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Material Information 

Title: 
Coordinatized Hjelmslev planes 

Alternate Title: 
Hjelmslev planes 

Physical Description: 
xxxiii, 267 leaves. : ; 28 cm. 

Language: 
English 

Creator: 
Bacon, Phyrne Youens, 1936 

Publication Date: 
1974 

Copyright Date: 
1974 
Subjects 

Subject: 
Geometry, Projective ( lcsh ) Mathematics thesis Ph. D Dissertations, Academic  Mathematics  UF 

Genre: 
bibliography ( marcgt ) nonfiction ( marcgt ) 
Notes 

Thesis: 
Thesis  University of Florida. 

Bibliography: 
Bibliography: leaves 263266. 

General Note: 
Typescript. 

General Note: 
Vita. 
Record Information 

Bibliographic ID: 
UF00098680 

Volume ID: 
VID00001 

Source Institution: 
University of Florida 

Holding Location: 
University of Florida 

Rights Management: 
All rights reserved by the source institution and holding location. 

Resource Identifier: 
alephbibnum  000580698 oclc  14074879 notis  ADA8803 

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COO2DIINATIZED HJSLZ'SLEV LANES
By
PHYRNE YOUENS BACON
A DISSLRTATIO;N RLSE:' D TC :EE GRADULI.TE COUNCIL, OF
THE UNIVERSITY CF ELCiz']. A
IN PARTIAL FULFILlMENT OF THE REUI1RE: ::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 mymother, 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 130131.
ACKNO'WLEL DGEMENTS
I would like to express my thanks and appreciation to
my advisor, David A. Drake, for his superior example, for his
contagious enthusiasm, and for his excellent suggestions.
I would like to thank the other members of my committee,
Ernest E. Shult, Kermit N. Sigmon, Mark L. Teply, Mark P. Hale,
Jr., and Billy Thomas, for their comments and suggestions.' I
would also like to thank George E. Strecker for his comments.
I would like to thank Benno Artmann, N. D. Lane, and
Willian J. LeVeque for having indirectly provided copies of
papers referenced in my bibliography, [Cyganova (1967)],
[Lorimer (1971)] and CSkornjakov (1964)] respectively, and I
would like to thank Wladimiro Scheffer for his translation of
[Cyganova (1967)].
I also wish to thank the many professors and fellow
students who have helped make my graduate study an interesting
and challenging experience. I would especially like to thank
Richard D. Present, William M. Bugg, W. Edward Deeds, E. G.
Harris, Harold C. Schweinler, Don D. Miller, David R. Hayes,
John G. Moore, Dennison R. Brown, George E. Strecker, W.
Edwin Clark, and Charles I. Babst.
TABLE OF CONTENTS
ACKNOWLEDGEM!E NTS V
LIST OF DEFINED TERMS viii
LIST OF FUNCTORS xvii
LIST OF SPECIAL NAPS xx
KEY TO CATEGORIES xxii
KEY TO SYMBOLS xxv
ABSTRACT xxxi
Sections
1. INTRODUCTION 1
2. HJELMSLEV PLANES 7
3. BITERNARY RINGS 44
4. SEMITRANSLATIONS AND GEOMETRY 69
5. SEMITRANSLATIONS AND ALGEBRA 86
6. PREQUASIRINGS AND QUASIRIT:GS 106
7. KERNELS OF QUASIRINGS 116
8. OTHER CENTRAL AXIAL AUTOMORPHIS:S 123
9. AHRINGS 139
10. HJELMSLEV STRUCTURES 150
11. DESARGUECIAN PHPLA.ES 199
12. PAPPIAN CONFIGURATIONS 208
Appendices
A. RESTRICTED BITERNARY RINGS 222
TABLE OF CONTENTSS continued
Appendices continued
B. QUASICONGRUENCES 243
BIBLIOGRAPHY 263
BIOGRAPHICAL SKETCH 267
LIST OF DEFINED TERMS
Term Subsection
natural isomorphism, functor 2.1
function, graph, compose, identity function,
underlying set, concrete morphism,
underlying set function, natural composition,
natural identity morphism, identity,
concrete category, homomorphisms, map, map 2.2
generated by, generated by, of, with 2.3
surjective, injective, bijective 2.4
equivalence, isomorphism 2.5
reciprocal equivalences 2.6
preserve, reflect 2.7
incidence structure, points, lines,
incidence relation, is incident with, lies on,
is a point of, is on, goes through,
is a line through, join, collinear, copunctal 2.8
incidence structure homomorphism 2.9
the category of incidence structures 2.10
the incidence structure induced from S by ,
the induced incidence structure 2.12
projective plane, ordinary projective plane 2.14
parallel, affine plane, ordinary affine plane 2.15
viii
LIST OF DEFINED TERS.S continued
Term Subsection
neighbor, projectively neighbor,
projectively neighbor, projective Hjelmslev plane,
PHplane 2.17
projective Hjelmslev plane homomorphism,
the category of projective Hjelmslev planes,
the category of projective planes 2.19
parallel, affinely neighbor, affinely neighbor,
affine Hjelmslev plane, AHplane 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
nondegenerate, 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 PHplanes,
the category of lined projective planes,
the category of lined incidence structures 2.39
the generalized incidence structure,
the lined generalized incidence structure, 2.40
LIST OF DEFINED TERMS continued
Term Subsection
generalized point, generalized line,
generalized incidence structure, line, point,
incidence structure 2.40
neighbor, generalized neighbor relation,
neighbor relation, neighbor, neighbor relation,
generalized neighbor relation 2.43
derived from, derived from,
derived from H by use of, derived, extended to,
extended to, extensions of 2.51
embedding, incidence structure embedding of S
into S' 2.52
generalized AHplane,
generalized AHplane homomorphism,
the category of generalized AHplanes 2.58
AHplane embedding 2.62
ternary field, zero, one, symbols,
ternary field homomorphism,
the category of ternary fields 3.1
neighbor, not neighbor, biternary ring,
right zero divisor 3.2
dual 3.3
symbols, zero, one, biternary field 3.4
biternary ring homomorphism 3.9
the category of biternary rings,
the category of biternary fields 3.10
x
LIST OF DEFINED TERMS continued
Term Subsection
coordinatized AHplane, coordinatization,
symbols, ycoordinate, xcoordinate,
representation, representation, representations,
xaxis, vaxis, origin, unit point 3.11
xyduals 3.12
coordinatized AHplane homomorphism,
coordinatization homomorphism, neighbor map 3.14
the category of coordinatized affine Hjelmslev
planes, the category of coordinatized affine plnes 3.15
dual 3.20
the biternary field associated with 3.29
generated by,
the AHplane generated by a biternary ring 3.31
dilatation, trace, semitranslation,
semitranslation with direction P, translation 4.1
(P,g)endomorphism, central axial endomorphism,
center, axis, (P,g)endomorphism,
central axial endomorphism, center, axis 4.3
neighbor, neighbor endomorphisms 4.6
jangle, vertices, sides, triangle, frelated,
(r,g,)related, (P,g.)jDesarguesian,
(rP,g)HDesarguesian 4.10
the canonical expansion of a (jl)angle to
* jangle 4.12
(r',)mimetic 4.13
LIST OF DEFINED TERMS continued
Term Subsection
(P,g)transitive 4.17
order, infinite order 4.20
Taddition, Tmultiplication, 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
Cregular, regular, axially regular,
regular in the direction r 5.13
the category of axially regular biternary rings,
the category of axially regular coordinatized
AHplanes 5.21
the category of coordinatized translation AHplanes,
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
LIST OF DEFINED TER5S continued
Term Subsection
the AHplane generated by a prequasiring 6.19
biquasifield, the category of biquasifields,
the category of coordinatized translation affine
planes 6.20
kernel, the kernel of a quasiring 7.1
local ring 7.2
trace preserving, a trace preserving endomorphism
of the translation group 7.5
left modular for s, left modular,
strongly left modular, T'weakly left modular for s 8.1
strongly (P,g.)transitive 8.6
Prelated, (P,g.)related, (P,g,)HDesarguesian,
strongly (P,g,)HDesarguesian 8.8
((0),[0,0]')normal for s, ((0),[0,0]')normal,
T'weakly ((0),[0,01)normal for s,
Tweakly ((0),10,03')normal for s 8.10
((0)',[0,01')normal for s, ((0)',[0,01')normal 3.15
affine Hjelmslev ring, AHring, Hjelmslev ring,
Hring 9.1
AHring homomorphism, the category of AHrings 9.2
kernel quasiring, the category of kernel quasirings 9.3
Desarguesian, the category of coordinatized
Desarguesian AHplanes 9.6
the AHplane generated by an AHring,
generated by 9.8
xiii
LIST OF DEFINED TERMS continued
Term Subsection
Hjelmslev Desarguesian,
strongly Hjelmslev Desarguesian 9.9
the category of division rings 9.12
the category of coordinatized Desarguesian
affine planes 9.13
the Hjelmslev structure of an AHring S 10.3
near, near, near, near 10.6
the lined affine Hjelmslev structure of S 10.9
extended to, extended to (H(S),rgQ) through ,
extended to, extended to H(S) through 10.10
projectively Desarguesian 10.16
Desarguesian 10.17
(P,g)automorphism, (P,g)transitive 10.19
full jvertex 10.26
Hjelmslev structure, near,
Hjelmslev structure homomorphism, gross structure,
neighbor map, the category of lined Hjelmslev
structures 10.36
full, a full lined Hjelmslev structure homo
morphism 10.39
basis triple 10.46
Klingenberg coordinatization .10.47
Klingenberg coordinatization 10.49
induces % through k and X',
is induced through X and X' by a, extension, 10.52
xiv
LIST OF DEFINED TERI;S continued
Term Subsection
extension of v through X and X' 10.52
the canonical basis triple,
canonical coordinatization, basis triple 10.59
the category of Desarguesian AHplanes with
nondegenerate AHplane hcmonorphisms 10.63
the category of lined Hjelmslev structures with
full lined Hjelmslev structure homomorphisms 10.66
the category of projectively Desarguesian
AHplanes with nondegenerate AHplane hononorph
isms, the category of lined Desarguesian rHplanes
with full lined Desarguesian PHplane homomorphisms 10.67
triangle, vertex, side, full 11.3
Hjelmslov Desarguesian 11.5
(L, 1g,g',Pl',P)Pappian configuration,
Pappian configuration, (A,l1g,g',p?',P)Pappian,
Pappian, (g,g',P ',P",g")Pappian, Pappian,
Pappian for the full triangle J 12.1
restricted biternary ring, right zero divisor,
symbols, zero, one A.1
restricted biternary ring homomorphism,
restricted biternary field, the category of
restricted biternary rings, the category of
restricted biternary fields A.6
the category of quasifields A.30
components, precongruence B.1
LIST OF DEFINED TERKS continued
Term Subsection
congruence B.2
semicongruence, projection map, quasicongruence B.3
quasicongruence homcmorphism B.4
pointed AHplane, base point,
pointed AHplane homomorphism, the category of
pointed translation AHplanes, the category of
pointed translation affine planes B.8
parallel B.9
quasicongruence cocrdinatization,
the canonical base point, the canonical quasi
congruence coordinatization B.19
the category of translation AHplanes with
nondegenerate AHplane homomorphisms B.20
LIST OF FUNCTCRS
Functor
G:A S
G :A H S
g 9
H* :A 
9 9
A:H A
:A  A
A :A  A
2:C 2 B2
B:r T r
B' :* 4 B'
C:B2 
v,: . F'
C_* :F' 
F
22
:C 2 2
B:0 * V
2
E':V >. 8
V 
Subsection
2.40
2.40
2.42
2.48
2.47
2.59
2.59
3.16
3.18
3.28
3.28
3.29
3.29
5.21
5.21
5.26
5.26
6.1
6.14
6.16
6.16
xvii
LIST OF FUNCTORS continued
Functor
Q:C Q
T
G':Q T
R*: C* *
QY: Dn  T
T T
2 s
K2 :(C) L
D,:(T s) L
X:Q * R
K
X':R 
R:C R
R':R  D
D
R.:C  R,
R*':R' C
D
H:A 4
Y:(B)n 48
D:((A ) )f (D)n
Sg
J :K A
J:K T
AP
K:Tp * K
K :(T) .
Z:B .Z
~ Z
CZ :Z *Z C
^:z
Subsection
6.18
6.18
6.20
6.20
7.6
7.6
9.4
9.4
9.11
9.11
9.16
9.16
10.2
10.63
10.66
10.66
B.9
B.9
B.14
B.20
A.10
A.8
A.13
A.17
xviii
LIST OF FUNCTORS continued
Functor
Z .:F* + Z.
P *:Z* ?*
~Z
Subsection
A.7
A.7
LIST OF SPECIAL MAPS
Symbol In use Subsection
f w = (A,A',fQ) 2.2
K K:A A* 2.25
K K:H  H* 2.25
TT TT(g) 2.35
g g(g ) 2.40
P P(TT) 2.40
S S(H,g) 2.47
T T(x,m,b) 3.2
T' T'(y,u,v) 3.2
; :OE  M 3.11
+ a + b 5.1
x ab = ab 5.1
* a b 5.1
a.b 5.1
Z Z(x,m,a) 5.4
Z' Z'(y,u,b) 5.4
i rx. 10.2
1
i yis 10.2
G G(H(S),h) 10.9
G G 10.9
Kern Kern Q 7.1
xx
LIST OF SPECIAL MAPS continued
Symbol
t
z
ij
6, #
T"
P
r
In use
Y:S  S/N
v:H(S)  H(S/r;)
(v,0,0)t
oz
cZ, tZ
Z.i
13
(a H(L))t, (tr aj(C)) #
za
GA
3s
T"(y,u,v)
P
A
r(x,x',x"), rx.
(y,y',y")s, yis
b, a b
J7(gD)
(OP3)
(OA4)
(PH3), ('2)
(AH4), (LF3)
(Ell)
(QF4)
( 11)
(R5), (RH6)
(R16)
J0
(OP )
(OA)
(PH ), (1 )
(AH ), ( )
(B )
(F )
(VJ )
(R ), (RH )
(R )
Subsection
10.12
10.12
10.21
10.21
10.43
10.44
10.44
10.53
10.59
10.62
A.1
B.19
10.2
10.2
5.3
B.9
2.14
2.15
2.17
2.21
3.2
6.2
6.3
9.1
A.1
KEY TO CATEGORIES
Equivalent categories are joined by [
Category Subsection
S incidence structures 2.10
S lined incidence structures 2.39
g
* affine planes 2.23
lined projective planes 2.39
g
H* projective planes 2.19
AAHplanes 2.23
generalized AHplanes 2.58
9
H PHplanes 2.19
H lined PHplanes 2.39
g
coordinatized AHplanes 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
AHplanes
axially regular biternary rings
prequasirings
coordinatized translation
AHplanes
regular biternary rings
quasirings
coordinatized translation affine
planes
regular biternary fields
biquasifields
quasifields
coordinatized Desarguesian
AHplanes
kernel quasirings
AHrings
coordinatized Desarguesian affine
planes
kernel biquasifields
division rings
Hjelmslev structures
translation AHplanes 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 AHplanes with non
degenerate homomorphisms 10.63
lined Hjelmslev structures with
full lined Hjelmslev structure 10.66
homomorphisms
quasicongruences B.6
pointed translation AHplanes 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
AHplane
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 AHplane
AB the AHplane generated by a
biternary ring B
the AHplane generated by a
prequasiring V
the AHplane generated by an
AHring S
B, B', (M,T,T')
C, C', (A,K)
D
Dr
E
E (A), E
biternary ring
coordinatized AHplane
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
xaxis
yaxis
PHplane
the gross structure of H
projective plane
lined PHplane
lined projective plane
incidence relation
coordinatization
quasicongruence
congruence
direction containing [m,01
a line not quasiparallel to gy
side of a jangle
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 jangle
jangle
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
Hring, AHring
restricted biternary ring
lined incidence structure
AHring
semicongruence
ternary operation
partial ternary operation
pointed translation AHplane
direction containing [u,03'
line not quasiparallel to gx
prequasiring
the set of translations
the group of translations.
the xcoordinate of a point
the ycoordinate 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 (AHplanes) are coordinatized by using biternary rings;
translation AHplanes by using quasirings; Desarguesian AHplanes
by using AHrings; and Desarguesian projective Hjelmslev planes
(Desarguesian PHplanes) by using Hrings.
An affine plane homomorphism <:A A' is an incidence
structure homomorphism which preserves the parallel relation.
If is nondegenerate (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 AHplane
homomorphism w:A A A' is an incidence structure homomorphism
which preserves the parallel and neighbor relations. If w is
nondegenerate (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 AHplane homomorphisms
which take xaxis to xaxis, yaxis to yaxis and unit point to
unit point, then the following pairs of categories are equivalent:
biternary rings and coordinatized AHplanes, quasirings and
coordinatized translation AHplanes, AHrings and coordinatized
Desarguesian AHplanes.
The category of quasicongruences is equivalent to the
category of pointed translation AHplanes, and the category of
Desarguesian AHplanes with nondegenerate AHplane homomorph
isms is equivalent to the category of lined Hjelmslev structures
with full lined Hjelsmlev structure homomorphisms. Desarguesian
PHplanes are Hjelmslev structures in which every two lines meet.
The directions of the xaxis and the yaxis are denoted by
(0) and (0)'. The translations of a coordinatized AHplane 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 xm + 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 ab = T'(a,b,0) for all a,b in M.
A coordinatized AHplane whose biternary ring satisfies the
conditions listed above is a translation AHplane if and only if
the addition + is abelian.
Translation AHplanes, Desarguesian AHplanes, Pappian
xxxii
translation AHplanes, Desarguesian :Hplanes and Pappian
Desarguesian PHplanes are each characterized geometrically,
in terms of algebraic properties of their coordinatizations, and
(for all except the sappian planes) in terms of properties of
their endomorphisms.
Algebraic characterizations are given of those coordina
tized AHplanes which have a semitranslation (in an affine plane
a semitranslation is a translation) with direction (k) which
moves the origin to (s,sk), and of those coordinatized AHplanes
which have a ((0,0),g,)endomorphism which moves (1,1) to (s,s).
There are similar results concerning ((0),[0,01') and
((0)',[0,0]')automorphisms where [0,0]' is the yaxis and (0)'
is the direction of the yaxis.
If H is a PHplane; if s s2, s3 are the sides of a tri
angle whose image in the gross structure of H is nondecenerate,
and if each of the three AHplanes A,, A2, A3 derived from H by
use of one of the sides sl, s2, s3 is Desarguesian, then H is
Desarguesian. There exists a Desarguesian 'AHplane which cannot
be derived from any Desarguesian PHplane.
xxiii
1. INTRODUCTION
A coordinatizaticn may be thought of as an isomorphism
between a geometric structure and a geometric structure which
has been constructed from an algebraic structure. Klingenberg
[(1955)] began the solution of the coordinatization problem for
Hjelmslev planes by constructing a projective Hjelmslev plane
from an Hring, and he showed that this constructed FHplane has
a number of properties. Lineburg [(1962)3 defined an algebraic
structure (which is here called a quasicongruence) and showed
that any translation affine Hjelmslev plane (translation AHplane)
can be coordinatized (in the sense mentioned above) by using a
quasicongruence. Lcrimer [(1971)3 continued work on the coordi
natization problem by constructing an affine Hjelmslev plane
from an AHring and then giving various theorems relating these
constructed planes to the class of Desarguesian AHplanes.
Cyganova [(1967)3 also did considerable work on the coordi
natization problem: she undertook to define an algebraic
structure which would have essentially the same relation to
affine Hjelmslev planes that ternary fields have to affine
planes. Unfortunately, her arguments contain a number of
serious omissions, not all of which I have been able to repair.
Her algebraic system has a ternary operation and a partial
ternary operation. In Definition A.1, a similar (but different)
algebraic system is defined which is called a restricted
biternary ring.
In Definition 3.2, an algebraic system with two (complete)
ternary operations is defined: this system is called a biternary
ring. An AHplane homomorphism which takes xaxis to xaxis,
yaxis to yaxis and unit point to unit point is called a
coordinatization hcmomorphism (Definitions 3.11 and 3.14). In
Theorem 3.27, it is shown that the category of biternary rings is
equivalent to the category of coordinatized affine Hjelmslev
planes (with coordinatization homomorphisms). This result is
used (along with others) to show that the category of quasirings
(these have two multiplications) is equivalent to the category
of coordinatized translation AHplanes (Corollary 6.18), and
that the category of AHrings is equivalent to the category of
coordinatized Desarguesian AHplanes (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 AHplane (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 AHplanes whose automorphisms are
(P,g.)transitive for any given direction V is given. In
Proposition 8.9, a geometric characterization of those AHplanes
whose automorphisms are (F,g.)transitive for any given point P
is given. These results are used (together with some algebraic
ones) to give geometric characterizations of translation AHplanes
(Theorem 5.25), of Desarguesian AHplanes (Froposition 9.10) and
of Desarguesian PHplanes (Theorem 11.6).
An AHplane homomorphism w:A  A' is required to preserve
the incidence, parallel and neighbor relations. An affine plane
homomorphism c:A  A' (c is an AHplane homomorphism between
affine planes) is said to be nondegenerate if it does not map
all the points of A onto a single line of A'. A nondegenerate
affine plane homomorphism is an incidence structure embedding
and reflects both the incidence and parallel relations (ag
implies g II h) (Theorem 2.63). If Q:A  A' is a nondegenerate
AHplane homomorphism (that is, e does not map all the points of
A into points of A' which are all neighbor to points on some
line k' of A'), then c reflects the quasiparallel and the
neighbor relations (Corollary 2.65).
In Theorem 5.11, an algebraic characterization is given of
those coordinatized AHplanes which have a semitranslation with
direction (k) which moves the origin to (s,sk) (in an affine
plane a semitranslation is a translation); in Proposition 8.2,
of those coordinatized AHplanes which have a ((0j0),g.)endo
morphism which moves the unit point to the point (s,s).
Propositions 8.11 and 8.16 give similar results for ((0),[0,0]')
and ((0)',[0,03')automorphisms respectively: here [0,01' is the
yaxis and (0) and (0)' are the directions of the xaxis and the
yaxis respectively.
4
Theorem 5.29 shows that the translations of a coordinatized
AHplane C are ((0),g ) and ((0)',g )transitive if and only if
the biternary ring (N,T,T') of C satisfies the following
conditions:
1) T and T' are linear.
2) The T and T'additions are equivalent: a + b = a b for
all a,b in M; that is, T(a,l,b) T'(a,l,b) for all a,b in R.
3) (M,+) is a group.
4) xm + sm (x + s)m and x.m + s.m = (x + s).n for all
x,s,m in M where the two multiplications are defined by
ab T(a,b,0) and ab T'(a,b,0) for all a,b in M.
Theorem 5.25 shows that a coordinatized AHplane whose
biternary ring satisfies the conditions listed above is a
translation AHplane if and only if the addition + is abelian.
Klingenberg [(1955)1 attempted to characterize in terms of
their automorphisms those PHplanes which are isomorphic to some
PHplane constructed from an Hring. His argument fails however.
In Theorem 11.6, such a characterization is given. This theorem
also indicates a geometric characterization of these PHplanes
(they are called Desarguesian PHplanes).
Lorimer [(1971)3 generalizes part of what Artin C(1957)]
calls "the fundamental theorem of projective geometry" by proving
some results relating the automorphisms of a Desarguesian AHplane
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 AHplanes with nondegenerate AHplane homo
morphisms to the category of AHrings; if t:A  A' is a morphism
in the first category, and if (S,8) and(S',9') are canonical
coordinatizations of A and A', then can be defined by ')P =
(Y(p)6P)Z' for some nonsingular matrix Z' with first column
(1,0,0) ; if there is an AHring homomorphism 4 and a nonsingular
matrix Z such that p can be defined by O'lP = (?GP)Z, then 2 = eZ'
and Y is defined by 4(s) = e(Y( )s)e for some e 6 S'.
In Appendix B, a number of theorems of Lineburg's are
used (along with other arguments) to show that the category of
pointed translation AHplanes is equivalent to the category of
quasicongruences (Theorem B.16). In Proposition B.20, it is
shown that there is a functor K from the category of translation
AHplanes with nondegenerate AHplane homomorphisms to the
category of quasicongruences which has the following property:
if :T T T' is a nondegenerate AHplane homomorphism between
translation AHplanes, and if (K,9) and (K', ') are the
canonical quasicongruence coordinatizations of T and T', then
p is defined (in terms of the algebras associated with the
coordinatizations) by yQ = KT(t)Q + O0.
In Proposition 11.4, it is shown that if H is a PHplane;
if sl, s2, s3 are the sides of a triangle whose image in the
gross structure of H is nondegenerate, and if each of the three
AHplanes Al, A2, A3 derived from H by use of one of the sides
sl, s2, 53 is Desarguesian, then H is Desarguesian.
In Proposition A.23, it is shown that the category of
restricted biternary rings is isomorphic to the category of
biternary rings; that is, there is an equivalence between them
which produces a onetoone correspondence between the objects.
The development here depends only on wellknown 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 AHplanes
given here resembles some of the recent work of J. U. Lorimer
and N. D. Lane C(1973)l and of J. W. Lorimer [(1973)a and (1973)b],
there are discussions of their recent papers and a brief history
of this research in Appendix A: A.24 through A.2S.
If it is desired, Appendix A (restricted biternary rings)
can be read immediately following Section 3 (biternary rings) and
Appendix B (quasicongruences) can be read immediately following
Section 5 (semitranslations and algebra). Nothing in any of the
sections depends on anything in the appendices.
2. HJEL:MSLEV PLANES
In this section we give a number of definitions (some of
which differ slightly from the usual ones and some of which are
entirely new), prove a few basic propositions, and discuss the
relationship between affine and projective HIjelmslev planes.
Proposition 2.31, Theorem 2.63, and Corollaries 2.64 and 2.65
concern nondegenerate 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 15, 49, 52, 591. We will
commit a common notational inconsistency by usually denoting a
category with objects A and morphisms I by simply A; our names
for the various categories will reflect this notation. We will
use the term 'natural isomorphism' in plane of Mitchell's term
'natural equivalence', and the term 'functor' in place of
Mitchell's term 'covariant functor'.
2.2 Definitions. A function f from a set A to a set B is an
ordered triple (A,B,G ) where Gf is a subset of AxB such that,
if ais an element of A, there is a unique element b of B such
that (a,b) 6 Gf G is called the graph of f. If (a,b) E Gf,
we write f(a) = b. Two functions f = (A,B,Gf) and g = (C,D,G )
compose to give gf = (A,D,G ) if and only if the domain C of
g is equal to the codomain B of f, and, if C = B, then gf is
defined by gf(a) = g(f(a)) for every a in A. We call the
function (A,A,G1 ) defined by 1A(a) = a for every a in A the
A
identity function on A.
Let A be a class such that for each A 6 A, there is a set
U(A), called the underlying set of A. An ordered triple ~ =
(A,B,f ) is said to be a concrete morphism (on A), and is
written w:A B, if A,B E A and f is a function from U(A) to
U(B); that is, f = (U(A),U(B),G ). If fj(a) = b for some a in
U(A), we write wa = b and w(a) = b. The function fL is said to be
the underlying set function of o, and we define U(w) to be f .
If < = (A,B,f) and p = (B,C,f ) are concrete morphisms on A,
we say that (A,C,f f ), denoted p(, is the natural composition
of p by c. 'e denote (A,A,lU(A)) by 1A and call 1A the natural
identity morphism on A or simply the identity on A.
A concrete catecorv A is a category whose morphisms are a
subclass M of the class of all concrete morphisms on A, whose
identity morphisms are the natural identity morphisms, and whose
composition is the natural composition. All of the categories
which we construct in what follows are concrete categories. Once
we have given a class A and a subclass M of concrete morphisms
called 'A homomorphisms', we will assume that the reader will be
able to identify the underlying sets of the objects in A and to
verify that the given subclass of concrete morphisms is closed
under natural composition and contains the natural identity
morphisms. Once this has been verified, it is immediate that A
is a concrete category since [A,B]A will always be a set; the
morphisms of A will always satisfy the requirement that (O) =
(p)C( where defined, and 1AC = W, plA = p where defined.
Instead of saying 'the concrete morphism t = (A,B,f )', we
will usually say 'the map w:A  B'. In an abuse of terminology,
we say 'the map F:A B' when A,B are classes and F is a
subclass of AEB such that for every A in A there is a unique
B in B, written F(A), such that (A,B) E F.
2.3 Definitions. If there is a map F:A * B which 'forgets'
structure, or which is the functor we construct from A to B,
we frequently say that a concrete morphism w = (A,A',f,) on
is a B homomorphism if (F(A),&(A'),f ) is a B homomorphism. For
example, if a = (A,A',f ) is a concrete morphism on AHplanes,
and if A = (S,U1), A' = (S',11) then we say 0 is an incidence
structure homomorphism if (S,S',f ) is an incidence structure
homomorphism. We abuse our terminology in other similar respects;
for example, we speak of the 'lines' of an AHplane A = (S,I\)
when we mean the lines of S.
Once we have constructed a functor G:C  D and shown that
G is an equivalence, we will call G(C) where C is.an object in C
the Dobject generated by C. If F:D  E is an obvious functor
which 'forgets' structure, we will call FG(C) the Eobject
generated by C. For example, in Section 3 we construct an
equivalence C:B  C, and if B is a biternary ring, we call C(B)
the coordinatized AHplane generated by B, and we call the
10
AHplane of C(B) the AHplane generated by B. We will frequently
say 'of' in place of 'generated by', and 'with' in place of
'which generates the'.
2.4 Definitions. Let <:A  B be a concrete morphism. If the
underlying set function of N is onto, a is said to be suriective.
If the underlying set function of c is onetoone, m is said to
be injective. If a is both injective and surjective, it is said
to be bijective.
2.5 Definitions. A functor F:C * D is said to be an equivalence
if and only if there is a functor G:D C together with natural
isomorphisms ::1  FG and %:GF 1. If an equivalence
F: " d.produces a correspondence between the objects of C and
D, then it is said to be an isomorphism if it is onetoone, onto.
2.6 Definition. If F:C * 5 and G:D  are functors, and if
q:l  FG and I:GF  1 are natural isomorphisms, we say that
F and G are reciprocal ecuivalences.
2.7 Definitions. If A is a class of objects such that a binary
relation PA is defined on the underlying set of each object A of
A, then a concrete morphism w:A  B is said to preserve p if
whenever (a,b) PA for some a,b in U(A); it is also true that
((a),(b)) C< We say that 4 reflects p if whenever
(w(a),w(b))e p. for some a,b in U(A); it is also true that
(a,b) .A
2.8 Definitions. Let (CP,o,I) be a triple of sets. Then (CP,,I)
is said to be an incidence structure if % and i have no elements
in common and I r 1 X a. The elements of 'Y are called points,
the elements of o are called lines, and if P 6 ', g E 0 we write
P I g whenever (P,g) 6 I. The set I is called the incidence
relation. If P I g, we say P is incident with g, P lies on g,
P is a point of g, or P is on g. '.e also say g goes through P,
or g is a line through P. We use other similar geometric
language to designate incidence or nonincidence. For example,
g is said to join P,Q if [,Q I g; that is, if P I g and Q I g.
The capital letters P, 0, G, K will be used to designate points
unless otherwise indicated; similarly, the small letters g, h, k
will be used to designate lines unless otherwise indicated.
Points which all lie on a common line are said to be collinear,
and lines which all go through a common point are said to be
copunctal. We say Ig A hi = n if the cardinality of the set of
points incident with both g and h is n. We let g n h denote the
point or set of points incident with both g and h. We define
Ig q h h 1 kl and g h h 1 k similarly. We write PQ to denote the
line or set of lines joining P and 0. Occasionally we write
P v Q instead of PQ.
2.9 Definition. Let S = (r,j,I) and S' = '(C',oj',I') be
incidence structures. An incidence structure homomorphism
w:S  S' is a concrete morphism which satisfies the following
conditions.
(1) fw is a function from 2 U O to 2' U o'.
(2) w(I) C qP'.
(3) (o) cj'.
(4) W preserves incidence; that is, if F I g, then wP I wg.
2.10 Definition. The class of incidence structures, the class
of incidence structure homomorphisms and the natural composition
of incidence structure homomorphisms form a category S which we
call the cateoryv of incidence structures.
Remark. We usually denote the incidence relation of any in
cidence structure by I.
2.11 Proposition. In the category of incidence structures, a
morphism % is an isomorphism if and only if is a bijective
homomorphism which reflects incidence. //
2.12 Definitions. Let ~ be an equivalence relation on the points
and lines of an incidence structure S = ( ,O,I) such that no
point is equivalent to any line. Let 'f, 9' be the sets of
equivalence classes of points, lines. Let P*, g* be the equiva
lence classes containing 2, g respectively. Let P* I* g* if and
only if there exist Q, h equivalent to P, g respectively such
that Q I h. Then S' = (f',*,I*) is called the incidence
structure induced from S by , or the induced incidence structure.
2.13 Notation. Let S be a set and let ~ be an equivalence
relation on the elements of S. The ~equivalence class contain
ing an element s will frequently be written s*; and an arbitrary
*equivalence class will frequently be denoted by a theretofore
unused letter with an asterisk; for example, t'. Once t' has
been used, however, t will denote an element of t'. Asterisks
used as superscripts do not necessarily denote equivalence
classes; it should be clear from the context what is meant in
each case.
2.14 Definition. Let H* be an incidence structure with
incidence relation I. One calls H* an (ordinary) protective plane
if the following three conditions hold.
(OP1) Whenever P" and Q' are distinct points of H*, there
is exactly one line g* such that P* I g* and Q* I g'.
(OP2) Whenever g* and h* are distinct lines of H*, there is
exactly one point P* such that P* I g* and P* I h*.
(OP3) There exist four points, no three of which are col
linear; that is, no three of the four points lie on a common
line.
2.15 Definitions. Let S* be an incidence structure with
incidence relation I. Let i be an equivalence relation defined
on the lines of S*. If g* \ h', we say g* is parallel to h'.
One calls A* = (S*,1) an (ordinary) affine'olane if the following
four conditions hold.
(OAl) Whenever P* and Q* are distinct points of S*, there is
exactly one line g' such that P* I g* and Q* I g*.
(OA2) Whenever P* is a point and g* is a line such that P*
is not incident with g*, there is exactly one line h* such that
F* I h* and h* and g" have no point in cormon.
(OA3) S* has three points which are not collinear; that is,
S* has three points not all on the same line.
(OA4) Two lines g* and h* are parallel, g' 1 h*, if and only
if.g* and h* have no point in corm.on or g* = h*.
*Remark. It is wellknown that if S* is an incidence structure
satisfying conditions (OAl), (CA2) and (OA3), then there is a
unique equivalence relation II such that (S*,l) is an affine plane
by our definition [Pickert (1955), pages 710).
The following result is well known.
2.16 Prooosition. If H is an affine plane projectivee plane),
then each line of H goes through at least two (three) distinct
points, and each point of H is incident wih at least three
distinct lines.
Proof. This is essentially shown in Pickert [(1955), pages
7, 9111. //
2.17 Definitions. Let H = ('~,I,I) be an incidence structure.
Points P and Q are said to be projectivol. neighbor and one
writes P ~ Q whenever there are distinct lines g and h such that
P,Q I g and P,Q I h. Lines g and h are said to be protectively
neighbor and one writes g h whenever there are distinct points
P and Q such that P I g,h and Q I g,h. One calls H a
projective Hjel.slev clano (abbreviated PHplane) whenever the
following three conditions are satisfied.
(PHI) If P and Q are points of H, there is at least one line g
such that P,Q I g.
(PH2) If g and h are lines of H, there is at least one point P
such that P I g,h.
(PH3) There is a surjective incidence structure homomorphism
f:H  H' from H to a projective plane H' such that the following
two conditions are satisfied.
( l) If P,Q < '; then PP = 4Q 0 P ~ Q.
(92) If g,h a oa; then 4g = ?h t> g h.
If P is a point of K and if g is a line of H, one denotes the
~equivalence classes containing P and g by P* and g' respective
ly.
2.18 Proposition. Any projective plane is a projective
Hjelmslev plane. //
2.19 Definitions.' Let H,H' be PHplanes. A projective Hjelmslev
plane homomorohism w:H  H' is an incidence structure homo
morphism which preserves the relation 'projectively neighbor'.
We denote the category of projective Hielnslev lanes by H,
and we denote the category of projective planes with PHplane
homomorphisms by H'.
2.20 Proposition. If H is a PHplane, and if q:H  H' is a map
satisfying conditions (PH3) of the definition of PHplane, then
4 is a PHplane homomorphism. //
2.21 Definitions. Let S = (,o,I) be an incidence structure,
and let 1 be an equivalence relation called the parallel
relation which is defined on the lines of S. Points P and Q are
said to be affinelv neighbor and one writes P ~ Q whenever
there are distinct lines g,h such that P,Q I g,h. Lines
g and h are said to be affinely neighbor and one writes g h
whenever to each point of each there corresponds a point of
the other which is affinely neighbor to it. One calls A = (S,t)
an affine Hjelmslcv plane (abbreviated AHplane) whenever the
following four conditions are satisfied.
(AHl) If P and Q are points of S, there exists at least one
line g such that P,Q I g.
(AH2) Let P 1 g,h. Then P is the only point on both g and h
if and only if g is not affinely neighbor to h.
(AH3) If P is a point and if g is a line, then there is exactly
one line h such that P I h and g \ h.
(AH4) There is a nap 4:A  A' from A to an affine plane A'
which is a surjective incidence structure honomorphism such that
the following three conditions are satisfied.
(1l) If P,Q e E; then qP = Q40 P ~ Q.
(f2) If g,h a 03; thenqh = 4g <> g h.
(93) If Ig .h\ = 0 in A, then 4g ,Wh in A'.
If P is a point and if g is a line of A, one denotes the
equivalence classes containing ? and g by P" and g"
respectively.
We usually denote the parallel relation of an AHplane
by U.
2.22 Proposition. Any affine plane is an AHplane.
Proof. Let A' be an affine plane. By Proposition 2.16, any line
g" of A* goes through at least two distinct points. In A* the
relation 'affinely neighbor' reduces to the relation 'is equal
to'. Using these results, one can easily show that A* is an
AHplane. //
2.23 Definitions. Let A and A' be AHplanes. An affine Hjelmslev
plane homomorphism ":A  A' is an incidence structure homomorph
ism from S to S' which preserves the relations 'parallel' and
'affinely neighbor'.
We denote the category of affine Hjelmslev planes by A, and
we denote the category of affine planes with AHplane homomorph
isms by A'.
*Remark. We have shown [Bacon (1971), page 21, Corollary 3.12]
that there exist two nonisomorphic AHplanes which have
isomorphic incidence structures.
2.24 Propositon. If A is an AHplane, and if J:A > A' is a
map satisfying condition (AH4) of the definition of AHplane,
then T is an AHplane homomorphism. //
2.25 Definitions. If A = (S,U) is an AHplane, we denote the
incidence structure induced by the equivalence relation
'affinely neighbor' by S*, and we denote the induced parallel
relation by II*: we say g* I* h* if and only if there are lines
k,m such that k ~ g; m h and k 11 m. We call A* = (S',I\*)
the aross structure of A. We call the concrete morchism
K:A  A* defined by KP = P*; Kg = g" the neighbor man of A.
If H is a PHplane, we denote the incidence structure
induced by the equivalence relation 'projectively neighbor' by
H', and we call H* the cross structure of H. We call the
concrete morphism K:H  H* defined by KP = P*; Kg = g* the
neighbor map of H.
2.26 Definitions. Once the gross structures have been defined,
we extend the relations 'projectively neighbor' and 'affinely
neighbor' in the following way. Let H be a PHplane or an
AHplane. Let P be a point and g a line of H. We say P ~ g
and g P whenever the image of P is incident with the image
of g in the gross structure of H. Hereafter, the symbol ~ is
to be read 'is neighbor to' except where otherwise specified.
Its negation is written 9 and is read 'is not neighbor to.'
The negation of the symbol U is written V, and is read
'is not parallel to.'
2.27 Proposition. If w:A * A' is an AHplane homomorphism,
then P I wg 4> 3 h such that P I h and wh = ig.
Proof. Let h be a line through P, h I g; then .h = wg. //
2.28 Proposition. If w:H  H' is a surjective PHplane homo
morphism, then wP I wg <> 3 h such that P I h and h = wg.
Proof. Assume oP I wg. Let '':H'  H" be a map satisfying
condition (PH3) There is a line wk in H' such that 4'(GP) is
not on P'(uk). Since wk 4 tg, k 4 g. Since (wk C\ tg) ()P),
(k R g) 4 P. Thus, if we let h = (k n g) v P, then wh = wg. //
2.29 Proposition. If H is an AHplane (PHplane), and if
t:H  H' is a map satisfying condition (AH4) (condition (PH3)),
then the gross structure H* of H is an affine plane projectivee
plane), and is isomorphic to H'; also, the neighbor map
K:H  H* is a surjective AHplane (PHplane) homomorphism
satisfying condition (AH4) (condition (PH3)).
Proof. Define Q:H*  H' by G9P =
welldefined bijection. If P* I g', then there are Q,h, Q ~ P,
h g, such that Q I h; hence GP* I Gg'. Since I is a surjection,
6 reflects incidence by Proposition 2.27 or 2.28: if OP' I Bg',
then PP I Sg and hence there is an h, h ~ g, such that P I h;
thus, P* I g'. Thus, 9 is an incidence structure isomorphism
and P* Q* < GP ~ 6Q; g* ~ h'* t 8g* Oh'. Hence S pre
serves and reflects the neighbor relation on points (lines).
If H is an AHplane, and if g* I* h*, then there are g' in
g' and h' in h* such that g' I h'. Then, since qg' = Gg' and
Th' = Gh*, we have that Gg* i Bh*, and hence 9 preserves the
parallel relation.
Assume that H is an AHplane and that tg tI h. Let Lk be
a line not parallel to th. Since k ph, \k \ h\ 0. Let
P E k A h. Let h' be a line such that h' U g; P I h'. Then
since h' II g, 4h' I g. Since qP I fh',Jh and Ih,4h' IL qg,
we have that qh' = (h. Thus h' h and h' I1 g; so that,
h* U* g*. Thus, 6 reflects the parallel relation.
Consequently, H* is an affine plane projectivee plane)
isomorphic to H'.
Since K = 619, K is a surjective AHplane (PHplane)
homomorphism satisfying condition (AH4) (condition (PH3)). //
2.30 Definition. Let t:A * A' be an AHplane (PHplane) homo
morphism. Let K' be the neighbor map of A'. If there are
three points R, R', R" of A whose images K'pR, K',R', K'RR" in
the gross structure of A' are not collinear, then we say that
Sis nondecenerate; otherwise we say that p is degenerate.
2.31 Proposition. If f:H * H' is a nondegenerate AHplane
(PHplane) homomorphism, then pP Irg 4: 3 Q such that Q I g
and P = rQ; andpP I g 3h such that P I h andrg = h.
Proof. Assume H is a PHplane and rP Ig. Let K' be the
neighbor map of H'. Since r is nondegenerate, there is a point
S such that A'S is not on K'Ig. Then P 4 S and IPS gl = 1,
since pP/S i g implies PS g. Let Q = PS n g; then /Q = pP.
Since is nondegenerate, there are three lines in H whose
images in (H')" are not copunctal. Thus, there is a'line pm in
H' such that (PP). is not on ( m)*. Let M = m R g. Since (pP)*
is not on (pm)*, P j t.M, and hence P L M. Let h = MP; then,
rh = r P = g and P I h.
21
Assume H is an AHplane and pP I g. Let K, K' be the re
spective neighbor maps of H,H'. By Proposition 2.27, there is a
line h such that P I h and g = ph. Let m be a line such that
K'rm R K'lg. Let m' and g' be lines such that m' R\ m, g' I g and
P I m',g'. If \m' n g\ = 0, Km' = Kg'; so that g' m'. Hence
K'pm II K'pg, a contradiction. Thus, \m' 0 g\ / 0. If m' g,
then K'pm 11 K'g, a contradiction. Hence rm' 0 g\ = 1. Let Q =
m' A g. Then, if pm' g, K'm' 11 K'pg, and hence K'pm \\ k'Ig,
a contradiction. Hence im' 4 g and Q = //
2.32 Proposition. If ,:H H' is an AHplane (PHplane)
homomorphism which induces an isomorphism between the gross
structures, then r is nondegenerate. //
2.33 Proposition. If p:H * H' is a surjective AHplane (PH
plane) homomorphism, then r is nondegenerate. Hence, if H is
an AHplane (PHplane), and if P is a point and g, a line of H,
then P ~ g if and only if there is a point Q on g such that
P Q. //
*Remark. Lineburg [(1962); pages 263, 264, 265; Satze 2.1, 2.3,
2.43 states the first two sentences of the following proposition
for AHplanes. Klingenberg [(1955), page 101, S 5] states the
first sentence of the following proposition for PHplanes.
2.34 Proposition. Let H be an AHplane (PHplane); then, there
are at least three pairwise nonneighbor lines through each
point of H, and at least two (three pairwise) nonneighbor
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 AHplane 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 AHplane. A \Iequivalence 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 AHplane. 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 AHplane 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 AHplane. 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 PHplanes 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
AHplane. 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 AHplane.
If M:A * A' is an AHplane 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 AHplanes 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 AHplane. 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 AHplane. 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 AHplane.
(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 AHplane homomorphism, < preserves
the generalized neighbor relation defined above as well as the
quasiparallel relation.
Proof. Since < is an AHplane 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 PHplane
(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 PHplane 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 910] 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 AHplane 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 PHplane, 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 PHplane, then
A(H,g) is an AHplane. This is stated by Lneburg [(1962), page
260, second paragraph], and is essentially proven by Klingenberg
[(1954), pages 390392, S 1.11 and S 3.63.
Proof. Let (H,g) be a lined PHplane 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 AHplane. Observe that A(H*,g*) is equal to the gross
structure of A(H,g).
If w:(H,g) (H',g') is a lined PHplane homomorphism,
it is easily seen that A(a) preserves the neighbor and parallel
relations and hence is an AHplane 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 AHplane and let (H,g) be a
lined PHplane. 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 AHplane. 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 AHplane which cannot be extended to a PHplane.
Drake [(1967), page 198, Theorem 3.1] states that every
finite uniform AHplane can be extended to a finite uniform
PHplane, and [in Bacon (to appear), Theorem 2.1] we state
that every projectively uniform AHplane can be extended to a
uniform PHplane, and we use the argument given by Artmann
[(1970), pages 1301343 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 PHplane. 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 295296] 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
AHplane 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 PHplane.
Klingenberg [(1954), page 390, D 61 calls S(Hg) an
'affine incidence plane with neighbor elements'. He then
shows [(1954), pages 391392, 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 PHplane as A(H,g) is.
2.55 Proposition. If A and A' are AHplanes 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 AHplane 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 AHplane homomorphism. //
2.56 Proposition. If c:H  H' is an AHplane (PHplane)
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
AHplane (PHplane) homomorphism, and o is an isomorphism. //
2.57 Proposition. If ,:A * A' is an AHplane (PHplane) 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 nondegenerate if and only if p* is nondegenerate. //
2.58 Definition. We say a lined incidence structure which is
j omorphic to the lined generalized incidence structure G (A)
g
of some AHplane A is a generalized AHplane. If
w;(Ag) ~ (B,h) is a lined incidence structure homomorphism,
if (A,g) and (B,h) are generalized AHplanes, and if w preserves
the (welldefined) induced neighbor relations, then we say that
wis a generalized AHplane homomorphism. We denote the category
of generalized AHolanes 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 AHplane. 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 AHplane 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 AHplanes
and generalized AHplanes except to aviod confusion. We will
say '(A,g) is isomorphic to the AHplane B' when we mean that
(A,g) is isomorphic to the lined generalized incidence structure
of B, and so on.
2.62 Definition. An AHplane homomorphism' is said to be an
AHplane 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 nondegenerate affine plane
homomorphism. Then, r is an AHplane 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 nondegenerate 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*' tR*'*; 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 AHplane
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 nondegenerate
AHplane homomorphisms, then i is a nondegenerate AHplane
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 nondegenerate AHplane
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
PHplane 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 nondegenerate AHplane
homomorphism such that pg 01 ph = g U h for all lines g,h; then
r is an AHplane 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 AHplane
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 nondegen
erate. Then p(g n k) = pg ) rk = p(h 0 k) and thus g = h. Hence
tis injective. Thus, p is an AHplane 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 nondegenerate AHplane
(PHplane) 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 nondegenerate AHplane
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 nondegenerate PHplane 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 nonneighbor, 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 AHplanes (PHplanes) the action of a nondegener
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 solvablefor 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 = Sl(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 AHnlane
and K = (g ,g ,E,,S:OE  :) is a coordinatization of A
x y
whenever A is an AHplane, gx,gy are nonneighbor 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 AHplane. 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 ycoordinate of P and
that a is the xcoordinate 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 xaxis of C; g is called the vaxis 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 xyduals. The functions 8, 6' given
above are xyduals, as are A and X'.
3.13 Proposition. Let g,h be lines of a coordinatized AHplane
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 AHplanes. A
coordinatized AHplane honomorDhism or coordinatization homo
morchism u:C  C' is a map a which is an AHplane homomorphism
such that w(gx) gx ', (g ) g and w(E) = E' where g gy
and E' are the xaxis, yaxis and unit point of C' respectively.
If C = (A,K) is a coordinatized AHplane, 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 xaxis 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 AHplanes 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 AHplane.
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
AHplane, 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 AHplane. 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
xyduality, [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 nonneighbor
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
xyduality, [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 xyduality N = n e M 3 k e6 k 4 0, T'(k,n,0) = O].
(Proof) Assume n C N\103. Then n o 0 and (l,n) ~ (1,0);
so that [n,01 [0,01. But [n,0o] [0,0o; so that there is a
point (k,0) on both such that (k,0) Z (0,0). Hence,k 4 0 and
T(k,n,0) = 0. If n = 0, then (1,0) I (0,01; hence T(1,0,0) = 0.
L1
Assume there is an element k, k > 0, such that T(k,n,0) =
0. Then both (k,0) and (0,0) are on [n,0] and o0,0]; so that
[n,0] ~ [0,0]. Thus (l,n) ~ (1,0) and n o 0, n E N. /
(Lemma 5) a o b if and only if every x which satisfies the
equation a = T(x,l,b) is an element of N.
(Proof) Since O  E, 0 i 1. Hence, by Proposition 3.13,
[0,a] is not cuasiparallel to [l,b]. Thus, the equation a =
T(x,0,a) = T(x,l,b) has a unique solution x.
Assume a o b. Then (0,a) ~ (0,b); so that [o,al [0,bl.
Thus (10,a \ [l,b1) ~ (0,b (\ [l,b]). Thus, if (x,a) =
[0,a] f [l,bl then (x,a) (0,b) and x o 0. Hence a = T(x,l,b)
and x E N.
Assume every x which satisfies the equation a = T(x,l,b)
is in N. .Let (x,a) = [l,b] 0 [0,al'; then a = T(x,l,b); x o 0.
Since l[l,b3 g \ = 1, (0,b) = [l,b] g g Observe that
t0,x1' 0,01' where g = [0,0]' and hence
([0,xl' [l,bl) ~ ([0,03 [l,b]); (x,a) (0,b), and a o b. /
(BO) The xydual to the proof of (Lemma 5) above shows
that a o b if and only if every y which satisfies the equation
a = T'(y,l,b) is an element of N. Thus in B = B(C) the
relations ~ and ~' of the definition of biternary ring are equal
to the relation o defined here. By (Lemma 4) N ='NB = 'B so
that condition (BO) of the definition of biternary ring holds in
B(C).
(Bl) The relation o on M is obviously an equivalence
relation.
(B2) Since (0,d) I [m,d], T(0,m,d) = d. Since [0,d] lgx,
(a,d) I [O,dl. Hence T(a,0,d) = d.
56
(B3) Since (l,a) I Ca,0o T(l,a,0) = a. Since [1,0] = OE,
we have that (a,a) I [1,0] and T(a,l,0) = a.
(B4) There is a unique line parallel to [E,0o through the
point (a,b): say [E,rl. Then T(a,m,z) = b. If T(a,m,z') = b,
then (a,b) I [m,7'] ; hence z = z'. Thus, z is the unique solution
to the equation T(a,m,z) = b.
(B5) The equation T(x,m,d) = T(x,m',d') is uniquely
solvable for x if and only if there is a unique point
(x,y) I [m,d],[m',d']; hence if and only if [m,d] is not quasi
parallel to [m',d'] and, by Proposition 3.13, if and only if
m j m '.
(B6) Look at the system [T(a,m,d) = b, T(a',m,d) = b'1.
If a a', then the unique line g joining (a,b) and (a',b') is
not quasiparallel to g Hence g = (m,d2 for some n,d and the
pair m,d is the unique solution to the system. If a a a', b o b',
then Cm,d] gx and hence by (Lemma 3) m o 0. If a o a', b b',
then, if (a,b),(a',b') I g, g I g and hence the system has no
solution Vm,d].
(B7) Let g,g' be any two distinct lines joining (a,b) and
(a',b'), (a,b) / (a',b'). Then g p g'. Observe that (g U gy
< g g y) and that g,g' II gy p g = [u,v3', g' = [u',v' for
some u,u' 4 N, v,v' e M. The lines g,g' are not quasiparallel
to g if and only if g = [m,d], g' = [m',d'] for some m,m',d,d'
in M. Hence condition (B7) holds in B(C).
(B8) Any line of the form [u,v]', u 6 N, is quasiparallel
to gy and hence meets any line [m,d] (not quasiparallel to g )
in exactly one point (x,y).
(B9) T(u,m,0) = 1 if and only if (u',l) I [m,0]. Hence
T(u,m,0) = 1 if and only if Cu,0]' = [m,0], and by xyduality,
Cu,0]' = [m,03 if and only if T'(m,u,0) = 1. If T(u,m,0) = 1,
and (a,b) I (m,e]J,u,v]', then [m,e] = [u,v]', and hence
(x,y) I [m,e] '=> (x,y) I [u,v]'.
(B10) The construction of the ternary operation T maps,
under the neighbor map K, to the construction of T* which is the
usual construction of the ternary field of a coordinatized affine
planesee (Hall (1959), pages 353355, Section 20.31and
(* (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 xyduals 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(ll(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) ( Ix {( ),< )>')/=. 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 ring 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 AHplane. 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 AHplane, 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.
Assue 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 nequivalence 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 AHplane. 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 AHplane.
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 xaxis, the yaxis and the unit
point of C(B'), C(w) is a coordinatized AHplane 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 AHplane. 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 xaxis of C, the yaxis 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 nonn2ichbor 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,

