to be a concrete moronism (on A), and is written w:A Â— B, if A,B e A and f^ is a function from U(A) to U(B); that is, f^ = (U( A) , U(B) ,G ). If fja) = b for some a in U(A), we write wa = b and to(a) = b. The function f is said to be the underlying set function of to, and we define U(u>) to be f . If tf = (A,B,f^) and a = (B,C,f ) are concrete morphisms on A, we say that ( A,C , f.f^ ) , denoted /Sot, is the natural composition of & by oc. We denote (A,A,1 ) by 1 and call 1. the natural 1 U v A I A A identity morphisn on A or simply the identity on A. A concrete category A is a category whose morphisms are a subclass M of the class of all concrete morphisms on A, whose identity morphisms are the natural identity morphisms, and whose composition is the natural composition. All of the categories which we construct in what follows are concrete categories. Once we have given a class A and a subclass M of concrete morphisms called *A homomorphisms ' , we will assume that the reader will be able to identify the underlying sets of the objects in A and to verify that the given subclass of concrete morphisms is closed under natural composition and contains the natural identity

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9 morphisms. Once this has been verified, it is immediate that A is a concrete category since [a,BJ-a will always be a set; the A morphisms of A will always satisfy the requirement that *(&o0 = U&)<< where defined, and 1 where defined. Instead of saying 'the concrete morphism u = (A,B,f )', we will usually say 'the map co:A Â— *Â• B'. In an abuse of terminology, we say 'the map F:A Â— * B' when A, 3 are classes and F is a subclass of A*B such that for every A in A there is a unique B in B, written F(A), such that (A,B) Â€ F. 2.3 Definitions . If there is a map F:A Â— Â» B which 'forgets' structure, or which is the functor we construct from A to B, we frequently say that a concrete morphism oo = (AjA'jf^) on A is a B homomorphism if (F( A) , F( A ' ) ,f ) is a B homomorphism. For example, if w = (A,A',f ) is a concrete morphism on AH-planes, and if A = (S,ll), A' = (S',||) then we say uj is an incidence structure homomorphism if (S,S',f ) is an incidence structure homomorphism. We abuse our terminology in other similar respects; for example, we speak of the 'lines* of an AH-plane A = (S,t\) when we mean the lines of S. Once we have constructed a functor G:C Â— * D and shown that G is an equivalence, we will call G(C) where C is. an object in C the D-object generated by C. If F:D Â— Â»Â• Â£ is an obvious functor which 'forgets' structure, we will call FG(C) the E-object generated by C. For example, in Section 3 we construct an equivalence C:3 Â— Â» C, and if 3 is a biternary ring, we call C(B) the coordinatized AH-plane generated by B, and we call the

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10 AH-plane of C(B) the AH-plane generated by B. We will frequently say 'of' in place cf 'generated by', and ' with ' in place of 'which generates the'. 2.4 Definitions . Let *:A Â— Â» B be a concrete morphism. If the underlying set function of <* is onto, * is said to be surjective . If the underlying set function of * is one-to-one, Â«* is said to be i"jective . If Â« is both injective and surjective, it is said to be bi jective . 2.5 Definitions . A functor F:C Â— D is said to be an equivalence if and only if there is a functor G:D Â— " C together with natural isomorphisms *?:1Â— Â»Â• FG and *:G? Â— * 1Â». If an equivalence F:C Â— * D. produces a correspondence between the objects of C and D, then it is said to be an isomorphism if it is one-to-one, onto. 2.6 Definition . If F:C Â— * D and G:D Â— C are functors, and if ^ : ift Â— * Â£G and ^ : Â£Â£ Â— *1a sre natural isomorphisms, we say that F and G are reciprocal equivalences . 2.7 Definitions . If A is a class of objects such that a binary relation P is defined on the underlying set of each object A of A, then a concrete morphism w:A Â— * B is said to preserve p if whenever (a,b) Â« f for some a,b in U(A); it is also' true that dÂ»>(a) ,w(b) ) < * , We say that uj reflects p if whenever (w(a) ,w(b) ) * P.. for some a,b in U(A); it is also true that (a,b) fi P.

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11 2.8 Definitions . Let (I 8 , o],.I) be a triple of sets. Then (^,0],!) is said to be an incidence structure if *Pand o] have no elements in common and I c. *)a x 03. The elements of fare called points , the elements cf oj are called lines , and if P Â€ IP, g e <7j , we write Pig whenever (P,g) 6 I. The set I is called the incidence relation . If P I g, we say P is incident with g, P lies on g, P is a point of g, or P is on g. We also say g goes through P, or g is a line through P. We use other similar geometric language to designate incidence or non-incidence. For example, g is said to join P,Q if P,Q I g; that is, if P I g and Q I g. The capital letters P, C, G, K will be used to designate points unless otherwise indicated; similarly, the small letters g, h, k will be used to designate lines unless otherwise indicated. Points which all lie on a common line are said to be collinear , and lines which all go through a common point are said to be copunctal . We say I g A h I = n if the cardinality of the set of points incident with both g and h is n. We let gH h denote the point or set of points incident with both g and h. We define Ig (\ h A kl and g f\ h fi k similarly. We write PQ to denote the line or set of lines joining P and Q. Occasionally we write P v Q instead of PQ. 2.9 Definition . Let S = (T*,flj,I) and S' = Vp.' , oj' , I Â• ) be incidence structures. An incidence structure hcmomorphism u:S Â—+ S' is a concrete morphism which satisfies the following conditions. (1) f w is a function from I 1 U oj to V U o,'. (2) wCP) C Ti'.

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13 "Â•-equivalence class will frequently be denoted by a theretofore unused letter with an asterisk; for example, t*. Once t* has been used, however, t will denote an element of t*. Asterisks used as superscripts do not necessarily denote equivalence classes; it should be clear from the context what is meant in each case. 2.14 Definition . Let H* be an incidence structure with incidence relation I. One calls H* an ( ordinary ) projective Plane if the following three conditions hold. (0P1) Whenever P* and 0* are distinct points of H* f there is exactly one line g* such that P* I g* and Q* I gÂ». (0P2) Whenever gÂ» and hÂ» are distinct lines of HÂ», there is exactly one point P* such that P* I g* and ?* I h*. (0P3) There exist four points, no three of which are collinear; that is, no three of the four points lie on a common line. 2.15 Definitions . Let S* be an incidence structure with incidence relation I. Let \\ be an equivalence relation defined on the lines of S*. If g* \\ h* , we say g* is parallel to h*. One calls A* = (S*,\l) an ( ordinary ) affine' plans if the following'' four conditions hold. (0A1) Whenever P* and Q* are distinct points of S*, there is exactly one line g* such that P* I g* and Q* I g*. (0A2) Whenever P* is a point and g* is a line such that P* is not incident with g*, there is exactly one line h* such that

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14 F* I h* and h* and g* have no point in common. (0A3) S* has three points which are not collinear; that is, S* has three points not all on the same line. (0A4) Two lines g* and h* are parallel, g* H hÂ», if and only if g* and h* have no point in cordon or g* = h*. " Remark . It is well-known that if S* is an incidence structure satisfying conditions (OAl), (CA2) and (OA3), then there is a unique equivalence relation 11 such that (S*,\i) is an affine plane by our definition [Pickert (1955), pages 7-10]. The following result is well known. 2.16 Proposition . If H is an affine plane (projective plane), then each line of H goes through at least two (three) distinct points, and each point of H is incident with at le-st three distinct lines. Proof. This is essentially shown in Pickert [(1955), pages 7, 9-111. // 2.17 Definitions . Let H = CP,oj,I) be an incidence structure. Points P and C are said to be pro jcctivcly. neighbor and one writes.? C whenever there are distinct lines g and h such that P,C I g and P,Q I h. Lines g and h are said to be projectively neighbor and one writes g ~ h whenever there are distinct points P and Q such that P I g,h and G I g,h. One calls K a projective Hjel-slev plane (abbreviated PH-plane) whenever the

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15 following three conditions are satisfied. (PHI) If P and Q are points of K, there is at least one line g such that P,Q I g. (PH2) If g and h are lines of H, there is at least one point P such that ? I g,h. (PH3) There is a surjective incidence structure homomorphism g ~ h. If P is a point of H and if g is a line of H, one denotes the ' Â— equivalence classes containing P and g by ?* and g* respectively. 2.18 Proposition . Any projective plane is a projective Hjelmslev plane. // 2.19 Definitions .' Let H,H' be PH-planes. A projective Hjelmslev plane homomorphism w:H Â— * H Â« is an incidence structure homomorphism which preserves the relation ' pro jectively neiahbor'. We denote the category of projective Hjelmslev planes by H, and we denote the category of projective oianes with PH-plane homomorphisms by H*. 2.20 Proposition . If H is a PH-plane, and if ^:H Â— * H' is a map satisfying conditions (PH3) of the definition of PH-plane, then *? is a PH-plane homomorphism. //

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16 2.21 Definitions . Let S = (*P,oj,I) be an incidence structure, and let \\ be an equivalence relation called the parallel relation which is defined on the lines of 3. Points F and G are said to be affinely neighbor and one writes ? ~ Q whenever there are distinct lines g,h such that P,Q I g,h. Lines g and h are said to be affinely neichbor and one writes a ~ h whenever to each point of each there corresponds a point of the other which is affinely neichbor to it. One calls A = (S,U) an affine Hjelrr.slev plane (abbreviated AH-plane ) whenever the following four conditions are satisfied. (AK1) If P and Q are points of 5, there exists at least one line g such that P,Q I g. (AH2) Let P I g,h. Then P is the only point on both g and h if and only if g is not affinely neighbor tc h. (AH3) If P is a point and if g is a line, then there is exactly one line h such that F I h and g \\ h. (AH4) There is a nap ^f:A Â— Â» A' from A to an affine plane A' which is a surjective incidence structure homomorphism such that the following three conditions are satisfied. ( F ~ Q. (?2) If g,h <= 03; then^h = vfg ^ g ~ h. (h in A'. If P is a point and if g is a line of A, one denotes the "-equivalence classes containing P and g by P* and qf respectively. 'We usually denote the parallel relation of an AH-plane by U.

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17 2.22 Proposition . Any affine plane is an AH-plane. Proof. Let A* be an affine plane. By Proposition 2.16, any line g* of A* goes through at least two distinct points. In A* the relation 'af finely neighbor* reduces to the relation 'is equal to'. Using these results, one can easily show that A* is an AH-plane. // 2.23 Definitions . Let A and A' be AH-plane s. An affine Hjelmsle\ plane homomorphism A Â• is an incidence structure homomorphism from S to S ' which preserves the relations 'parallel' and 'affinely neighbor'. We denote the category of affine Hjelrr.slev planes by A, and we denote the category of affine planes with AH-plane homomorphisms by A*. ' Remark . We have shown [Bacon (1971), page 21, Corollary 3.121 that there exist two non-isomorphic AH-planes which have isomorphic incidence structures. 2.24 Propositon . If A is an AH-plane, and if
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1( relation by II*: we say g* U* h* if and only if there are lines k,m such that k ~ g; m ~ h and k II m. We call A* = (S*,H*) the gross structure of A. We call the concrete morphism K:A Â— * A* defined by *P = P*; Kg = gÂ» the neighbor map of A. If H is a PH-plane, we denote the incidence structure induced by the eguivalence relation 'protectively neighbor' by H*, and we call H* the gross structure of H. We call the concrete norphism k:H Â— Â»Â• H* defined by kP = P*; *g = g* the neighbor map of H. 2.26 Definitions . Once the gross structures have been defined, we extend the relations 'projectively neighbor' and 'af finely neighbor' in the following way. Let H be a PH-plane or an AH-plane. Let P be a point and g a line of H. We say P ~ g and g ~ P whenever the image of P is incident with the image of g in the gross structure of H. Hereafter, the symbol ~ is to be read ' is neighbor to ' except where otherwise specified. Its negation is written Â•/Â• and is read ' is not neichbor to . ' The negation of the symbol W is written tf, and is read ' is not parallel to . * 2.27 Proposition . If Â«*>:A Â— Â» A* is an AH-plane homornorphism, then <*>p I wg <Â£> 3 h such that P I h and Â«h = uig. Proof . Let h be a line through P, h l\ g; then uh = ug. // 2.28 Proposition . If w:K Â— * H ' is a surjective PH-plane homornorphism, then up I uig <Â£Â£. 3 h such that P I h and "h = uig.

21 Assume H is an AH-plane and mP Kg. Let *, w' be the respective neighbor maps of H,H*. By Proposition 2.27, there is a line h such that P I h and pq = Â»*h. Let m be a line such that K'*m -fr KVg. Let m Â• and g' be lines such that m ' l\ m, g* \\ q and P I m',g'. If lm' (\ q\ = 0, Km' = Kg'; so that g' ~ m'. Hence KVm U K.Vg, a contradiction. Thus, \m' g\ ^ 0. If m Â• ~ g, then K'^m II KVg, a contradiction. Hence lm' C\ g\ = 1. Let Q = m 1 A g. Then, if ^m ' ~y*g, nVm' II Kyg, and hence KVm II k'^g, a contradiction. Hence M.m* + A*g and Â«Q = ^P . // 2.32 Proposition . If ^:H Â— * H Â• is an AH-plane (PH-plane) homomorphism which induces an isomorphism between the gross structures, then v. is non-degenerate. // 2.33 Proposition . If u.:H Â— * H ' is a surjective AH-plane (PHplane) homomorphism, then Â«. is non-degenerate. Hence, if H is an AH-plane (PH-plane), and if P is a point and g, a line of H, then P ~ g if and only if there is a point Q on g such that P -~ Q . // ' Remark . Luneburg [(1962); pages 263, 264, 265; S'atze 2.1, 2.3, 2.4l states the first two sentences of the* following proposition for AH-planes. Klingenberg [(1955), page 101, S 5] states the first sentence of the following proposition for PH-planes. 2.34 Proposition . Let H be an AH-plane (PH-plane); then, there are at least three pairwise non-neighbor lines through each

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22 point of H, and at least two (three pairwise) non-neighbor points on each line of H. Hence each line is uniquely determined by the set of points on it. Also, each point is uniquely determined by the set of lines through it. Proof . The first two (all three) sentences of the proposition follow easily from Propositions 2.33 and 2.31. Let H be an AH-plane and let (g \ P I g] = \ g I Q I g"} . Let g,h be lines such that P,Q I g,h and g + h. Then by (AH2), Ig H hi = 1. Thus, P = Q. // 2.35 Definition . Let A be an AH-plane. A U-equivalence class of lines is called a direction . We denote the set of directions by g^. If g is a line, the direction containing g will be denoted by TUg). Arbitrary directions will be denoted by "ft, Â£, T or some other capital Greek letter. 2.36 Definition . Let A be an AH-plane. If P is a point and g a line of A, we denote the unique line of A through P parallel to g by L(P,g). If P is a point and V is a direction, we denote the unique line of P through P by L(P,D. 2.37 Definition . If g,h are lines of an AH-plane A, and their images under the neighbor map of A are parallel, we say g and h are quasiparallel , and write g U h. If h is a line and P is a direction, we write h 11 V and rih and say 'h is quasiparallel to f" and ' f is quasiparallel to h ' if there is a line g in

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23 r isuch that h \\ g. Similarly, two directions T,l. of A are said to be quasiparallel , r \\ 2. $ if they map into the same parallel class under the neighbor map of A. The negation of the symbol W is written -It. 2.38 Proposition . Let A be an AH-plane. Two lines h,k of A have exactly one point in common if and only if they are not quasiparallel. Also, \\ is an equivalence relation. Proof. Assume h X k. Since the images of the two lines are not parallel, the lines are not neighbor. If I h f\ k \ =0, then their images would be parallel. by (AH4). Hence by (AH2), \h k\ = 1, and h and k have exactly one point in common. Assume |h kl =1. Then h 4* k. Hence h* ^ k*. But lh* fl kÂ»| ^ 0; hence h* is not parallel to kÂ». Thus, h is not quasiparallel to k. // 2.39 Definitions . If S is an incidence structure, and if g is a line of S, we say (S,g) is a lined incidence structure with base line 9Â« If (S,g) is a lined incidence structure, the points of S which are not neighbor to g are called the affine points of (S,g). Any line of S which goes through an affine point is called an affine line of (S,g). We say that ui:(S,g) Â— Â» (S',g*) is a lined incidence structure homomorphism if
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24 structure or some special type of incidence structure homomorphism, we modify our terminology accordingly. We denote the category of lined PK-planes by H , the category of lined projective planes by H* , and the category of lin ed incidence structures by S . g 2.40 Construction of G:A Â— * S and G :A Â— > S . Let A be an _ ~ ~g g AH-plane. Let g w be the set of parallel classes of A. For every parallel class TT in g , let P(TT) be a new point, and adjoin P(TT) to each line in IT. Let the P(TY)'s be different for different TT's. Let g(gj be a new line incident with each of the new points. Choose the P(TT)'s and g(g w ) in such a way that the new point set IPU 1 P (TT) I TT e gj and the new line set Â«7j U ^g(g^)} are disjoint. Let G(A) be the incidence structure obtained by adjoining the new points P(TC), the new line g(g 0O ) and the. new incidences to the points, lines and incidences of A. Define G (A) to be (G(A) ,g(g w ) ). G(A) is called the ceneralized incidence struct ure of A. G (A) is called the lined aeneralized ~g ! incidence structure of A. A point of G(A) is called a generalized point , and a line of G(A) is called a generalized line . The incidence relation of G(A) is called the generalized incidence of A. We call the original points, lines and incidence structure of A, the affine points , affine lines and affine incidence structure of A. Unless otherwise specified, line ( point ; incidence structure ) will mean affine line (affine point; affine incidence structure) in an AH-plane. If w:A Â— * A' is an AH-plane homomorphism, then u> can be

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25 extended in an obvious natural way to an incidence structure homomorphism G(u>):G(A) Â— *G(A ? ), and to a lined incidence structure homomorphism G (>*>):G (A) Â— * G (A'). ~g ~g ~g Remark . The definition of an affine point (affine line) of A agrees with the definition of an affine point (affine line) of G (A). ~9 2.41 Proposition . The map G:A Â— S constructed above is a functor from the category of AH-planes to the category of incidence structures, and G :A Â— * S is a functor from A to the ~g g category of lined incidence structures. // 2.42 Proposition . The map H* :A* Â— * H* defined by H* (A*) = ~ g g ~ g G (A*), H* (to) = G (w>) is a functor from the cateqory of affii ~g g -g a i planes to the category of lined projective planes. Proof. Pickert [(1955), page 11, Satz 7l shows that if A* is an affine plane, then G(AÂ») is a projective plane. Hence H* (A*) is a lined projective plane. Since the neighbor relation in a projective plane is trivial, if w:A* Â— * B* is a morphism in A", then H* (w) i s a g lined projective plane homomorphism. Thus, H* is a functor. // y 2.43 Definitions . Let A be an AH-plane. Let K:A Â— * A* be the neighbor map. Let R be a point or a line in G (A), and let S be a point or a line in G (A). We say R is neighbor to S, and

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26 write R ~ S, whenever G (k)R = G (k)S, G (k)R I G (k)S or ~g -g -g ~g 5 (K)S I G (k.)R in the lined projective plane H* (a*) = G (AÂ»). y 9 ~ g ~g We call the relation ~ thus defined the generalized neighbor relation of G (A), or the neiohbor relation of G (A). One g ~g can show that restricted to the affine points and affine lines of S q ( A )Â» the generalized neighbor relation agrees with the neighbor relation induced from A. Once this has been shown (see Proposition 2.44), extend the neighbor relation of A in the obvious way: we say 'R is neighbor to S ' in A (where R,S can be a point, a line, a direction or g w ) whenever R ~ S in G (A). If R is neighbor to S we v/rite R ~ S; otherwise we write R * S. We call ~ the (generalized) neiqhbor relation of A. 2.44 Proposition . Let A be an AH-plane. The restriction of the generalized neighbor relation of G (A) to the affine points and affine lines of A is the relation 'neighbor' of A. Proof. Observe that P ~ Q kP = kQ; g ~ h c^ ng=Kh; P ~ g<=> KP I Kg; g~ P <=> kP iKg. // Remark . Hereafter we will frequently not distinguish between g(TT) and TT; g(g,J and g : G (A), G(A) and A; G (w) , g(u>) and oo. ~g Â»g Â» 2.45 Proposition . Let A be an AH-plane. (1) Let h be a line and let T be a direction of A. The following are equivalent. a) h II T.

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27 b) h ~ r. c) If P I h, there is a line g <= P such that g 11 h and P Â€ g A h. (2) Let Â£ ,V be directions in A. The following are equivalent. i) ijr. ii) l ~ r. iii) If h el, then h U T. Proof. Part (1). Let h be a line, and let T be a direction of A. Assume h 1\ ?. Then there is a line g in T such that g \\ h. Hence Kg U Kh in A*, and TT(kq-) I wh in HÂ» (A*). Thus, h ~ T. g Assume h ~ T. Then k(T) I K(h). Let P I h, and let g = L(P,D. Then since Kg = kP v k(T) h, we have that g 1\ h. Assume that if P I h, there is a line g fc T such that g l\ h and Pig. Let P I h. Then P I g, g ft h; hence h 11 T . Part (2). Let Â£ ,T be directions in A. Assume I II P. Then *(Â£) = k(D, and hence 1 T. Assume Â£ T. Let h eÂ£. Let P I h. Let g = L(P,D. Then since k(Â£) = k(H), we have that Kh = Kg; hence h H g. Thus, h u r. Assume that for every h 6 Â£ , h II P . Let h Â£ Â£. Then there is a g 6 T such that h II g. Thus, kKTTCg)) = K (TT(h)), and 2.46 Proposition . If <* is an AH-plane homomorphism, <* preserves the generalized neighbor relation defined above as well as the quasiparallel relation.

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28 Proof . Since Â« is an AH-plane homomorphisra, * preserves the Â•affinely neighbor' relation. Hence by Proposition 2.34, Â« preserves the relations P ~ g and g ~ P. If T ~ Â£ , then there are lines g fc P , h t 1 such that g ~ h. Hence *g .Â«h, and Â«P ~
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29 [(1955), pages 9-10] shows that S is the incidence In ,g ; structure of an affine plane. Two lines of A*(HÂ»,g*) are parallel if and only if they fail to meet in S. (and hence (H* ,gÂ» ; meet at a point on gÂ».) Thus, A*(H*,gÂ») = (S. > ft) is an ~ In , g ) affine plane. If uj:(H*,g*) Â— Â» (H*,g*) is a morphism in H* , then uÂ»{p*i P* I g*~\ c {?* \ p* i g*"5, and u> maps the affine points (lines) of H* into the affine points (lines) of H* . Hence A*(ui) is an affine plane homomorphism. It is easily seen that A* is a functor. Let A* and A* be affine planes. Observe that A*H* (A*) = Â— g A*. If w:A* Â— *Â• A* is an affine plane homomorphism, then A*H* (uj) = w. Define y to be the identity map on A*. Then ~ g a T:A*H* Â— Â» 1* is a natural isomorphism. ~ ~ g -A* c If (H*,gÂ») is a lined projective plane, define a map ^ (H , j:(HÂ«,gÂ«) Â— * HÂ» A*(H*,g*) by letting it be the identity on the affine points and the affine lines of (H*,gÂ»); by letting it take a point P* on g* to POT) where TT is the set of affine lines through P% and by letting it take g* to g(g o0 ). It is easily seen that OC. t is an isomorphism. If I n , g ) w:(H*,gÂ») Â— > (H*,gÂ») is a morphism in H* , then, since a lined projective plane homomorphism is completely determined by its action on the affine points and the affine lines, we have that g natural isomorphism. Thus, H* and A* are reciprocal equivalences. //

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30 2.49 Proposition . If (G,gJ is the lined generalized incidence structure of an AH-plane A, then A can be obtained from (G.a ) by a construction identical to that used to obtain the affine plane A(H*,gÂ») from a lined projective plane (K*,g*). // 2.50 Proposition. The map A:H Â— * A constructed in Construction 2.47 is a functor from the category of lined projective Hjelmslev planes to the category of affine Hjelmslev planes. If (H,g) is a lined PH-plane, two affine points (affine lines)' are neighbor in (H,g) if and only if they are neighbor in A(H,g). if HÂ« is the gross structure of H, and if g* is the class of lines neighbor to g in H, then A(H*,gÂ») is equal to the gross structure of A(H,g). Remark . It is well known that if (H,g) is a lined PH-plane, then A(H,g) is an AH-plane. This is stated by Luneburg 1(1962), page 260, second paragraph], and is essentially proven by PClingenberg [(1954), pages 390-392, S 1.11 and S 3.6], ' Proof. Let (H,g) be a lined PH-plane and let S = S, x . Then (h,g) A(H,g) = (S,l|); S is an incidence structure, and II is an equivalence relation on the lines of S. P, a point of H, is a point of S if and only if P + g; and h, a line of H, is a line of s if and only if h Â•jt g. . Â• We use the symbol to denote the relation ' af finely neighbor' in A = A(H,g) in order to avoid confusion with the symbol ~ which we use to denote the relation 'projectively

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31 neighbor' in H. If P,Q are. points of A, P ~ Q in H if and only if P D Q in A since a line is removed only if all the points on it are also removed. If h,k are lines of A, we wish to show that h ~ k in H if. and only if h k in A. Assume h ~ k. Let P be any point of A which is on h. By Corollary 2.34, there is a line m of H through P such that m f h. Hence m f k. Let Q = m (1 k in H. Then in H*, C* = ?Â• = m* A k* by (0P2). Thus, C is in A; P Q, and Q I k. By symmetry, the corresponding statement holds for an arbitrary point of A on k. Thus, h D k. Conversely, assume h h'. In H* there are at least two points P*,Q* on h* but not on g* by (CP2) and Proposition 2.16. By Proposition 2.31, there are points R,S on h such that R Â£ P*, S Â€-0*. Let R',SÂ» I h' such that R' ~ r, s< ~ S. Then in H*, h* = (h')Â» by (0P1); hence h ~ h'. Hereafter we will use ~ to indicate 'is neighbor to' in both A and H. Any two points of A are joined by at least one line; that is, (AH1) holds in A. If P I h,k; we wish to show that h k = P if and only if h Vk in A. Assume that in A, h H k = P. Then h ^ k in H. Hence, hf k in A. Conversely, assume F I h,k; h ^ k in A. Then h + k in K and ? = h (\ k. Thus, P = h k in A, and A satisfies (AH2). Let P and h be a point and a line of A. Let Q. =' h g in H. Then there is a unique line k joining F and G in H, and we have k II h in A. If k Â• is any line such that k* U h and P I k'; then Q I k'. Thus, k = k', and (AH3) holds in A.

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32 By Proposition 2. 43, A(H*,gÂ») is an affine plane. Define Â«f:A Â— * A(HÂ»,gÂ») by fP = PÂ», ) preserves the neighbor" and parallel relations and hence is an AH-plane homomorphism. It is also . easily seen that A(aÂ«) = A(a)AU) and that A(l,Â„ .) = 1 , ~~ I "Â• I ~ ~ ( H , g ) A ( H , g ) Hence, A:H Â— * A is a functor. // 2.51 Definitions . Let A be an AH-plane and let (H,g) be a lined PH-plane. If A is equal to A(H,g), then we say that A is derived from (H,g), or we say A is derived from H ( by use of the line g), and we say A is a derived AH-plane. If A is isomorphic to A(H,g), we say A can be extended to (H,g), or we say A can be extended to H; we also say that (H,g) and H are extensions of A. Remarks . Drake [(to appear), Corollary 6.2] states that there is an AH-plane which cannot be extended to a PH-plane. Drake [(1967), page 198, Theorem 3.1] states that every finite uniform AH-plane can be extended to a finite uniform PH-plane, and [in Bacon (to appear), Theorem 2.l] we state that every protectively uniform AH-plane can be extended to a

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33 uniform PH-plane, and we use the argument given by Artmann [(1970), pages 130-134] to show this. 2.52 Definition . An injective incidence structure homomorphism Â«o:S Â— S' which reflects the incidence relation is called an ( incidence structure ) embedding ( of s into 5*). 2.53 Proposition . Let (H,k) be a lined PH-plane. The map ^ : i?g ( Â£ (H ' k)) * (H Â» k} defined by X(P) = P, X(h) = h for all affine points and lines and by X(TUh)) = h A k and X(g ) = k for all affine lines h is a lined incidence structure embedding. // Â» 2.54 Remarks . Dembowski [(1968), pages 295-2961 and Artmann [(1969), page 175, Definition 6] have given definitions of Â•affine Hjelmslev plane 1 which they assert are equivalent to that given by Liineburg [(1962), page 263, Definition 2.3]. In [Bacon (1972), page 3, Example 2.1] we give an example of an incidence structure and a parallel relation on the lines of the incidence structure which satisfies the definitions given by Dembowski and Artmann, but not that given by Luneburg. We repeat this example here. ' Example . Take any affine plane A. Keep the same lines and the same parallel relation. Choose one point P of A, and adjoin a new point Pto the point set of A. Let the incidence relation be the same for the old points and lines, and let P* be incident with precisely the lines which go through P.

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34 Remarks . This example fails to satisfy the definition of AH-plane given here (which is essentially equivalent to that given by Liineburg). It can easily be shown that this example cannot be derived from a lined PH-plane. Klingenberg [(1954), page 390, D 61 calls S, , an (H,g) 'affine incidence plane with neighbor elements'. He then shows [(1954), pages 391-392, S 3.6] that A(H,g) = (s, , , H) ~ (TTg) ~ w (tfh) and u, P ~ W Q, we see that ^g ~ wh, and hence w is an AH-plane homomorphism. // 2.56 Proposition . If
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35 Proof. Let g Â• be a line of HÂ». Let = L(P,h). Then k' \\ h ;
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36 2.59 Construction of A :A Â— > a and AÂ„ :A Â— A. We define ~g g ~ II g A :A Â— Â» A by letting A (A) = G (A) and A (oo) = G (to) for everv 9 g -g -g -g ~g cvciy A A and w in A. Let (A,g) be a generalized AH-plane. We let I be the a restriction of the incidence structure of A to the affine points and affine lines of (A,g). If 1* and (* are the sets of affine points and of affine lines of (A,g) respectively and if a relation II is defined on oj^ by k II h 4=* k,h and g have a point in common in (A,g), then we denote ( (*Â£ ,o, ,1 ) , i\ ) by A (A. a). If a J a a ~ll y w is a generalized AH-plane homomorphism, we define A (w) in the obvious way. 2.60 Proposition . The maps A :A Â— + A and A ,:A Â— Â» A are ~ g g ~ u g reciprocal equivalences. // 2.61 Remark . Hereafter we will not distinguish between AH-planes and generalized AH-planes except to aviod confusion. We will say '(A,g) is isomorphic to the AH-plane B* when we mean that (A,g) is isomorphic to the lined generalized incidence structure of B, and so on. 2.62 Definition . An AH-plane homomorphismÂ•< is said to be an AH-plane embedding whenever tx is injective and reflects the incidence, neighbor and parallel relations. ' Remark . V. Corbas' argument for the validity of his Teorema C(1965), page 375 1 inspired the following proposition. Corbas'

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37 Teorema deals with surjective morphisms between affine planes. 2.63 Theorem . Let u:A* Â— Â» A* be a non-degenerate affine plane homomorphism. Then, u. is an AH-plane embedding of A* into A*; hence i*. is infective and reflects the incidence and parallel relations. Also, K* OO :Â£Â• (A*) Â— * H* (!Â•) is a lined projective plane embedding; thus, H* (u) preserves and reflects the incidence relation. Thus, u induces a projective plane embedding of the projective plane associated with A* into that associated with A*. Proof. Assume j*:A* Â— * A* is a non-degenerate affine plane homomorphism and that G", KÂ» and M* are points of A* whose images under yu are not collinear. We wish to show that yu is injective with respect to parallel classes. Let P* f Â£Â• be distinct directions in A*. Let g* 6 C Â• . since g* meets every line of Â£Â•, ug* meets every line of {j*s*\ s* e 1Â»"J. In particular , u g* meets |Â»L(G*,2Â»), Â«.L(K*,?*) and y*L(MÂ»,2Â»). But by our assumption, at least two of these three lines are distinct. Thus, i*g* cannot be parallel to all three; hence -T* 4 w%* . Hence i* is injective on directions. Let P* f R* be distinct points, and let g* be the line joining them. Let Q* be a point such that^Q* is not onM-g'; such a point exists by our assumptions. Then P*QÂ« JrRÂ»Q*; hence y-(P*Q*) tt^(R*Q*); so that mP* ^ j*R* . Hence u, is injective on points. Let pg m = ^h* and let Q* be a point such that **Q* is not on

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38 /*g*. Let P* be a point of g*. Since ^h* -ft i*(P*QÂ»); h* -K P*Q*; hence P*C* meets h* at some point R*. Since l^gÂ» A i*(P*Q*)l = 1; ^PÂ« = /*R*, and PÂ« = R*. Since ^gÂ» il^h*; g* l\ h* and g* = h*. Thus, p is infective. By Proposition 2.31, j^ reflects incidence and hence is an incidence structure embedding. If uh* II /*g* f then either ^h* =^g* and h* = g* or Lh' A ug'l =0 and |hÂ« l\ g*( , 0. In either case, h* \\ g*. Thus, j* is an AH-plane embedding. We wish to show that H* ( u ) reflects incidence. Obviously, H* (/*) reflects incidence for affine points and lines. If h R ' J 9Â«o* in H* (A ~*), then R* I g Â• in HÂ« (A*). Let V* be a ' g "Â• ~ g direction in A*; let g* 6 r* and let h* be an affine line such that^f Iy*h* in H* (A*). Then, y-h* 11 j+ gÂ« in A*, and by our earlier argument, h* l| g* in A*. Thus, h* * V in A* and r* I h* in S* g tA * J * Thus, {j* U) reflects incidence and is a lined incidence structure embedding. // 2.64 Corollary . If X:A -+ A' and u:A' Â— ^ A" are non-degenerate AH-plane homomorphisms , then yuX is a non-degenerate AH-plane homomorphism. // ' Remark . The following corollary was inspired by Lorimer's argument for the validity of his Lemma 4.4 ni973)b, page 10"] which deals with surjective morphisms and the neighbor relation on points and on lines. See Discussion A. 27. 2.65 Corollary . If **:A Â— * A Â• is a non-degenerate AH-plane

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39 homomorphism, then m. preserves and reflects both the generalized neighbor relation and the quasiparallel relation. Thus, P ~ Q <=> mP ~ y^Q, and so on. Proof . By Proposition 2.46, ^ preserves both relations. Bv Theorem 2.63, the induced lined projective plane homomorphism H* (u*):H* (A') Â— * H* (A'*) (where A* and A" are the cross g ; g g Â» structures of A and A 1 ) is a lined projective plane embedding, and hence Â»*. reflects both relations. // 2.66 Corollary . If co*:(H*,g*) Â— + (H*,gÂ«) is a lined projective plane homomorphism, then wÂ» is either an incidence structure embedding or there is a line k* 4 g* such that if P* is not on gÂ», then wÂ«PÂ« I *i*k*. Thus, if w:(H,g) Â— > (H,g) is a lined PH-plane homomorphism, then either there is an affine line k such that if F is an affine point of (H,g), then <*>P ~ w k, or u> preserves and reflects the neighbor relation (thus, P ~ g <=^>u i P ~ ujg, and so on). Proof . One can easily see this by looking at A*(u>Â»). // 2.67 Proposition . If w:H* Â— * H* is a projective plane homomorphism, and if g* is a line of H* such that toh* = ug' implies h* = g* and such that there are two points P*,Q* on g* such that uiP* Â£ ioq* } then io is an incidence structure embedding or there is a line k* 4 g* such that if P* is not on g*, then uPÂ« I wk*.

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40 Proof. Assume Â»:HÂ» Â— * H* is such a morphism. if r* i s not on g', and if wR* I Â«ogÂ», then there is a point wsÂ« on wg* by our hypotheses such that wR* Â£ wS*. Thus, Â«:(H*,gÂ«) Â— * (HÂ» , gÂ«) defined by us' = ( (H*,g*) , (H*,wgÂ«) , f ^ ) i s a lined projective plane homomcrphism. The result follows from the corollary above. // Remark. The following proposition was inspired by Lorimer's argument for the validity of his Theorem 4.5 C(1973)b, page 101 which deals with morphisms which are surjective with respect to points: see Discussion A. 27. 2.68 Proposition. If ^:A Â— A' is a non-degenerate AH-plane homomorphism such that ^g II ph =5g U h for all lines g,h; then p is an AH-plane embedding; that is, ^ is injective and preserves and reflects the incidence, neighbor and parallel relations. If, in addition, p is surjective on points, then ^ is an AH-plane isomorphism. Proof. Assume that ^:A Â— A' is such a morphism and that -P = r Q. Let g Â€ PQ. Let R be a point such that yuR is not neighbor to jÂ»g. Let h = PR, k = QR. Observe that h ft k c=* P = Q since h,k + g, and P = g (\ h, Q = g (\ k. Hence ^h U Â«*k <Â£Â» P = Q. By our assumption above, r P = ^Q; so thatch = ^k and hence P = Q. Thus p. is injective on points. Then by Proposition 2.31, i* reflects incidence. By Corollary 2.65, yu reflects the neighbor relation.

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41 Assume /-g = ^h. Since^g lUh; g \\ h. Let k be a line such that TU^k) l> TY(^g): such a line exists since ^ is non-degenerate. Then ^(g f\ k) = ^g A ^k = j^ih (\ k) and thus g = h. Hence p. is infective. Thus, ^ is an AH-plane embedding. If in addition ^ is surjective on points, then by Proposition 2.56, p. is an isomorphism. // 2.69 Proposition . Let yu.:A Â— A' be a non-degenerate AH-plane (PH-plane) homomorphism. Then there are at least two points P,Q on each line k whose images y*P,*Q are not neighbor in A'. Thus, the action of yu. is uniquely determined by its action on the points of A. Proof. Assume first that ^*:A Â— * A is a non-degenerate AH-plane homomorphism. Let k be a line of A. Let R,S,T be points whose images under k'^Â» (where *' :A* Â— > (A')* is the neighbor map of A') are not collinear. At least one of the directions ^TT(RS ) , r TT(RT) and ^TT(ST) is not quasiparallel to^k since otherwise the lines xy(k), K y(RS), K'f(RT) and k'^(ST) would all be parallel, and hence k'^R, kVS and k'^T would be collinear. Let P be a direction such that r H n^k. The lines L(R,D, L(S,D and L(T,D all meet k in a single point: say R', S', TÂ», respectively. Observe that at least two of the points k'^R ' , k'^S Â• , *VT' are not equal, since otherwise kVR, kVS and K yT are collinear. Thus, there are points P,Q on k such that Â«*P f uQ. Now assume ^:A Â— > A' is a non-degenerate PH-plane homomorphism. Let k be a line of A and let R,S,T be points

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42 whose images under k'/* are not collinear. Then RS, RT and ST each meet k in, say, P, P', P". If K ' r K = tyP, k'^P Â• , k'^P", then Ky(RS), K'^(RT) and Ky(ST) are copunctal. Since K Â»yuR, kVS and k'^T are pairwise non-neighbor, we may assume ^M 4uR,uS without loss of generality. Then (k'/.M), (x'^T) I ( K VTS),( *y.TR) and K'^TS ^ KyTR; so that K'^M = n'yuT. But then k'mT I kV(SR), a contradiction. Thus, at least two of ^P,~P Â• , uPÂ» are not neighbor. Thus, there are points V,W I k such that kV ^ i*W. Thus, in AH-planes (PH-planes) the action of a non-degenerate homomorphism * is uniquely determined by its action on points. // 2.70 Proposition . If w:A* Â— > A* is a degenerate affine plane homomorphism, then there is a line k* such that ooP* I Â«ik* for every point P* of A*, and exactly one of the following three conditions holds. (a) There is a point Q* such that uÂ»PÂ» = W Q* for all points P* of A*. (b) For all lines g* of A', tog* = wk* and there are points P*,Q* such that wP* A wQ*. (c) There is a direction ?Â• not containing k* such that wm* = wk* for every m* $ V*, such that wk' Â£ u>V , and such that cog" 4 wh* for some lines g*,h* e r*. Moreover, if A* is an affine plane, there is at least one endomorphism of A* of each of the three types: (a), (b) and (c). Proof. Assume that ui:A* -> A* is a degenerate affine plane

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43 homomor phi sin. Hence, the images of any three points of A* are collinear in A*. Let G* be a point of A*. If Â«*}?Â• = ooG* for every point P* in A*, then every line ug* goes through toG* and hence case (a) holds and the other cases do not hold, and we can let k* be any line of A*. Assume that there are points G*, H* such that wH' / wG'. Let k* = G*H*. Then, by our assumptions, wF' Iwk' for every point P*. There are two remaining subcases. If cog* = wk* for every line g* of A*, then case (b) holds and the other cases do not hold. If there is a line g* such that wg* / tok*, then, since P* I g* implies toP* Iu>g*,ok* which implies tog* 4f iok* , we have that w(L(H*,g*)) ^ w(L(G*,g*)). If m* Vr g*, then m* meets both L(H*,g*) and L(G*,g*); hence iurn* = tok*. Hence case (c) holds and the other cases do not hold. Let A* be an affine plane. Let Q* be a point of A*. Define u>:A* Â— * A* by uoP* = Q*; u>g* = L(Q*,g*): to is a type (a) homomorphism. Let k* be a line and let Q* be a point on k*. Define k :A* Â— > A* by Â« P* = P* if P* Ik*, by Â«P* = Q* if P* is not on k*, and byocm* = k* for all lines m* of A*. Then <*. is a type (b) homomorphism. Let k* be a line and let P* be a direction such that k* ^ P*. Define V:A* Â— * A* by v(F*) = L(P*,P*) A k* for all points E* of A*, by v(g*) = g* for g* e P*, and by \>(h*) = k* for h* ^ T*: C is a type (c) homomorphism. //

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3. BITERNARY RINGS In this section we define 'coordinatized affine Hjelmslev plane' and 'biternary ring 1 , construct the related categories, and show that they are eguivalent. 3.1 Definitions . Let T* be a ternary operation defined en a set M* with distinguished elements 0* and 1* with 1* Â£ 0*. Then (H*,T*) is said to be a ternary field if it satisfies the following five conditions: (TF1) T*(x*,0*,cÂ») = TÂ»(0*,m*,cÂ») = cÂ» for all x*,m*,c* in M*. (TF2) T'(l',m*,0Â«) = TÂ» (mÂ« ,1* ,0* ) = m* for all m* in H". (TF3) For any x*,m*,c* in M", there exists a unique z* in M* such that T*(x*,m*,z*) = c*. (TF4) For any m*,d*,n*,b* in N* such that m* 4 n*, there is a unique x* in M* such that T*(xÂ»,m*,dÂ») = T* (x* ,n* ,b* ) . (TF5) For any x* ,C , x ' * ,c * * in K* such that x* ^ x'Â», there exists a unique ordered pair (m*,d*) such that T* (x* ,m* ,d* ) = c* and T*(x' * ,mÂ« ,d* ) =Â• c * * . We say that 0* is the zero and that 1* is the one of (M*,T*). We call the elements of H* symbols . If (MÂ»,TÂ») and (Q*,S*) are ternary fields, an ordered triple uj = ( (M*,TÂ») , (Q*,s*) ,fj is said to be a ternary field homomorphisrr, if f^M* Â— * Q* is a function such that 44

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45 u>('T*(xÂ»,m*,e*)) = S* (wx*,uim* ,we* ) and such that wO* = 0*, u>l* = 1*. We denote the category of tern ary fields by F* . 3.2 Definitions . Let M he a set with distinguished elements and 1, and with two ternary operations defined on M. Let N = {n 6 Ml 3 k Â€ M, k t 0, 3 T(k,n,0) = o"5 , and let N' = (n'Hia k t, K, k ^ 0, * T'(k,n,0) = 0"5. Define a relation ~ on M by a ~ b (read 'a is neighbor to b') if and only if every x which satisfies the equation a = T(x,l,b) is an element of N. Define a relation Â«*' on M by a ~' b if and only if every y which satisfies the equation a. = T'(y,l,b) is zv. element of ."."Â•. The negation of a ~ b is written a Â£ b and is read 'a is not neighbor to b'. Then, (!:,T,T f ) is said to be a biternary ring if the following twelve conditions are satisfied. (BO) N = N', and a necessary and sufficient condition that a Â— ' b is that a ~ b. (Bl) The relation is an equivalence relation; that is, the relation ~ is reflexive, symmetric and transitive. (B2) T(0,m,d) = T(a,C,d) = d for any a,m,d from K. (B3) T(l,a,0) = T(a,l,0) = a for any a from M. (B4) T(a,m,z) = b is uniquely solvable -for z for any a,m,b from M. (B5) T(x,m,d) = T(x,m',d') is uniquely solvable for x if and only if m /m' for any m,d,m',d' from M. (B6) The system T(a,m,d) = b, T(a',m,d) = bÂ» with a ^a' is uniquely solvable for the pair m,d; if a 4> a ' , b ~ b 1 , we

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46 have m Â€ N; if a Â« a 1 and b + b', the system cannot be solved. (B7) If a ~a', b~ b', and if (a,b) 4 (a',b'), then one and only one of the systems Â£T(a,m,d) = b, T(a',m,d) = b Â• "5 and $T'(b,u,v) = a, T'(b',u,v) = a* where u <= N*} is solvable with respect to rn,d correspondingly u,v (where u fe N) , and it has at least two solutions; and we have m' ~ m", d' ~ d" or u Â• ~ u" v' ~j v" respectively for any two solutions. (B8) The system ly = T(x,m,d), x = T'Cy.u.v)"} where u e N, m,d,v (, M, is uniquely solvable for the pair x,y. (B9) For any m,u G M, T(u,m,0) = 1 if and only if T'(m,u,0) = 1. If T(u,m,0) = 1, if T(a,rr.,e) = b, and if T'(b,u,v) = a for some n,u,a,b,e,v Â€ M, then (T(x,m,e) = y <=$> T'(y,u,v) = x) for every x,y 6 M. (BIO) The function T induces a function T* in M/~ , and (K/~,TÂ») is a ternary field with zero 0* = {z lz ~ 0] and one 1* = \ e I e ~ 1^ . (BID Conditions (30) through (BIO) hold with T and T' interchanged throughout; the new conditions will be called (BO)' through (BIO)'; condition (BIO)' states that the function TÂ« induces a function T'* in MA', and that (K/~',T'Â») is a ternary field with zero 0* and one 1*; of course, N and N', ~ and ~* are interchanged throughout also. Each element of N is said to be a right zero divisor . 3.3 Definition. If (M,T,TÂ») is a biternary ring, then (M,T',T) is a biternary ring by the symmetry of the definition of biternary ring: (M,T*,T) is said to be the dual of (M,T,TÂ»).

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47 3.4 Definitions. Let (B,T,T<) be a biternary ring. We will frequently write B to denote (B,T,?'). We will frequently write N B or simply N to denote the set of right zero divisors in B. The elements of the set 3 are called sv-bcls ; is called the zero of 3 and 1 is called the one of 3. If N = {o] we say B . that B is a biternary field . 3.5 Proposition. Let (2,T,T') be a biternary ring and let u e B. Then u if and only if u 6 N. Proof. Assume u ~ 0. By (33), u = T(u,l,0), and hence u fe N by the definition of neighbor in 3. Assume u Â€ N. Then there is a k in K, k 4 0, such that T(k,u,0) = 0. Since x = and x = k are both solutions to the equation T(x,u,0) = T(x,0,0), we have by (35) that u ~ 0. // 3.6 Prop osition . If (E,T,T') is a biternary ring, then 1*0. Proof. By (E10), 1* ^ 0Â»; hence 1*0. // 3.7 Proposition. In a biternary ring (3,T,TÂ«), the equation a = T(x,l,b) has a unique solution x for each pair (a,b). In addition, a ~ b if and only if x 6 N, Proof . Let a and b be elements of the set 3; that is, let a and b be symbols. Since 0*1, by (B5) there is a unique solution x to the equation T(x,l,b) = T(x,0,a). By (32),

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48 we have that T(x,0,a) = a. Hence, a ~ b if and only if x G N. // 3.8 Proposition . Let (3,T,T') be a biternary ring and let m 6 B. There is a u Â€ B such that T(u,n,0) = 1 if and only if m Â£ M. If m ^ NÂ» then the solution u is unique and u ^ N. Moreover, the map S:M\N Â— * M\N defined by T(J(m),m,0) = 1 is a bisection. If u Â€ M\N, then T(u, i" 1 (u) ,0) = 1. Proof . If m Â€ N, then m ~ and there is no element u e B such that T(u,m,0) = 1 since 0* ^ 1Â« and T*(u*,0*,0*) = 0Â» for every u* in MA,. If m ^ N, then m -f by Proposition 3.5, and by (B5) there is a unique u such that T(u,m,0) = T(u,0,l). If u were in N, then u ~ and, by (B5) , T* (u* ,m* ,0Â» ) = T* (0* ,mÂ« ,0* ) = 0*, a contradiction. Thus, we can define a map S :M\N Â— M\N by T(S(m),m,0) = 1. If u <Â£ N, then the system (T(u,m,d) = 1, T(0,m,d) = 0l is uniquely solvable for the pair m,d by (B6) since u y0. By (B2), d = 0, and, since TÂ»(u*,0*,0Â«) = 0*, m j N. Thus, I is surjective. If m' satisfies the equation T(u,m',0) = 1, then the pair m',0 is a solution to the system above and hence m' = m. Thus, the map S is bijective.' If u Â£ M\N, then there is an m fe KAN such that T(u,m,0) = 1. Hence S(m) = u, and we have that m = %" (u) and that T(u,J>~ 1 (u),0) = 1. // 3.9 Definition . Let (3,T,TÂ») and (M,S,S') be biternary rings.

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49 A biternary ring ho-onorohism *:B Â— * M is a concrete morphism such that *(N B ) Q N^-, *(0) = 0; oUl) = 1; *(T(x,in,e) ) = S(Â«x,*m,e(e) and tf(?'Â£y,u,v)) = S ' (oty ,*u , *v) for all x,n,e,y,u,v in B. 3.10 Definitions . It is easily seen that the class of biternary rings and their homomorphisms form a category. We denote this category by 3 and call it the category of biterr.a rv rings. The full subcategory of B whose objects are biternary fields we denote by B*. 3.11 Definitions . We say C = (A,K) is a coordinatized AH-nlane and K = (g^g ,E,K,^:CE Â— * K) is a coordinsticaticn of A whenever A is an AH-plane, g^ ,g are non-neighbor lines of A, x y E is a point of A not neiahbor to either a or g M is a set x 3 y' with distinguished elements and 1, CS is the line joining = 9 V ft g,. to E, and f :CE -^ M is a bisection such that (0) = 0, X y ) " (E) = 1. Let C = (A,K) be a coordinatized AH-plane. If ? is a point of A, define 6(P) = (x,y) = C^(0S A L(P,g )),f;(CE fl L(P,g ))); y x the construction is indicated in Figure 3.1; and define fi'(P) = (y,x)'. If 0(F) = (a,b), we say b is the y-ccordinate of P and that a is the x-coordinate of P; let 1r F = a , -It P = b. x y If k is a line of C = (A,K), and if k -tf g , define X(k) = ~ " Y rm,d] = [ir y (L(0,k) ft LCE.g )),Â«* (k f\ g )3 ; the construction is indicated in Figure 3.2. Whenever k -fr g , we interchange the roles of g and g in the definition of X to define V (k) = x y [u,v]' = fir (L(C,k) A L(E,g )),1T (k f\g )]'.

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50 Figure 3.1 . g y

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51 It is easily seen that the maps 9, &< (X, X') are welldefined functions from 1* (from {k \ k -tf g } . from fk Iklo]) ~ y J ~ y x J into MXT-1 and that they are bisections. If g is a line of C such that X(g) = tm,d], then Um,d3 is said to be a representation of g; similarly, if \'(g) = Uu,vl', then [u,vl' is said to be a representation of g. If 0(?) = (x,y), then (x,y) and (y,x) Â• are said to be representations of F. Since 0, 6', A, X* are bijections, we can, without fear of confusion, identify a point or line with each of its representations or with its one representation. The line g is called x the x-axis of C; g is called the y-axis of C; is called the origin of C and E is called the unit point of C. Let X denote the point g f\ L(E,g ), and let Y denote the point g (\ L(E,g ). x y y x 3.12 Definition . Any pair of statements or functions which can be gotten one from another by interchanging the roles of g and x g throughout are said to be xy-duals . The functions 0, 0' given above are xy-duals, as are' A and X'. 3.13 Proposition . Let g,h be lines of a coordinatized AH-plane C, and let g = [m,e]. Then h is quasiparallel to g, h 11 g, if and only if there are m',e' such that h = tm',e'] and (l,m) ~ (l,m' ) . Procf_. Assume h 11 g. Then, since Ig* g *1 = 1 implies lh* H g *1 = 1, we have that lh g 1 = 1. Thus, for some m',e' y y we have that h = [m'.a']. Since h 11 g, L(0,g) II L(0,h) and

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52 hence L(0,g) ~ L(0,h). Thus, since L(E,g ) ^ L(0,g) ,L(0,h) , we have that (L(0,g) t\ L(E,g )) v (L(C,h) A L(E,g )), and hence y y (l,m) ~ (l,mÂ«). Assume h = [m ',e'] and (l,m) ~ (l,m'). Then L(0,g) 11 L(0,h). and g \\ h. // 3.14 Definitions . Let C and C* be coordinatized AH-planes. A coordinatized AH-plane honomorohism or coordinatization homomorphism u:C Â— * C is a rr.ap uo which is an AH-plane homomorphism such that w(g ) = g ' , *>(q ) = c < and 10(E) = E' where q '. g Â• x x y ~y xy and E' are the x-axis, y-axis and unit point of C respectively. If C = (A,K) is a coordinatized AH-plane, then the neighbor map K:A Â— A* induces a coordinatization homomorphism from C to C* = (A*,?;*) where K* is the coordinatization of A* whose x-axis is (g )'Â• and so on; we denote this induced map by K:C Â— C* and call K the neichbor man of C. 3.15 Definition . It is easy to see that the class of coordinatized AH-planes together with their coordinatization homomorphisms form a category. We denote this category by <:, and call it the category of coordinatized affine Kjelmslev planes . We denote the full subcategory of t whose objects are coordinatized affine planes by C* . 3.16 Construction of B:C Â— * B. Let C be a coordinatized AH-plane. Define a ternary operation T:iM Â— * M by T(x,m,e) = y if and only if there exist a point P and a line g, Pig, such that fl(P) =

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53 (x,y), X(g) = Cm, el. Define a second ternary operation T':M Â— * M by interchanging the roles of q and q in the x y definition of T; that is, let T'(y,u,v) = x if and only if there exist Q, h, Q I h, such that 8'(Q) = (y,x)* and h' (.h) = Cu,v"}'. Let B(C) = (X,T,T'). Given a morphism us:C Â— Â•> C * in C, define a map B(ui):B(C) Â— * B(C) by B(uj)m = Â£' (u( " (m))) for all m in M. By M we mean the set of symbols of C. Â• Remark . Many of the intermediate steps in the proof of the following proposition are stated in Lcyganova (1967)] (see our Remarks A. 2, A. 15 and A. 16 in Appendix A): she states (Lemma 1), part of (Lemma 2), part of (Lemma 4), (Bl), (B2), (B3), (B4), (B5), (B6), (B7) and (B3). 3.17 Proposition . The map B:C Â— B defined above is a functor from the category of coordinatized affine Kjeimslev planes to the category of biternary rings. If C* is a coordinatized affine plane, then B(C*) is a biternary field. If C is a coordinatized AH-plane, then (a,b) ~ (a',b') in C ^ a ~ a 1 , b ~ b' in B(C); [m,dl ~ [m'jd'l in C ^ m ~ mÂ», d ~ d' in B(C) ; [u,v) ' ~ [u',v'] ' in C <=Â£ u ~ U', v ~ v' in B(C) . Proof . Assume C is a coordinatized AH-plane. Define a o b if and only if (a, a) ~ (b,b). Let N = \n e m\ n o o}. ' Observe that if C is a coordinatized affine plane then N = ^.o} . (Lemma 1) (a,b) ~ (a', fa') <Â£> a o a', b o b'. (Proof) Assume (a,b) ~ (a',b'). Then

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54 L((a,b),g ) ~ L((a',b'),g ); so that, since OE is not quasiparallel to g , (L((a,b),g ) H OE) ~ (L((a',b'),g ) H OE) and hence (a, a) ~ (a', a'), and a o a'. Similarly, b o b'. Assume a o a', b o b'. Then L((a,a),g ) ~ L((a',a'),g ); so that, since g is not auasiparallel to q , (a.b) ~ (a'.b). Y x ' Also L((b,b) t g ) ~ L((b',b'),g ); so that (a' ,b) ~ (a',b'). Thus, (a,b) ~ (a' ,b' ) . / (Lemma 2) [m,e"J ~ [m',e'] <=} m o m', e o e'; hence by xy-duality, [u,v]' ~ [u',v']'
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55 Assume there is an element k, k Â£ 0, such that T(k,n,0) = 0. Then both (k,0) and (0,0) are en [n,o] and [o,o]; so that [n,0] ~ [o,o]. Thus (l,n) ~ (1,0) and n o 0, n Â€ N. / (Lemma 5) a o b if and only if every x which satisfies the equation a = T(x,l,b) is an element of N. (Proof) Since -yE, 4> 1. Hence, by Proposition 3.13, [0,a] is not quasiparallel to [l,bj. Thus, the equation a = T(x,0,a) = T(x,l,b) has a unique solution x. Assume a o b. Then (0,a) ~ (0,b); so that [o,a] ~ [0,b]. Thus ([0,al f\ [l,b]) ~ ([0,bl f\ [l,b]). Thus, if (x,a) = Co, a"] H [l,b] then (x,a) (0,b) and x o 0. Kence a = T(x,l,b) and x Â£ N. Assume every x which satisfies the equation a = T(x,l,b) is in N. . Let (x,a) = [l,bl [o,a]'; then a = T(x,l,b); x o 0. Since l[l,bj (\ g \ =1, (0,b) = [l,b] A g . Observe that l0,xl' ~ 10,0]' where g = [o,0]' and hence (l0,xl ' fl [l,b]) ([0,0] ' A [l,b]) ; (x,a) ~(0,b), and a o b. / (B0) The xy-dual to the proof of (Lemma 5) above shows that a o b if and only if every y which satisfies the equation a = T'(y,l,b) is an element of N. Thus in B = B(C) the relations ~ and ~' of the definition of biternary ring are equal to the relation o defined here. By (Lemma 4) N = 'N = N* so B B that condition (B0) of the definition of biternary ring holds in B(C). (Bl) The relation o on M is obviously an equivalence relation. (B2) Since (0,d) I [m,d], T(0,m,d) = d. Since [o,d] llg , (a,d) I Co,d]. Hence T(a,0,d) = d.

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56 (B3) Since (l f a) I [a,0'J, T(l,a,0) = a. Since [l,0] = OE, we have that (a, a) I [l,cl and T(a,l,0) = a. (B4) There is a unique line parallel to [m,o] through the point (a,b): say [m,z]. Then T(a,m,z) ,b. If T(a,m,z') = b, then (a,b) I fm,zO; hence z = z'. Thus, z is the unique solution to the equation T(a,m,z) = b. (B5) The equation T(x,m.d) -_Kx.m'jd') is uniquely solvable for x if and only if there is a unique point (x,y) I Cm,d],r m ',dO; hence if and only if ["m.d] is not quasiparallel to [mÂ»,d'] and, by Proposition 3.13, if and only if m 4> m' . (B6) Look at the system $T(a,rr.,d) = b, T(a',m,d) = b'^. If a Â£ a', then the unique line g joining (a,b) and (o',b') is not quasiparallel to g y . Hence g = [m,d3 for some m,d and the pair m,d is the unique solution to the system. If a cS a' bob* then [m,d] 11 g^ and hence by (Lemma 3) m o 0. Ifa oa'.b/b', then, if (a,b),(a',b') I g, g li g and hence the system has no solution fm,d]. (B7) Let g,gÂ» be any two distinct lines joining (a,b) and (a',b'), (a,b) /. (a',b'K Then g ~g'. Observe that (g (\ q ~ s y ^ 9' ! V and that g.g' !g y <^ g [u.v]-, g Â« = [u', V ']< for some u,u' 6 N, v,V e M . The lines g,g' are not quasiparallel to g y if and only if g = [m,d], gÂ« = [m',d-] for some m,m',d,din M. Hence condition (B7) holds in B(C). (B8) Any line of the form [u,v]' f u Â£ N, is quasiparallel to g y and hence meets any line fm,d] (not quasiparallel to g ) in exactly one point (x,y).

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57 (39) T(u,m,0) = 1 if and only if (u",l) I [m.O']. Hence T(u,m,0) = 1 if and only if [u,0"]Â» = [m,0l, and by xy-duality, fu,0]' = [m,0] if and only if T'(m,u,G) = 1. If T(u,m,0) = 1, and (a,b) I [m,e] , [u, v] ' , then [m,e~\ = [u,v]', and hence (x,y) I [m,e] <^> (x,y) 1 [u,v3*. (BIO) The construction of the ternary operation T maps, under the neighbor map K, to the construction of T* which is the usual construction of the ternary field of a coordinatized affine planeÂ— see [Hall (1959), pages 353-355, Section 20.3]Â— and $***," (0) = 0Â», ^'K^ (1) = 1*, where 0* is the zero and 1* the one of (M*,TÂ»). (BID Observe that since we have made no special requirements on C, the xy-duals of conditions (BO) through (BIO) also hold in 3(C) . Thus we have shown that B(C) is a biternary ring. Assume that u>:C Â— * c ' is a morphism in C, and that 3(C) = (;-;,T,T'), B(C) = (Q,S,S'). We wish to show that B(w) :B(C) Â— *B(C) defined by B(<-j)m = t,'^ 10 ^ (m))) is a biternary ring homomorphism. Define us' :M Â— > Q by B(ui)m = uj'm. Then it is easily seen that w(x,y) = (w'x, m'y) , u>Cm,d3 = [co'm,*>'d3 and uÂ»fu,v]' = r^'u , * (T(x,m,e) ) = S(u)'x, u>'m, w'e) and u)' (T' (y,u,v) ) = S ' (w'y, w'u.w 1 v) . Since (n,n) ~ (0,0) implies (w'n,w'n) ~ (0,0), w'N C N . Observe that ui'(0) = 0, ui* ( 1 ) = 1. Hence B(<*Â») is a morphism in B. Obviously B(l c ) = 1 B(C) If Â«-C 1 -^ C 2 and p:C 2 Â— C 3 are morphisms in C, then if m 6 M , 3(AÂ«)m = $ Ce*($ Â„ ~~ n ; 3 ^ 3 ( p ( f 2 t2 U( S!~ CmJ>))) = B(j5)(B(oO(m)), so that B(pec) = B(/5)B(Â«) and 3:C Â— * B is a functor.

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58 If C is a cocrdinatized affine plane-, recall that N , . = B(C) {0] and hence observe that 3(C) is a biternary field. // 3,18 Construction of C:5 Â— Â» C and A : 0] Â— * (M x { < >,< >Â•])/=. Given a biternary ring B = (K,T,T') we construct an incidence structure 5 = CP,oj,I) and a parallel relation W in the following way. Let V= Mx M and let Oj = M * M *tOl and 0} = M < M * til. For convenience we denote an element (m,d,0) of 0] by [m,d] and an element (u,v,l) of m by [u,v]'. We define U ji A -g T ^ Â«5 T , -*H*H >,< Yl by A F (g) = , if g = [m,d] f and 6 (g) = ', if g = [u,vl'. We define incidence by (x,y) I [m,d] <=} y = T(x,m,d) and by (x,y) I Cu,v]' <Â£> x = T'(y,u,v). We identify and ' 4=> T(u,m,0) = 1 = T'(m,u,0). If = ', and if there is a point (a,b) on both [m,d"] and [u,v]', then by (B9) and (B9)Â«, ((x,y) I [m,d"] <=> (x,y) I [u,v]'), and we identify [m,d] and [u,v]'. Denote the set of lines by a] where 0j = (0] T U 0j T ,)/=. Define A : 0] Â— * (M Â» t< >,< >'})/= in the obvious way. We define g II h if and only if &(g) = A(h). We denote the incidence structure and the parallel relation thus defined by A fi . We let 5 B :[l f ol Â— * H be defined by $ g (a,a) = a. Let K = ([0,0],[0,0]',(1,1),K,C, ). we define C(B) to be (A.K ). o /v B B Given a biternary ring homomorphism uj:B Â— > B' we define C(w):C(B) Â— *C(BÂ«) by C(u>)(a,b) = (oja.cob), C(w)[r.,d] = [>om,ujd] and C(lo)[u,v1' = [ w u,wvl' for all points (a,b), lines [m,d], [u,vl' in A . Observe that C(uj) is well defined. 3.19 Lemma . Let (B,T,T') be a biternary ring. Then the

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59 following conditions hold in the construction given above. (a) If m,d 6 K, m Â£ N, then there is a unique line Cu,v3' of oj T , such that [m,d] . [u,v]'. If m 6 N, there is no such line Cu,v]'. (b) If u,v 6 M, u ^ N, then there is a unique line [m,d] of 0] T such that [u,v]' = [m,d]. If u 6 N, there is no such line [m,d]. (c) The function &:0]-^ (HÂ«t( >,<. >'})/= constructed above is well defined. Proof. (a) Assume g = [m,d], tn 4 N. Then (0,d) I g. There is a unique u Â€Â• N such that T(u,m,0) = T(u,0,l) since + m. Hence T'(m,u,0) = 1, and there is a unique v such that (0,d) I Cu,v]'. Hence [m,d] = [u,v]Â«. If [m,d] = [w,z]', then T(w,m,0) = 1 and u = w. Also since = T'(d,u,z), z = v. Thus [u,v]' is unique. Assume g = [m,dl, m 6 N. Then, since m 0, 0^1, there dees not exist an element u e M such that T(u,m,0) = 1, and hence there is no [u,vl' fe oj t , such that [m,dl = [u.v]'. (b) This is the dual of case (a). (c) If [m,d] = [u,v]', then &[m,d] = (m*> = Â» = &[u,v]Â«; hence A is well defined. // 3.20 Definition. If B = (M,T,T') is a biternary ring and if B= (M,T',T) is the dual of B, then (x,y) I [m,d") in C(B) <=^ (y,x) I [m,dlÂ« in C(B'), and (x,y) I [u,v]' in C(B) <=> (y,x) I [u,v] in C(B'). The structure C(BÂ») is said to be the structure dual to C(B).

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fcviC 60 3.21 Proposition . Let 3 be a biternary ri-ng and let 3' be the dual of 3. The map 6:C(3) -*c(B'> defined by 0(x,y) = (y,x), 9[ra,d] = tra.dl' and 0[u,v]' = [u,v] is an incidence structure homoraorphism which preserves and reflects the parallel relation. // 3.2? Proposition . If B* is a biternary field, then C(B*) constructed above is a cccrdinatized affine Diane. Proof. If 3* = (M*,T*,T*Â«) is a biternary field, then (M*,T*) is a ternary field and it is easily seen that C(3*) is the coordinatized affine plane commonly constructed over the ternary field (K*,T*): see [Hall (1959), top of pace 356]. // 3.2 3 Prop osition . The map C:B -* C indicated above is a functor from the category of biternary rings to the category of coordinatized affine Hjelmslev planes. oof. Let B = (K,T,T') be a biternary ring. The structure A 3 .nstructed above is an incidence structure S D with a relation Parallel' defined on the lines of s_. We wish to show that A Â» B s an AK-plane. We start by proving a number of lemmas. To void confusion, throughout the remainder of this proof we will ise the symbol ~ to indicate the neighbor relation in B, and :he symbol Q to indicate the relation 'af finely neighbor' in A : B see the definition of AH-plane, Definition 2.21. (Lemma 1) Whenever P,Q are points, there is a line g such that P,Q I g.

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61 (Proof) Let P = (a,b), Q = (a',b'). There are four cases. (Case 1) Assume P = Q = (a,b). Then by (E2), (a,b) I [0,bl. (Case 2) Assume P 4 Q, a ^ a ' . The system T(a,m,d) = b, T(a',m,d) = b 1 has a unique solution m,d by ( B6 ) , end hence P,Q I [m,d].

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62 (Proof) (<= ) Assume g h t P I g,h. We wish to show Ig A hi = 1. Let P = (a,b). (Case 1) Assume g = Â£m,el, h Â«= [m^e'], and that Q I g,h; Q ^ P. Let Q = (a',b' ). By Lemma 2, a ~a', b -v b ' . Hence by (B7), m ~ m', e ~ e ' . Let (c,d) be a point on one of the lines; say g. Then d = T(c,m,e). Let d' = T(c,m',e l ). By (BIO), d' ~ d. Thus by Lemma 2, (c,d) D (c',d'). By symmetry, g h, a contradiction. Hence Ig A h\ = 1.

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63 (Case 3) Assume g,h e 0J T ,Â» g ^ h. This is Case 2 in the dual structure, hence g j3h. (Case 4) If none of the preceding cases occurs, by Lemma 3.19 we may assume g = [m,e~], h = [u,vl' where u Â£ N. There is a symbol w such that w -/v. Assume g h. Let z = T(w,m,e); Q = (w,z). There is a point R = (w',z') on h such that R Q. By Lemma 2, w ~ w'. Since u Â€ N, u ~ C, and since w' = T'(z',u,v) by (BIO), w" = v* ; so that w' ~ v. Thus, since ~ is an equivalence relation on B; w ~ v, a contradiction. Hence g $ h. / (Lemma 4) If g = [m,e]; then g h if and only if h = [m',e'] and m m', e ~e'. If g = [u,v]'; then g h if and only if h = [u'jV'l' and u ~ u Â• , v ^v'. (Proof) (Case 1) Assume g = [m,e]. (Â£=) Assume h = [m'je'l, where m m', e ~ e'. Let (a,b) be a point on one of the lines: say g. Let b' = T(a,m , ,e'). By (BIO), b ~ b'. Hence (a,b<) I h and (a,b') D (a,b) by Lemma 2. By symmetry, g h. ( =^) Assume g D h. If g = h we are done. Assume g I h. By Lemma 3, g h implies Ig hi 4 1; hence by ( E3 ) , h 4 [u,vV, u 6 N; hence by Lemma 3.19 (b) , h = [m',e']for some m',eÂ» in K. If m * m', then by (B5), there is a unique point of intersection, a contradiction by Lemma 3. Hence m ~ m*. The point (0,e) is on g. Let (u,f) I h such that (0,e) D (u,f). Then, ~ u, e ~ f. By (BIO) there is a unique z* such that T*(0*,m*,zÂ«) = e*. Then e* = e'* = z* and e e'. Thus, m ~ mÂ», e ~ e ' .

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64 (Case 2) Observe Case 2 reduces to Case 1 in the dual structure. / (Lemma 5) The relation 'af finely neighbor' is an equivalence relation on the points and lines of A . 3 (Proof) The relation 'af finely neighbor' is an equivalence relation on the points of A and is a reflexive and symmetric relation on the lines of A_ by Lemmas 2 and 4. Assume q D h B 3 1 h P k. If g e oj t , then h 6 0j t ; so that k 6 oj by Lemma 4. Hence g D k by Lemma 4 and (31). Similarly, if g 6 Oj ( , then h,k Got and g D k. Thus 'af finely neighbor' is an equivalence relation on the points and lines of A . / B (Lemma 6) If P is a point; q, a line, then there is a unique line h such that P I h, h II g. (Proof) (Case 1) Let P = (x,y) and g = [m,b]. Then, by (B4) there is a unique symbol z such that y = T(x,m,z). Kence lm,2 J is the unique line of 0| parallel to [m,bl containing F. Assume P I [u,vl' and [u,v]' (l I>.,b]. Then * u y =^m> so that since P I [u,v"]Â« ,U,zl f and Cu,v]' = [m,z"]. Thus, [ m,z] is the unique line through F parallel to [m,b], (Case 2) This case reduces to Case 1 in the dual structure, / (Lemma 7) There is a map ^:A^ Â•Â— Â» A_. which' is a surjective incidence structure homomorphism such that Ig f\ hi in A implies E Â«fg H ^h in A fi< , and such that (Â»? F = if Q <=^ p D Q) and ( g O h.) (Proof) If a & M, denote the Â— equivalence class of a in B by aÂ». Then define Â«?(a,b) = (a*,bÂ«), *ffm,d] = m%d* and

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65 *f[u,v"]' = [0*,v*]' when u Â€ N. Then, since the map v:B Â— Â» B* induced by ~ is a biternary ring homomorphism, ^ preserves the incidence relation. The remainder of the lemma is immediate by Lemmas 2, 4 and 5. / Thus, A g is an AH-plane. Observe that [0,01^ Co,Ol'; that (1,1) [o,0],[o,0 "]'; and that %_ is a bijection. Thus C(B) = (A B , ( [0,01, [0,0]' , (1,1) fM,Â§ ) ) is a coordinatized AH-plane. If w:B Â— *Â• B ' is a biternary ring homomorphism, then C(ui) defined earlier is an incidence structure homomorphism which preserves the parallel relation. Recall that u>(N ) C N B B ' Hence, if a bin 3, the uia Â«-wb in 3'. Thus, (a,b) D (a',b') in C(B) implies (u>a,) is a coordinatized AH-plane homomorphism. L c(B) and S^/ 30 ^ = x ( /^Â£ (<<) ' 2 : ^ ~^ ^ is a Since C(1J = ].Â„,Â„, and C(ftx) = C(ft)C(Â«), C:B ~ B functor. If B* is a biternary field, then N = to! and C(B*) is a coordinatized affine plane since the neighbor map of C(3*) is essentially the identity map. // 3.24 Proposition . The functor BC is the identity functor on B. Proof. Let (B,T,T') be a biternary ring, and let (Q,S,S') = BC(B). By the constructions, Q = 3, and y = S(x,m,d) in 3C(B) O (x,y) I [m,dl in C(3) <Â£=$> y = T(x,m,d) in 3. Similarly, x = S'(y,u,v) in BC(B) <*=> (x,y) I [u,vl* in C(B)<$=> x = T'(y,u,v) in B. Hence BC(B) = (B,T,T').

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66 Let ui:3 Â— *Â• 3" be a morphism in 2. Then CU)(a,a) = (*ja,wa) for every a e 3, end B(cU))a = -r"CU)Â« a = {"(we,wa) = wa. Thus BC:B Â— Â•> 3 is the identity functor on 3. // 3.25 Construction cf p:l~ Â— Â•> CB. Let C = (A, (g ,a ,E,M,c)) be a coordinatized AK-plane. Then CB(C) = ( A_ ( ) , ( LO ,0l ,[0, 0] ' , ( 1 , i ) , M, ^r(D^ an " ^" t can eas ^-^y be seen b Y looking at the usual identification of points and lines of C with their representation; that there is an isomorphism P r :C Â— C3(C) v.'hich takes a point (x,y) of C to the point (x,y) of CB(C). 3.26 Proposition. The map Â»:l*s 1 t r _ c Indicated above is natural isomorphism. Proof . Let u>:C Â— * C ' be a morphism in C. Recall that B(w): ^'(uj(<^ m)). r.ence, if P is a point of CE in C, EU)(*P) = ^Â•UP). Thus if ? I CE in C, CB(u>)(o P) = CBUM^P^P) = (B(w)(^P),S(w)($P)) = C^'UP) , Â£,' UP)) = * (up). Thus the diagram: CB(m) -* CB(C') commutes with respect to the x-axis of C, the y-axis of C and all the points of GE, and hence it commutes with respect to all the lines which are parallel to either g or c , and hence with 'x ' y respect to all the points of C. The image of each line of C in CB(C') intersects either both Co,Ol and [0,ll or both [0,o]' and

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67 lOjlJ 1 in CB(C') in points which are non-neighbor images of points in C. Thus, since the diagram commutes with respect to all the points of C, it commutes with respect to all the lines of C. Thus fsljs, Â— * CB is a natural isomorphism. // We have shown the following theorem: 3.2 7 Theorem . The functors B:C Â— * B and C:3 Â— Â» C are reciprocal equivalences where C is the category of coordinatized affine Hjelmslev planes and B is the category of biternary rings. // 3.26 Corollary . The functors B*:C* Â— * 3* and C*:B* Â— Â» C* defined as restrictions of B and C respectively are reciprocal equivalences where B* is the category of biternary fields and C* is the category of coordinatized affine planes. // 3.29 Definition . There are reciprocal equivalences F*:C* Â— -> F* and C p *:F* Â— * C* (where F* is the category cf ternary fields) defined by letting F*(C) be (M,T) if B*(C) = (H,T,T'); by letting C F *(F) be the usual coordinatized affine plane constructed over a ternary field F (see the proof of Proposition 3.22); by letting F* take a morphism w:C Â— > C Â» to (F* (C) ,P* (C ' ) ,f _ , J and by letting C * take a morphism <* = (F.F'.f ) to (C Â•(F). C *(F').f ) -r ' Â•>. Â«. F Â»~p ' i Â«' where f ^ , is defined by f -t (x f y) = (f^x.f^y), f^^.d! = Um,Â«d] and f , rO,vl' = fdO,etv"]' . The composition of B*:C* Â— Â» B* and C *:F* Â— * c* is a functor B'C F *:FÂ» Â— B* from the category of ternary fields to the category of biternary fields. If (H*,T*) is a ternary field,

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63 then BÂ»C F *(M*,TÂ») = (MÂ»,T*,T") is called the biternary field associated with the ternary field (M*,T*). 3.30 Proposition . If u>:A Â— * A' is a non-degenerate AH-plane homomorphism; that is, if there are three points of A whose images under k'u> are not collinear (where k' denotes the neighbor map of A 1 ), then there are coordinatizations K, K' such that w" = ( (A,K) , (A' ,K' ) , f w ) is a coordinatizaticn homomorphism with 0, X and Y as three such points; hence B(w") is a biternary ring homomorphism. Conversely, if w":(A,K) Â— Â» (A', K ') is a morphism in C, then the images of P = 0, P = X and P = Y are not collinear in the gross structure of A ' , and hence u> = (A,A*,f f| ) is non-degenerate. // 3.31 Definition . If B is a biternary ring, we call A , the B AH-plane we constructed in Construction 3. IS, the AH-plane generated by B. Remark . Appendix. A can be read at this point if desired. However, nothing treated there is used in any of the proofs in the sections which follow.

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4. SEMITRANSLATICNS AND GEOMETRY In this section we define the concepts of semitranslation, j-angle, (P,g w )-j-Desarguesian, ( P,g w )-endomorphism, (F, A) -mime tic, translation and ( P, g^-transitive and prove that an AH-plane A is ^jgÂ«,i-H-Desarguesian if and only if the ser.itranslations of A are (PÂ»9 fc )-transitive as well as various other results. 4.1 Definitions . A homomorphism
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70 shown to be equivalent to ours by using Prepositions 4.5 and 4.7 (4) below. 4.3 Definitions . Let A be an AH-plane. An endonorphism
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71 4.5 Definition . V.'e say that an endomorphism
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72 If P ~ tp, then in the above argument L(F,PR) ~ L(*P,PR)-, and hence R ~
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73 is' not the identity, then all the directions of and (Q^J^J are ( T , g>> )related if they are T-relsted and * i(i+1) Â« k i(i+1) for each i, 1 Â•Â£ i < j and m i(i + l) * P f Â° r each i ' 1 Â£ i < J* We sa >' A is (rSgoJ-jPesarouecian , if whenever two j-angles (P.;m .) and l ef (Q i ;k ef } are (P >9jr elated, then Q I LCC^m^). We say that A is (r i ,g,)H-Desarauesiar. , if A is ( P,gÂ„)j-Desarguesian for every j Â£ 2. We will sometimes put a dash in place of m in gh J = ^ P i' P 2' " * ' F j ;n l2' " * * ' m il* if m ah Can be any appropriate line between ? and P h ; for example, the direction may be specified by some requirement on J. We sometimes write a direction Â£ in place of ra to indicate that m , = L(P .1) . gh gh 9 4.11 Prop osition . If an AH-plane A is (P,gJ-j-Desarguesian,

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74 j > 2, then A is (V,g _)-( j-1 )-Desarguesian. Proof . Let K = (P ..., P j-l ;rn l2'"* ,rn (j-l)l ) be a tj-D-angle. Then K. . 1^^ ^i^,. ) is a j-angle. j-1' 12'" 12'*"'"'(j-l)l Using this expansion, we see that the proposition follows easily. // 4.12 Definition . The expansion given above will be called the canonical expansion of a (j-l) -angle to a j-angle. 4.13 Definitions . An AH-plane A is said to be (P, 2.) -mime tic if whenever g, g Â« ,h,h Â« ,k,k ' are lines of A such that g,g' are in P; h 11 hÂ»; k,k' are in Â£ and 1 k A h A g | /Â„ , |k A h A g Â• 1 4 0, IkA h'ngl/O, then IkA h' A g'l 4 0. See Figure 4.1. We say A is ( P , g^ ) -mimetic if it is ( P,2 ) -mimetic for every direction 1. Fiaure 4.1.

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75 4.14 Propcsitlon . Let A be an AH-plane. Then A is (P, geo )-2Desarguesian if and only if it is (P,l)-mimetic for every Â£ 4F . Â£Â£Â£Â£| Â« <=>) Look at the biangles (g n k,g' A k;k,h) and (g n k Â• , g Â• k Â• ; k ' , k Â• ) . (Â«=) Let (P Â± ;m ef ) and (Q^k^) be (r.g. )-related biangles such that m 12 * 2 and i * T. Then, by (r,l)m i m i cry there is a point R I S 2 Â» k 12 .MQ 1? m 21 ). Hence R = Q^ Thus Q 2 I LCO^.m^). and A is (P,g 4<> )-2-Desarguesian. // 4.15 Proposition . Let A be an AH-plane and let PC H> L(P,P) for some points K,P and some direction P. Then there is a unique line KP joining K and P, and KP + P. Proof. Observe that K * L(P,P) implies K + p; hence K and P are joined by exactly one line. If KP ~ T, then k(KP) & k(D; hence K(KP) = K(L(?,D), a contradiction. Thus KP * T. // 4.16 Proposition . Let A be an AH-plane. If A is (P,l)-mimetic for some P, P <+Â• 1 , then every semitranslation with direction Â£ is a translation; that is, every (1 , g#o ) -automorphism is a translation. Proof. The identity map is a translation. Assume A is (P,Â±)mimetic for some P, P 4 g. , and let
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76 We wish to show that h' is a trace of
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77 L(K',k) = Jc' and cr(H) I g 2. By Proposition 4.18, A is (PÂ»9oJand (.i.,gÂ„) -mime tic. Let cr be a semitranslation with direction TT. Then A is (A, ID-mimetic for some /\ + IT. Thus, by Proposition 4.15, 0 )-4Desarguesian, then the semitranslations of A are (r,g (>0 )-transitive. If the order of A is greater than two and if A is (P,gÂ„)-3Desarguesian, then the semitranslations of A are (P,gÂ») -transitive. If the semitranslations of A are (P,gÂ») -transitive, then A is (n Â»gÂ«, ) H D esarguesian; that is, A is ( P,g eo )-j-Desarguesian for

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78 every integer j greater than or equal to two. Proof. Assume that the semitranslations of A are (T.g^J-transitive, that (F ;m ) and (Q ;k ) are any two (P ,g^)-related jangles, and that r is a (Pjg^-semitrar.slation taking P to Q . Thena-m^ = LCQ^m^) = k^; L(K,D and define T .by using the construction above with S, T (S) in place of K, K'. We wish to show that f (P) = T (?) for every ooint P in D H D . Let * Â•> K S P ^ Â°K A D S* If p = K or P = S, the equality is immediate. If P / S,K; look at Figure 4.2. Let k 6 K'T (P). We have that KS -f P, SP * P. Using the fact that A is (P.g^-O-Desargueslan,

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79 fÂ„ (K) . K' K P = K T K (S) = Q 2 S = P. Figure 4.2. we have that T (P) I L(K',PK). Hence T and T agree on S K , S D A D . Define L(Q,T) . Thus T US Q coincides with oon all of D . If Q + L(S,r), then Q L(K, ) and the above argument with the roles of K and S interchanged shows that T coincides with Â«" on D . 4 v_<

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so s-P Figure 4.3 . Now assume that A has order greater than two and that A is at least (r , ,g i>0 )-3-De3arguesian. This is the remaining case. TÂ„ (r Pick a point W such that W + L(K, D ,L(S,D . Then T and W w and T g ) agree on D, ; (\ D ?; (D {j (\ D^ by the argument for T K and V And hence Â°" P = T W (P) on D w Thu s,
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81 Let h be a line of A, and let G be a given point on h. Assume h + V. Then T_(H) =
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82 Thus
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83 Proof. Let g 6 T and Pig. Let h Â€ P such that o-F,-rcrP I h; such a line exists since T e D-. Since N g 6 r, g't Â£ and g" Â£ A such that F I g,g',g" for some point P. Then P + X P by Proposition 4.7 (2). Thus, g" = P(X?) and g" >,L g,g'. We wish to show that crX has no traces neighbor to either g or g*. By our hypotheses, cr X e D for some direction A.. If A~r,i,then h = L(?,Jl) is a trace of crX. Since Xcr =

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84 h,g', we have that o"P /X(irP), a contradiction to X + 1 by Proposition 4.7 (2). Hence, XL -/i -1 ,! and (\
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85 P which is not neighbor to h. Let R,S be points on k; R,S /. P. Let Â£ be a direction net neighbor to h,k. Let R' = L(R,Â£) A h; S' = L(S,1) H h; then since R,S + ?; and 1 + h,k, we have that R',S' -Y P. Let obe a ( P,g )-endomorphism which takes R' to S'. Then cr takes L(R,Â£) to L(s,l) and hence R to S. By repeating the argument, we can see that the endomorohisms of A are (P.q )transitive. Assume g is affine and that P = TT where TV Vg. Let A,l be directions; A ,1 4TT ,TT(g) . Let Q I g; Q + h. Let R' = L(G,M n h; S' = L(G,1) h; then, R',S' + g. Let cr be a (TT,g)endomorphism taking R' to S*. Then g. Let Gig; RÂ« = GR A h; S* = GS A h; then, R*,S' 4g. Let
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5. SEMITRANSLATIONS AND ALGEBRA In this section we relate properties of the biternary ring of a coordinatized AH-plane to the existence of certain semitranslations. We also show that the category of regular biternary rings is equivalent to the category of coordinatized translation AH-planes. 5.1 Definitions . Let (M,T,T") be a biternary ring. We define the Tadditi on + :M* M Â— * M by a + b = T(a,l,b). We define the Tmultiplication n:Hx M Â— K by axb = T(a,b,0). The T-multiplication is usually denoted by juxtaposition; that is, we usually write ab instead of axb. We define the T'addition Â»:MxM Â— Â»M by a Â• b = T' (a,l,b). We define the T 'multiplication Â• :Mx M Â— * M by a-b = T'(a,b,0). We say T is linear if T(a,m,b) = (am) + b for all a,m,b eM. We usually assume that multiplications occur before additions and write (am) + b as am + b, and write (a. in) * b as a.m * b. Similarly T" is said to be linear if T'(a,m,b) = a-m Â• b for all a,m,b 6M. Observe that the definitions of the Tand T'-additions are dual; as are the definitions of the Tand T '-multiplications. 5.2 Proposition (Compare Lorimer [(1971) , page 166, Theorem (6.1.2)1). Let (K,T,TM be a biternary ring with T-additon +. 86

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87 Then (K, + ) is a loop; that is, it is a binary algebra with a unit such that any equation x + a = y can be solved uniquely for any one of x, a, y once the other two are specified. Remark . If a loop is associative it is a group. Proof. By (B2), + a = a, and by (B3), a + = a for all a in M; hence is a unit of (:!, + ). Given x, a in M, x + a is welldefined. Given x,y in M, by (B4) there is a unique a such that x + a = y. Given a, y in M, by (B5) there is a unique x such that x + a = T(x,0,y) and by (32) T(x,0,y) = y. If z + a = y, then T(z,0,y) = y by (32); so that z = x, and the solution is unique. Hence (K, + ) is a loop. // 5.3 Definition . Let (K,T,TÂ») be a biternary ring whose T-addition (M,+) is a group. We will denote the T-additive inverse of an element b in M by b and write a b in place of a + (b). Also we will write a + b + c to mean a + (b + c). 5.4 Definitions . Let B = (M,T,T') be a biternary ring. If a, m, c are elements of M, then by condition (34) there is a unique z in K such that T(a,m,z) = c. Define a function Z:M 3 Â— * M by letting Z(a,m,c) = z whenever T(a,m,z) = c. Let Z' be the dual function whose definition has T 1 in place of T above. 5.5 Proposition. Let (M,T,TÂ») be a biternary ring, and let s be an element of H. If s * 0, then L :M Â— * M defined by L (m) =

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88 T(s,m,0) is a bijection which preserves and reflects the neighbor relation. Proof. Let t,s 6 M, s f 0. Since s Â± 0, there is an m such that t = T(s,m,0), = T(0,m,0) by (B6). Hence L is surjective. Assume L (z) = L (w) , z Â£ w. Let t = L (z). Then (z,0), s s s ' ' (w,0) are both solutions to T(s,m,d) = t, T(0,m,d) = ; so that t ~ 0, s ~ 0; a contradiction. Thus L is biiective. s If m ~ n, then sm ~ sn by (BIO); hence L preserves the s neighbor relation. If sm ~ sn, and if m /n , then by (BIO) s Â— 0, a contradiction. Hence L reflects the neighbor relation. // 5.6 Proposition . Let (M,T,TÂ») be a biternary ring, and let s 6 M. If s f 0, then the map R :M Â— M defined by R (m) = s s T(m,s,C) is a bijection which preserves and reflects the neighbor relation. Proof. Assume s ^ . Let k e M. There is a unique x such that T(x,s,0) = T(x,0,k) = k by (B5). Hence R is surjective. Assume R s (m) = R s (n). Let y = T(m,s,0) = T(n,s,0). Observe that y = T(m,0,y) = T(n,0,y).. Thus by ( B6 ) , m ~ n, and by.(B7) either m = n or s ~ 0. Thus, m = n and R is injective. By (B10), R preserves the neighbor relation. If R (m) ~ R (n), m f n, then by (B10), s ~ 0; hence R reflects the neighbor relation. //

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89 Remark . The following proposition closely resembles the first result that I proved in connection with ccordinatization. 5.7 Proposition . Let (A,K) be a coordinatized AH-plane. (Part I) If ois a dilatation on A, then the action of
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90 following are equivalent. a) <* induces a translation .on A. b) Conditions (D), (DÂ«), (T) and (T-) are satisfied. c) Conditions (D), (DÂ») , (T) and (?Â») are satisfied. Proof. (Part I) Assume
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91 *fu,v]Â« where u Â£ N to be [ u, Z ' ( f ( ) ,u, f ( v)] ' , then < preserves incidence and Â« g I! g for all lines g of A. We wish to show that el preserves the neighbor relation. Since Â«< preserves the neighbor relation for parallel classes, it suffices to show that * preserves the neighbor relation with respect to points. Ler (x,y) ~ (x',y'), and let h and k be distinct lines joining (x,y) and (x',y*). Then Â« h -ft' tik and hence
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92 5.S Corollary . If f ' is bijective, then condition (D) of Proposition 5.7 is equivalent to: (D) # f'(T(x,m,(f)" 1 (T(f(0),m,t))) = T(f(x),r.,t) for all x,m,t in K. So, by xy-duality, if f is bijective, then (D') is equivalent to: (D') f (T 1 (y,u,f~ 1 (T'(f '[0),u,s))l = T'(f (y),u,s) for all y,u,s in h. Similarly, if f is bijective, then (D") is equivalent to: (DÂ») f(T'(y,u,f -1 (T'(f '(C), u,s))) = T Â• ( f Â• ( y) , u , s) for all y,s in M and all u in N. Proof. Assume f is bijective. Then t = Z(f ( ) ,rr,f Â• (b) ) if and only if T(f (0) ,m,t) = f'(b), and if and only if b = (f) (T(f (0) ,m,t) ) for any m in K. Thus, it is easily seen that (D) and (D) are equivalent, and by xy-duality it is easily seen that if f is bijective, then (U') and (DÂ») are equivalent. The proof that if f is bijective, then (D") is equivalent to (D") 1 ' is similar to the dual to the first part of the proof given above. // 5.9 Definition . Let (A,K) be a coordinatized AH---lane. We denote the direction containing [m,0] by (m) , and, we denote the direction containing [u,o]' by (u)' for every m,u in M. 5.10 Definition. Let (M,T,T') be a biternary ring and let s and k be elements of M. We write T(s,x,0) as sx. We say (M,T,T') is (k)recular for s whenever (M,T,TÂ») satisfies the followinq

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93 three conditions for s and k, and for all x,m,b,y,u,v in M. 1) T(s,k,T(x,~,b)) = T(T'(T(s,k,x),l,Z'(sk,l,s)),m,Z(s,m,T(s,k,bÂ») 2) T'(T(s,k,T'(y,u,v)),l,Z'(sk,l,s)) = T Â• ( T ( s , k , y ) , u , Z ' ( Â£ k , u , T Â• ( T ( s , k , v ) , 1 , Z Â• ( s k , 1 , s ) ) ) J . 3) T(T'(T(s,k,x),i,Z'(sk,l,s)),k,d) = T(s,k,T(x,k,d)). We say (K,T,T') is (k)regular if it is (k)-regular for all s in M. We define (k) 'regular for g and (k) 'reoular dually. We say (H,T,T') is axially regular whenever it is both (0)and (0) '-regular. We say (M,T,T') is regular if it is' both (k>and (k) '-regular for all k in M. 5.11 Theorem . Let (A,K) be a coordinatized AH-plane with biternary ring (M,T,T'), and let s,k be elements of Z. Then A has a semitranslation with direction (k) taking (0,0) tc (s,sk) if and only if (Z,T,T") is (k)-regular for s. If tr is a se-.itranslation with direction (k) which takes (0,0) to (s,sk), then
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94 f (T(x,k,d) ) = T(f(x),k,d). With x = 0, this becomes f'(d) = T(s,k,d) for any d in M. with d = 0, this becomes f (0) = t = sk. From condition (D) of Proposition 5 . 7 we obtain: T(s,k,T(x,:r:,b)) = T (f (x) ,m, Z( s ,m,T( s,k ,b) ) ) for all x ,T.,b in K. And from condition (D-) we obtain: f(T'(y,u,v)) = T*(T(s,k,y),u,Z'(sk,u,f (v))) for all y,u,v in M. Letting v = 0, u = 1, we have: f(y) = T Â• (T( s,k,y) , 1 ,Z Â• ( s k, 1 , s ) ) for all y in M. Putting this result in the equations we cot from conditions (D) and (D') we get conditions 1) and 2) above. Given any x,d in M, there is a y in M such that (x,y) I [k,dl. Then since [k,d] is a trace, f'(T(x,k,d)) = T(f(x),k,d). Thus condition 3) holds for ail x,d in M. (Â«=) Assume that (M,T,TÂ») is (k)-regular for s. Define a map
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95 Hence, by Proposition 5.7, gÂ»'H Dss a-5uesian by Theorem 4.21.

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96 5.15 Corollary. Let A be an AH-plane. The foil owinq are equivalent. 1) A is regular. 2) For some ccordinatizaticn K, the biternary ring of (A,K) is regular. 3) For every cc: rdinatization K, the biternary ring of (A, JO is regular. 4) A is isomorphic to the AH-plane AÂ„ generated by a regular biternary ring B. Proof . 11 4*> 2) <Â£$ 3) Obvious. 2) =^7 4) Assume the biternary ring B of (A,K) is regular. Then (A,K) is isomorphic to C(3) by Proposition 3.26; hence A is isomorphic to the AH-plane A of C(B). / S ~ 4) =^2) Assume that 0:A_ Â— * A is an isomorphism from B the AH-plane generated by the regular biternary ring (B,T,T') to A. Let K B = ([0,0l,U0,03',(l,l),B,C ) where % is defined by 5 (a, a) = a fcr all points (a, a) on (0,0)v (1,1). Observe that (A D ,K ) = C(B) and that E(A p ,K) = (B,T,TÂ«). Let K = (etO.Ol.SCo.Ol'jOd.D.B^gG" 1 ). Then B(A,K) = (B.T.T'J. Hence A is regular. // Remark . Recall that we have defined a%b = ab = T(a,b,0); a-b = T'(a,b,0); a + b = T(a,l,b); and a Â» b = T'(a,l,b) for all a,b in M. 5.16 Corollary . A coordinatized AH-plane (A,K) has a semitrans-

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97 lation with direction (I) taking (0,0) to (s,s) if and only if the biternary ring 3(C) = (K,T,T') satisfies the following three conditions (here C = (A,K)): 1) s + T(x,m,b) = T(s + x,m,Z(s,m,s + b)) for all x,x,b in M. 2) s + T'(y,u,v) = T'(s + y,u,Z'(s,u,s + v) ) for all y,u,v in M. 3) (s+x)+d=s+(x+d) for all x,d in M. If
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98 condition: 1)" T(x,m,T(s,m,w) ) = T(x * s,rn,w) for ail x,m,w in M. Letting m = 1, w = in 1)", we get a) above. If condition a) holds it is easily shown that condition 1)" is equivalent to condition b) and that condition 2)' is equivalent to c). // 5.18 Corollary . The semitranslations of a coordinatized AK-plane (A,K) with biternary ring B(A,K) = (M,T,T') are ( (0 ) ,g rt )-transitive if and only if the following three conditions hold for all s,x,m,b in K: a) x+s=x*s. b) T(x,m,T(s,m,b)) = T ( x + s,m,b). c) T' is linear; that is, T"(x,m,b) = x-m * b. d ) ( M , + ) is a group:. Proof . (=Â») Condition c) of Corollary 5.17 with m = 1 implies the T'-addition is associative; with b = it implies that T* is linear; that is T'(x,n,s) = (x-m) * s. Since the Tand T 1 additions are equal by Corollary 5.17 condition a), by Proposition 5.2, (M,+) is a group. (4=) By c) and d) above, we have that T*(x,m,b) + s = ((xÂ«m) + b) + s = x-m + (b + s) = T"(x,m,b + s) for all s,x,m,b in M. Hence conditions a), b) and c) of Corollary 5.17 are satisfied for every s in H. Let (a,b) and (d,b) be any points on a line [0,b], Then for some s in M, T(a,l,s) = d and hence there is a "semitranslation Â«r taking (a,b) to (a + s,b) = (d,b); hence the semitranslations of A are ( (0 ) ^^-transitive. //

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99 5.19 Corollary . Let (A,K) be a coordinatized AH-plane with biternary ring (M,T,TÂ»). The semitranslations of A are ((0),g )and ( (0) 'jg^J-transitive if and only if the following four conditions are satisfied: 1) T and T" are linear; that is, T(x,m,b) = xm + b and T'(x,m,b) = x m * b for all x,m,b in M. 2) The Tand T'-additions are equivalent; that is, x + a = x * a for all x,a in M. 3) CM,+) is a grouc . 4) xm + sm = (x + s)m and x.m + s-ra = (x + s).m for ail x,s,m in M. Proof. (-Â£) Assume the semitranslations of A are ((0),gÂ„)and ( ( Â°J ',g w )-transitive. Then, by Corollary 5.18 and its xy-dual, the first three conditions above follow immediately. The remaining condition follows from condition 5.13 b) with b = and' its xy-dual. / (<=) Assume that conditons 1), 2), 3) and 4 3 hold in the biternary ring (H,T,TÂ«). Then apply conditions 1), 3), 4) and 1) to. get: T (x,m,T( s ,m, b) ) = xm + ( sm + b) = (xm + sm) + b = ((x + s)m) + b = T(x + s,m,b) for all x,s,m,b in M. Then, by Corollary 5.18, we havethat the semitranslations of A are ( (0) ,gÂ„)-tra .sitive, and by an xy-dual argument, they are also f(0) l t g_)-transitive. . // 5.20 Corollary . Let (A,K) be an axially regular coordinatized AH-plane. Then, any semitranslation of A is a translation, and

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100 if we let U = i or 1 v and T are translations with directions (0) and (0)' respectively!, then U = \
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101 5.24 Definition . Let A be an AK-plane. We say that A is a translation AK-olane if A has a group of translations which is transitive on the points of A. 5.25 Theorem . Let A be an AH-plane. The following are equivalent. 1) A is a translation AH-plane. 2) A is regular; that is, the semitranslaticns of A are (r,g^)-transitive for every direction V, 3) A is T-regular for three pairwise non-neighbor directions r. 4) For some cocrdinatization K, the biternary ring (!Â•'., T,T') of (A,F) is axially regular and (K,+) is abelian (see Corollary 5.23 for a characterization of axially regular). 5) The set of translations of A forms an abelian group which acts transitively on the points of A. 6) A is regular in two non-neighbor directions and the set of semitranslaticns of A is closed under composition. 7) A is regular in two non-neighbor directions and the set of translations of A is closed under composition. 8) For every coordinatization K of A, the biternary ring (M,T,T') of (A,K) is regular (hence, axially regular) and (M,+) is abelian; the set of translations W of A is an abelian group under composition and W = {
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102 Proof . 1) =Â£ 2). Assume A is a translation AH-plane. Let V be a direction; let g e V and let P,P' I g. There is a translation
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103 4) =Â£ 5). Assume that for some coordinatization K, the biternary ring (M,T,T I ) of (A,K) is axially regular and that (K,+) is abelian. Then Z(x,m,c) = -xm + c and Z'(y,u,w) = -y-u + w for all x,m,c,y,u,w in M. Using these two equalities, we can easily show that conditions 1), 2) and 3) of the definition of "(k)-regular for s" are satisfied for every k,s in M. Thus, every cr defined by (a,b) = (a + s,b + sk) for all points (a,b) in A and for some s,k in M is a ( (k ) ,g e0 )-semitranslation. By Corollary 5.12, these semitranslations are ( (k ) ,g oo )-transitive. By Proposition 4.19, each such cr is a translation. By xy-duality, each cr ' defined by
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104 1) and 2) => 7). Obvious. / 1) and 2) => S). Assume A is a regular translation AK-plane. Then A is axially regular, and the set of translations is an abelian group under composition. Let K be any coordinatization of A. If (;-'., T, 7') is the biternary ring of (A,K), then (MjT.T 1 ) is regular by Corollary 5.15. The remainder follows by Corollary 5.20. / S) => 4) . Obvious. / 9) <=> 2). This is immediate by Theorem 4.21. // 5.26 Definitions . Let C denote the category of coordinatized translation AH-pla.nes and let E denote the cateqory of rccular ~~ r * biternary rinqs. Let 3 :C m Â— * B and C :B Â— -Â»Â• C b'=> the i 2Â— ~r T r ~T r T restrictions of 3 and C to C m and B respectively. T r 5.27 Corollary . The maps B :C Â— * B and C :B Â— * C defined above are reciprocal equivalences such that B C = 1. // ~r~T ~B 5.28 Corollary . A biternary ring (M,T,T') is regular if and only if (M,+) is abelian and (M,T,T') is axially regular; that is, (K,+)"is abelian and (M,T,T') satisfies the four conditions of Corollary 5.23. // 5.29 Theorem . The translations of a coordinatized AH-plane C are ((0),g M )and ( (0) Â• ,g w ) -transitive if and only if the biternary ring (M,T,TÂ«) of C satisfies the following conditions: 1) T and T' are linear.

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105 2) The Tand T '-additions are equivalent; that is, a + b = a * b for all a,b in M. 3) (M,+) is a group. 4) xm+sm=(x+s)m and x-m + s-m = (x + s)-m for all x,s,m in M. Proof . This follows easily from Proposition 4.19 and Corollaries 5.12 and 5.23. // 5.30 Proposition. The maps B *:C * Â— Â» B Â• and C *:B* Â— C Â• i ^ ~r T r ~T r T defined by restricting B and C to the categories of coordinatized translation affine planes C * and regular biternary fields B * T r respectively are reciprocal equivalences. //

107 6.3 Definitions . Let M be a set with two distinguished elements 0, 1, and let three binary operations +, /, Â• be defined on M. We will write ab in place cf axb. The operation + is said to be the addition of M; X is said to be the first multiplication of M, and Â• the second multirlication of M. Let N = {n Â« H| 3 k Â« M, k 4 0, 3 kn Â» 0] and let N= {n fe K \ 3 k e M, k 4 0, -* k-n = ol. Define a ~ b if and only if every x such that a = x + b is in N and define a ~' b if and only if every x such that a = x + b is in N*. We say (M,+,X,Â«) is a precuasirlng if end only if the following twelve conditions are satisfied: (VWO) N = N';_ a b <-=$> a ~* b for all a,b in M. (VW1) (M,+) is a group with identity 0, and (N, + ) is a subgroup of (M,+). (VW2) 0m = mO = for all m in M. (VW3) I'm = ml = m for all m in M. (VW4) xm + sm = (x + s)m for all x,s,m in M. (VW5) xm = xm' + d is uniquely solvable for x if and only if ( m m ) ^ N . (VW6) The equation am = b with a Â± N is uniquely solvable for m. (VVJ7) if a Â€ N, b e N with a,b not both zero, then one and only one of the equations (am = b, b-u = a (with u 6 N)] is solvable for m Â€ M, u e M. If am = b, then there is an m' Â£ m such that am' = b. If. b-u = a , u <= N, then there is a u Â• & N, u' ^ u, such that b-u' = a. (VW8) The system (y = xm + d, x = y.u + v where u 'e n3 is uniquely solvable for the pair x,y.

109 skew quasirinq. If (K,+,x f .) is a skew quo siring, then <;Â•:, + ,-, x) is also a skew quasiring and it is called the dual of (M,+ x .). Remark. Condition (VW9s) is self-dual; that is, if we interchange the X and the in (Vâ€¢9s), the result is equivalent to (VW9s). 6.8 Proposition. Let (M,+,X,.) be a prequasiring or a skew quasiring; then -(ya) = (-y)a and dually -Cy-a) = (-y).a. Proof. Observe that ya + (-y)a = (y + (-y ) ) a = 0. // Remark. Letting y = 1 , a = -1 we observe that in a prequasiring or skew quasiring (-D(-l) = 1 and dually (-l)-(-l) = 1. 6.9 Proposition. The definitions of prequasiring and skew quasiring are equivalent. Proof. 1) Assume V = (M, +,*;Â•) is a prequasiring. If urn = 1 , then since 0m + = and -u + = , we have by (VW9) with e = 0, v = that xm = y if and only if y. u = x. If urn = 1, then xm = (xra) and hence (xm)-u = x for all x in H; similarly, (y.u)ra = y for all y in M. By duality, the same results hold if m-u = 1. Thus (VW9s) holds in V and V is a skew quasiring. / 2) Assume V = (M,+,X,.) is a skew quasiring. If um = 1, then (lm).u = 1 by (VW9s)j so that m-u = 1. Dually if m-u = 1, then um = 1. Assume um = 1, am + e = b and b. u + v = a f or some

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110 a,b,e,v in M. By (VW9s), (am)-u = a = (b e).u = b-u e-u. Thus v = -e-u. Similarly e = -vm. If xm + e = y for some x,y in M, then xm vm = y = (x v)m = y. Hence x v = y.u and y-u + v = x. 3y duality if y-u + v = x, then xm + e = y. Thus (VV/9) holds in V. 3y duality (VW9)' holds in V and V is a prequasiring. // 6.10 Proposition . In a prequasiring if m Â«Â£ N then there is a u Â£ N such that urn = 1; as one can see by applying' (VW5) to the equation xm = xO + 1. Dually, if u Â£ N, there is an m $ N such that m.u = 1. // 6.11 Definition. Let (V l+f *,.), ( V Â• , + Â• , x Â• , .Â• ) be prequasirings. We say ^:V Â— V is a pre qua siring homomorphism whenever u>(0) = io(l) = l, M (N V )
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112 Let w: D be a homomorphi; wnere a = (K,T,TM and D = (Q,R,R'). Then, u>(T(a,m,b)) = R(coa,oÂ»m,u>b) and w(T' (a,m,b) ) = R' (wa,:V Â— Â» w be a prequa siring homomorphism. Define EÂ»(io) by (E'(Â«o))m = u>m for all m in K. 6.15 Proposition . The map E':V Â— B defined above is a functor from the category of prequasirings to the category of regular biternary rings. Moreover E*E and ES' are identity functors. If V is a prequa siring, then N = N , as sets. Proof. Let (V,+,x,-) be a prequasiring . We wish to show that E'(V5 is an axially regular biternary ring. Let N be the set of right zero divisors of V. Then N = (n fe v|3 k, 'k / 0, i T(k,n,0) = C] = N E , (V) and similarly N' y = N' Pf( } . Let V

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113 Observe that a ~ b if and only if a b in N; hence, a ~ b if and only if the unique x which satisfies the equation a = T(x,l,b) is in N. Thus a v b if and only if every x which satisfies the equation a = T(x,l,b) is in N. Similarly, a ~ b if and only if every x which satisfies the equation a = T*(x,l,b) is in N. (VWO) implies (30). (VW ) implies (Bl) as can easily be checked. (VW1) and ( VW2 ) imply (D2). (VW3) implies (53). (VWl) implies (E4). (VW5) and (VWl) imply (E5). (VW6), (VWl) and (V./4) imply (B6). (VW7), (VWl) and (VW4) imply (E7). (VW3) implies (S3). (VW9) implies (B9). (VWlC) implies (BIO), and (VW11) implies (311). Hence E*(V) is a biternary ring. By Corollary 5.2 3, E'(V) is (0)and ( ) '-regular . It is easily seen that if u>:V Â— > W is a precua siring homomorphism, then E'(w) is a biternary ring hemomorphism, that S' (1 V ) = 1 E'(V) and that E Â• C are reciprocal equivalence such that W = l r -. //