- Email: [email protected]
Group Modifications of Some Partial Groupoids
319
Annals of Discrete Mathematics 18 (1983) 319-332 0 North-Holland Publishing Company
GROUP MODIFICATIONS OF SOME PARTIAL GROUPOIDS
A. Dr6pal and T. Kepka"
1. INTRODUCTION Let G
( 0 )
dist(G(O),G(n))
and G (") be twogroupoids w i t h t h e same u n d e r l y i n g s e t . =
c a r d {(a,b)
E
6';
a
0
b # a
A
b}.
For an i n t e g e r n
We p u t
> 2, l e t
g d i s t ( n ) = min d i s t ( Q , Q ( o ) ) where Q runs through a l l groups o f o r d e r n and through a l l quasigroups w i t h t h e same u n d e r l y i n g s e t such t h a t Q # Q ( 0 ) . d i v i d e s n then c l e a r l y g d i s t ( n )
Q
Q
m
I t seems t o be an open problem whether
2.
I n t h e present paper, a technique which m i g h t
be u s e f u l i n s o l v i n g t h i s problem i s developed. (i.e.
If 2
gdist(m), and t h e r e f o r e g d i s t ( n ) B g d i s t ( p ) , p
being t h e l e a s t prime number d i v i d i n g n. g d i s t ( n ) < g d i s t ( p ) f o r some n
Q(0)
I n p a r t i c u l a r , the modifications
r e l f e x i o n s ) o f c e r t a i n p a r t i a l g r o u p o i d s i n t o t h e category o f groups a r e i n -
vestigated.
L e t us n o t e here t h a t t h e numbers g d i s t ( n ) p l a y i m p o r t a n t r B l e i n t h e
enumeration questions concerning t h e number o f a s s o c i a t i v e t r i p l e s o f elements i n f i n i t e quasigroups (see [ 11). 2. PARTIAL GROUPOIDS By a p a r t i a l groupoid we mean a non-empty s e t t o g e t h e r w i t h a p a r t i a l b i n a r y o p e r a t i o n ( p o s s i b l y empty).
T h i s o p e r a t i o n w i l l be denoted m u l t i p l i c a t i v e l y i n
t h i s section. L e t K be a p a r t i a l groupoid. M(K,a) = {(b,c)
E
For every a E K we p u t
KL; a = bc}, so t h a t t h e o p e r a t i o n o f K i s d e f i n e d j u s t f o r o r -
dered p a i r s from t h e s e t M = M(K) =
U
M(K,a),
a
E
K.
Further, l e t
B(K) = { b E K; (b,c) E M I , C(K) = { c E K, (b,c) E M } , A(K) = B(K) D(K) = {bc; (b,c) E M I .
F i n a l l y , l e t m(K)
L e t K and L be p a r t i a l groupoids.
U
C ( K ) and
c a r d M.
By a homomorphism o f K i n t o L we mean any
mapping f o f K i n t o L s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n : ( 1 ) I f (a,b) E M(K) then ( f ( a ) , f ( b ) )
E M(L) and f ( a b ) = f ( a ) f ( b ) .
phism f i s s a i d t o be s t r o n g i f t h e f o l l o w i n g i s t r u e :
The homomor-
320
A . Drcipal and T. K e p h
( 2 ) I f (x,y) E M(L) and x,y
E
f ( K ) then t h e r e e x i s t s a p a i r (a,b) E M(K) such t h a t
f ( a ) = x and f ( b ) = y. P a r t i a l groupoids w i t h homomorphisms form a category which w i l l be denoted by
P. L e t f be a homomorphism o f a p a r t i a l groupoid K i n t o a p a r t i a l groupoid L. Put f(M(K)) = ( ( f ( a ) , f ( b ) ) ;
(a,b) E M(K)}
5M(L).
groupoid I ( 0 ) = F[ K ] a s f o l l o w s : I = f ( K ) and u i f f t h e r e a r e a,b
E
K such t h a t (a,b) E M(K),
(then (u,v) E M(L) and z = u v ) .
0
Further, d e f i n e a p a r t i a l v = z i s d e f i n e d f o r u,v,z E I
f ( a ) = u, f ( b ) = v and f ( a b ) = z
I t i s easy t o see t h a t f : K
homomorphism and t h e i d e n t i t y mapping i d : I(.)
-+
.+
I ( . ) i s a strong
L i s a homomorphism.
By a pseudoimmersion (resp. immersion) o f a p a r t i a l g r o u p o i d K i n t o a p a r t i a l
groupoidL we mean any homomorphism f o f K i n t o L such t h a t the r e s t r i c t i o n s f j
E(K) and
f
C(K) (resp. f
1 B(K),
L e t K be a p a r t i a l g r o u p o i d .
f
1 C(K)
and f
I D(K))
are i n j e c t i v e mappings.
By a congruence o f K we mean any equivalence
r e l a t i o n r d e f i n e d on K such t h a t :
( 3 ) I f (a,b),(c,d)
E
r and (a,c),(b,d)
E
M(K) then (ac,bd) E r.
I n t h i s case, we
can d e f i n e a s t r u c t u r e o f a p a r t i a l g r o u p o i d on the f a c t o r s e t K / r by xy = z f o r x,y,z
E K / r i f f t h e r e a r e a E x and b E y such t h a t (a,b) E M(K) and ab E z.
The
n a t u r a l p r o j e c t i o n o f K onto K / r i s then a s t r o n g homomorphism. An element e o f a p a r t i a l groupoid K i s s a i d t o be a p a r t i a l n e u t r a l element
of K i f the following three conditions are satisfied: ( 4 ) If (e,a) E M(K) then ea = a. ( 5 ) I f (a,e) E M ( K ) then ae = a. ( 6 ) (e,e)
E
WK).
A p a r t i a l g r o u p o i d K i s s a i d t o be reduced i f K = A(K)
u D(K).
R the f u l l subcategory o f P formed by reduced p a r t i a l groupoids.
We denote by
Obviously, every
p a r t i a l g r o u p o i d K can be expressed u n i q u e l y as t h e d i s j o i n t union K = L M(K)
5 LL
and
L
U
J where
i s reduced.
A p a r t i a l g r o u p o i d K i s s a i d t o be balanced i f t h e sets B ( K ) , C(K) and D(K) are p a i r - w i s e d i s j o i n t . A p a r t i a l g r o u p o i d K i s s a i d t o be c a n c e l l a t i v e i f t h e f o l l o w i n g two condi-
tions are satisfied: ( 7 ) I f (a,b),(a,c)
E
M(K) and ab = ac then b = c.
( 8 ) I f (b,a),(c,a)
E
M(K) and ba = ca then b = c.
A group i s a semigroup w i t h (unique) d i v i s i o n .
Thus a group i s a (reduced
32 1
Group modifcations of some partial groupoids
c a n c e l l a t i v e p a r t i a l ) g r o u p o i d and we denote by g t h e f u l l subcategory o f groups i n F u r t h e r , we denote by S ( T ) t h e f u l l subcategory o f R c o n s i s t i n g
t h e c a t e g o r y R.
o f reduced balanced (cancel l a t i v e ) p a r t i a l groupoids.
LEMMA 2.1:
of
Let K
E
R.
Then there e d s t L E S and a surjective strong immersion k
L onto K. Moreouer,L is canceZZative, provided K is
PROOF: L e t B,C,D h: D
+
D(K) b i j e c t i v e mappings.
f o r a l l b E 6, c
k = f
be p a i r - w i s e d i s j o i n t s e t s and f : B
U
g
U
E
+
SO.
B(K), g: C
C(K),
P u t L = B u C u D and d e f i n e bc = h - ' ( f ( b ) g ( c ) )
C such t h a t ( f ( b ) , g ( c ) ) E M(K).
Now, i t s u f f i c e s t o p u t
h.
L e t K be a p a r t i a l g r o u p o i d a n d x = (a,b) E M = M(K). t h e l e a s t congruence o f K such t h a t (a,b)
E
We denote by s = sx
s and a/s i s a p a r t i a l n e u t r a l element
o f K/s ( i t i s easy t o check t h a t t h e r e e x i s t s such a congruence). a r e l a t i o n s ' = s; ( 9 ) (c,c)
+
Further, define
on K as f o l l o w s :
E s ' f o r e v e r y c E K.
(10) I f (a,c) E M t h e n (ac,c),(c,ac)
E s'.
( 1 1 ) If (d,b) E M t h e n (db,d),(d,db)
E s'.
( 1 2 ) I f (a,c),(d,b)
E M and ac = db t h e n (c,d),(d,c)
LEMMA 2.2: Let K E T and x
E
Then sx =
M(K).
S' X
E
s'.
and the naturaZ homomorphism of
K onto Kfs is an immersion. PROOF: One may v e r i f y e a s i l y t h a t s;
i s a congruence o f K and t h e r e s t i s c l e a r .
LEMMA 2.3: Let h be a pseudoimmersion of a p a r t i a Z g r o u p o i d K into a canceZZatioe
partial groupoid L.
Then K is canceZlative.
PROOF: Obvious.
LEMMA 2.4: Let I,J,K be a p a r t i a l g r o u p o i & and 9: J
such that J E S , g(M(L))
5 f(M(1))
K, f: I + K homomovhisms
and f is an immersion.
homomorphism h of J into I such that f h = 9 . PROOF: Easy.
-t
Then there exists a
A , Drapal and T. Kepka
322
An element a o f a p a r t i a l g r o u p o i d K i s s a i d t o be idempotent i f (a,a) E M(K) and aa = a.
3. GROUP MODIFICATIONS OF REDUCED PARTIAL GROUPOIDS I n t h i s s e c t i o n , the operations o f general p a r t i a l g r o u p o i d s a r e denoted by
0
and the operations o f groups m u l t i p l i c a t i v e l y . Let K
E
R and l e t F(K) designate the f r e e group o f words over the s e t A(K). -ld-l
and L e t N(K) be the normal subgroup o f F(K) generated by a l l t h e words bce -1 where (b,c),(d,e) E M(K,a), a 6 D(K), and (u,v) E M(K,z), z E A(K) n D(K). uvz Put G(K) = F(K)/N(K) and d e f i n e a mapping g = gK o f K i n t o G(K) by g ( a ) = aN(K) and g(b
0
c ) = bcN(K) f o r a l l a E A(K) and (b,c) E M(K).
PROPOSITION 3.1: Lot K E R.
Then g = gK i s a we12 defined homomorphism of K i n t o
G{K) and
r pm&i g ( e ) = 1 f ~ every
Moreovoq
neutra2 element e E K.
PROOF: Using t h e d e f i n i t i o n o f N = N(K) and the f a c t t h a t K i s reduced, i t i s easy t o show t h a t g i s a homomorphism.
H.
a
Now, l e t h be a homomorphism o f K i n t o a group
There i s a homomorphism p o f F(K) i n t o H such t h a t p ( a ) = h ( a ) f o r every E
A = A(K).
Clearly, N
5 Ker
p, and hence we have a homomorphism f o f G = G(K)
i n t o H such t h a t f(aN) = h ( a ) f o r each a E A.
Then h = f g and the r e s t f o l l o w s
from the f a c t t h a t G i s generated by g(A). L e t K,L E R and l e t h be a homomorphism o f K i n t o L.
According t o 3.1,
t h e r e e x i s t s a unique homomorphism G(h) o f G(K) i n t o G(L) such t h a t G(h)gK = gLh. This homomorphism i s d e f i n e d by G(h)(aN(K)) = h(a)N(L) f o r every a E A(K). F u r t h e r , g K ( a ) g K ( b ) - l E Ker G(h) whenever a,b E K and h ( a ) = h ( b ) .
F i n a l l y , G(h)
i s s u r j e c t i v e , provided h i s so.
K,L E R an2 l e t h be u s u r j e c t i v e strong homomoqkism o f K onto L. -1 IIce,; Ker G(H) is j u s : tne tiormu2 s 5 g r m p of G(K) generated by a21 g ( a ) g ( b ) , K K a,b E K, h ( a ) = h ( b ) . LEMMA 3.2: L,t
PROOF: Denote by P t h a t normal subgroup.
c Ker G(h). Then P -
Put H = G(K)/P,
denote by p t h e n a t u r a l homomorphism o f G(K) onto H and d e f i n e a mapping f o f L i n t o H by f h ( a ) = pgK(a) f o r every a
K.
Then f i s a homomorphism, and hence
3 23
Group modifcations of some partial groupoids f = kg
f o r a homomorphism k o f G(L) i n t o H. L p = kG(h) and Ker G(h) 5 Ker p = P.
NOW, pgK = f h = kgLh = kG(h)gK,
L e t K E R , g = gK and x = (a,b) E M(K). G = G(K) generated by a l l g ( a ) - ’ g ( c ) ,
g(d)g(b)-’
Denote by H(K,x) t h e subgroup o f where (c,d) E M(K).
P(K,x) be t h e normal subgroup o f G generated by g ( a ) and g ( b ) .
Further, l e t
Denote by p x t h e
n a t u r a l homomorphism o f G o n t o G/P(K,x).
LEMMA 3.3:
Let K
E
R and x,y
px(H(K)) = G(K)/P(K,x).
E
M(K).
Then H(K,x) = H(K,y) = H(K) and
Moreover, i f K has a t l e a s t one idempotent element then
H(K) = G(K).
PROOF: Easy.
LEMMA 3.4: Let K,L E Rand l e t h be a homomorphism of K i n t o 1. G(h)(H(K))
5 H(L).
Then
Moreover, G ( h ) ( H ( K ) ) = H(L), provided h i s s u r j e c t i v e and
strong. PROOF: Easy.
Let K E R.
A normal subgroup R o f G(K) i s s a i d t o be pseudoregular ( r e s p .
r e g u l a r ) if t h e homomorphism f g K o f K i n t o G(K)/R, f b e i n g t h e n a t u r a l homomorphism o f G(K) o n t o G(K)/R,
i s a pseudoimnersion ( r e s p . an immersion).
The
g r o u p o i d K i s s a i d t o be pseudoregular ( r e s p . r e g u l a r ) i f t h e u n i t subgroup o f G(K) i s pseudoregular ( r e s p . r e g u l a r ) . LEMMA 3.5: Let K E R be pseudoregular.
Then K is c a n c e l l a t i v e .
PROOF: Every group i s c a n c e l l a t i v e and t h e r e s u l t f o l l o w s f r o m 2.3.
LEMMA 3.6: Let K E R.
Then K i s fpseudolregular i f f t h e r e e x i s t a group G and a
(pseudo)immersion of K i n t o G. PROOF: Obvious
LEMMA 3.7: Let K,L E R and l e t h be an (pseudolimmersion of K i n t o L.
If L is
324
A. Drdpaland T. Kepkn
:psel*dolregular then K i s so.
PROOF: Use 3.6.
4. GROUP MODIFICATIONS OF REDUCED BALANCED PARTIAL GROUPOIDS LEMMA 4.1 : Let K H( K) = 1
LP:G
6
S 3, such t h t curd K =3.
Then G( K) is a f r e e group of rank 2
.
PROOF: Obvious
Let K
LEMMA 4.2: k t K gK(K)
Denote by V(K) t h e normal subgroup o f G ( K ) g e n e r a t e d by H(K).
E S.
V(K) =
E S.
Then G(K)/V(K) is a f r e e group o f rank 2 , H(K)
5 V(K)
and
8.
PROOF: As one may see e a s i l y , t h e r e e x i s t L E S and a s u r j e c t i v e homomorphism h of The r e s t i s c l e a r : f r o m 3.2 and
K o n t o L such t h a t c a r d L = 3 and h i s s t r o n g .
4.1.
LEMMA 4.3: G(K)/P(K,x)
Then there e x i s t s an isomorphism f
Let K E S and x E M(K). onta H(K) snch t h a t f p x x
I H(K)
X
of
= id.
I H, G = G(K,x) and P = P(K,x). There i s a K’ p = pX s u r j e c t i v e homomorphism q o f F(K) o n t o H = H(K) such t h a t q ( c ) = g ( a ) - ’ g ( c ) and
PROOF: L e t x = (a,b),
g = g
I t i s easy t o check t h a t
q ( d ) = g ( d ) g ( b ) - l f o r each ( c , d ) E M = M(K).
N
= N(K) c Ker q and a,b E Ker q.
Hence P
homomorphism f = f x o f G/P o n t o H. and f p ( g ( d ) g ( b ) - l ) = g ( d ) g ( b ) - ’ ,
PROPOSITION 4.4: i ‘;
:f
L e t K E S,
q/H and q induces a s u r j e c t i v e ‘
F o r (c,d) E M, f p ( g ( a ) - ’ g ( c ) )
so t h a t f p = i d H .
x E M(K) a d U(K,x)
H(K) is i s o ~ o r p h i cto G(K)/P(K,x),
G(K) crnd G(K) is generated bg H(K) ( i G * lP(K,x)/U(K,x)
5 Ker
U
P(K,x).
is u f r e e group of r a n k
Similarly, p f = i d GfP‘
= P(K,x)
H(K)
2.
= g(a)-’g(c)
17 V(K).
P(K,x)
Then:
= 1, H(K) i s u r e t r a c t
Group modifcations of some partial groupoids
325
(iii)G( K)/U ( K,x) i s isomorphic t o the d i r e c t product H( K) x F where F i s
Q
free group of rank 2. PROOF: Apply 4.2 and 4.3.
L e t K E S, x E M(K) and t = tx = f x p x . phism o f G = G(K) o n t o H = H(K) and t and R = t - l ( S ) .
IH
= id.
By 4.3,
t i s a s u r j e c t i v e homomor-
L e t S be a normal subgroup o f
H
We s h a l l say t h a t S i s an x-(pseudo)regular subgroup o f H i f R i s
(pseudo)regular i n G.
Thus S. i s x-(pseudo)regular i f f q t g K i s a (pseudo)immer-
sion, q being t h e n a t u r a l homomorphism o f H onto H/S.
LEMMA 4.5: Let K E S and l e t S be a normal subgroup of H(K). x,y E M(K),S
Then, for a l l
i s x-(pseudolregular i f f it i s y-(pseudolregular.
PROOF: Suppose t h a t S i s x-pseudoregular.
L e t x = (a,b),
y = (c,d)
and l e t
u,v E B(K) be such t h a t q t g(u) = q t g ( v ) where g = gK and q i s t h e n a t u r a l Y Y E S, and hence homomorphism o f H = H(K) onto H/S. Then g ( c ) - ’ g ( u ) g ( v ) - ’ g ( c ) g ( a )-’g ( u ) g ( v )-’g ( a ) = g ( a I-’g ( c )g ( c I-’g ( u 1g ( v I-’g ( c 1g ( c I-’g ( a I E g(a)-’g(c)
E
H and S i s normal i n H.
S being x-pseudoregular.
s
s irice
Consequently, q t g(u) = q t x g ( v ) and u = v, X
S i m i l a r l y f o r u,v E C(K) and, provided S i s x - r e g u l a r ,
f o r u,v E D(K). L e t K E S and l e t S be a normal subgroup o f H(K).
We s h a l l say t h a t S i s
(pseudo)regular i f i t i s x-(pseudo)regular f o r some x E M ( K ) .
LEMMA 4.6: Let K
E
S be regular.
Then g i s i n j e c t i v e . K
PROOF: Easy.
5. GROUP HODIFICATIONS OF REDUCED BALANCED CANCELLATIVE PARTIAL GROUPOIDS L e t K E T and x E M(K).
Consider t h e congruence s = sx, p u t Kx = K/s and
denote by qx t h e n a t u r a l homomorphism o f K onto Kx.
Then Kx E R, Kx has a p a r t i a l
n e u t r a l element and qx i s a s t r o n g imnersion. LEMMA 5.1: Let K E T and x E M(K).
Then Ker G(qx) = P(K,X).
326
A . Drhpal and T. Kepka
PROOF: The a s s e r t i o n f o l l o w s e a s i l y f r o m 2 . 2 and 3 . 2 .
LEMMA 5.2:
H(K)
Ls.5 K E T
sh.2;~ thct
h x ( S ) c.:
an2 x
H(K)i;
Then there i s an isomorphism h of G(Kx) onto X
G(K ) i s (psedo)regular i f f the subgroup X
30.
is: K
t h e r e i s an isomorphism kx o f G(Kx) o n t o G(K)/P(K,x)
such
Now, i t s u f f i c e s t o p u t hx = f x k x .
= kxG(qx).
LEMMA 5.3:
M(K).
a tiormL aubgroup S of
PROOF: According t o 5.1, t h a t p,
E
T z d x,y
6
6
Then there e x i s t s an isomorphism h o f G( K )
M(K).
X
G ( K ) suc;. :;an a norm2 subgroup S of G(K ) i s (pseucb)reguZaz~i f f tize subY X ~ l l L r i i ) h(S) sf G ( K ) :‘s SO. Y -;::G
PROOF: T h i s i s an immediate consequence o f 5 . 2 .
PROOF: ( i ) i m p l i e s ( i i ) .
F o r , l e t x = (a,b)
(pseudo)regular.
P x g ( c ) = P x g ( d ) , 9 = gK. g(c)g(d)-’
We must show t h a t t h e subgroup P = P(K,x)
o f G(K) i s
and c,d E B(K) be such t h a t
Then f x p x ( g ( c ) g ( d ) - ’ )
= 19 g ( a ) - l g ( c ) g ( d ) - ’ g ( a )
= 1 , g ( c ) = g ( d ) and c = d, s i n c e K i s ( p s e u d o ) r e g u l a r .
= 1%
S i m i l a r l y the
rest. ( i i ) implies ( i i i ) . ( i i i ) implies ( i ) .
COROLLARY 5.5: r
ict K
T h i s i s an immediate consequence o f 5 . 3 . Use R . 7 and t h e f a c t t h a t q,
E
T and x E M(K).
:zLLc:
_6. COUPLES
.
i s an immersion.
?hen K i s lpseudolreguZar i f f t g i s a x K
OF COMPANIONS
I n t h i s s e c t i o n , we s h a l l use v a r i o u s symbols f o r p a r t i a l b i n a r y o p e r a t i o n s . Using a n o t h e r n o t a t i o n t h a n t h e m u l t i p l i c a t i v e one, we p u t t h e o p e r a t i o n symbol
327
Group modifcations of some partial groupoids i n t o t h e b r a c k e t s b e h i n d t h e symbol f o r t h e u n d e r l y i n g s e t . Let
K(Q) and K(a) be p a r t i a l g r o u p o i d s w i t h t h e same u n d e r l y i n g s e t .
s h a l l say t h a t t h e p a i r (K("),
K(n))
We
i s a c o u p l e o f companions i f b o t h t h e p a r t i a l
i g - o u p o i d s a r e reduced and c a n c e l l a t i v e , M ( K ( 0 ) ) = M(K(9:)) = M and f o r a l l (a,b) E M t h e r e e x i s t c,d,e,f (a,c),(d,b),(a,e),(f,b)
E
M such t h a t c
E
M, a
0
b = a
f:
c = d
# b # e, d # a # f, f:
b and a fi b = a
LEMMA 6.1 : Let ( K ( " ) ,K(fc)) be a couple of companions.
e = f
b.
0
Then there e x i s t a couple
and a s u r j e c t i v e mapping k of L onto K such t h a t both
o f companions ( L ( o ) , L ( a ) ) L(0)
0
and L ( n ) are balanced, k i s a strong i m e r s i o n o f L ( 0 ) onto K(0) and k i s a
strong immersion of L ( n ) onto K(f:). PROOF: Easy ( u s e t h e same t e c h n i q u e as i n t h e p r o o f o f 2 . 1 ) .
A partialgroupoid
K
i s s a i d t o be exchangeable i f i t i s reduced and
c a n c e l l a t i v e and i f i t possesses a t l e a s t one companion.
LEMMA 6.2: Let K be a p a r t i a l grupoid, L ( 0 ) an exchangeable p a r t i a l groupoid and f
a pseudoimersion of L ( o ) i n t o K.
Suppose t h a t e i t h e r f is an i m e r s i o n or
Then the partialgroupoid I(o) = f[ L(.)]
canceZZative.
PROOF: L e t L ( k ) be a companion o f L ( 0 ) .
I = f ( L ) by u
3:'
f ( b ) = v and f ( a
v = 9:
z
i s exchangeable.
Define a p a r t i a l binary operation
i f f t h e r e a r e a,b E L such t h a t (a,b) E M(L(f:)),
b) = z.
K is
sc
on
f(a) = u,
I t i s easy t o v e r i f y t h a t t h e p a i r (I(o),I(s'c))
is a
coup1 e o f companions.
A p a r t i a l g r o u p o i d K i s s a i d t o be ( s t r o n g l y ) p r i m a r y i f i t i s exchangeable and e v e r y p s e u d o i m e r s i o n o f an exchangeable p a r t i a l g r o u p o i d i n t o K i s a s u r j e c t i v e strong pseudoimersion (immersion).
LEMMA 6.3: Let f be a pseudoimmersion of an exchangeable p a r t i a l groupoid L i n t o a
p r i m a r y p a r t i a l groupoid K.
Then:
( i l f(M(L))
= A(K),
= M(K), f ( A ( L ) )
f ( B ( L ) ) = B(K), f ( C ( L ) ) = C(K) and
f ( D ( L ) ) = D(K).
( i f )The partiaZ groupoid L i s primary.
Moreover, L i s strongly primary Iresp.
balanced, provided K i s so.
(iii)I f K i s balanced and f is an immersion then f is an isomoprhism.
328
A . Drhpal and T. Kepko
PROOF: Easy. LEMMA 6.4: L,et K be a ( s t r o n g l y ) p r i m a r y partialgroupoid.
Then there e x i s t a
balance*! ( s t r o n g l y ) p G m r u partialgroupoid L cozd a s u r j e c t i v e strong i m e r s i o n k o f L onto K.
PROOF: Use 6.1 and 6 . 3 ( i i ) .
LEMMA 6.5: Let K be a f i n i t e exchangeable parsialgroupoid.
Then there e x i s t a
L i n t o K.
;,rlmaq. parcinlgroupoid L and mi i n j e c t i v e homomorphism f o f
PROOF: W e s h a l l proceed by induction on m ( K ) .
Suppose t h a t K i s not primary.
Then there a r e an exchangeable p a r t i a l groupoid J and a pseudoimmersion g o f J i n t o K such t h a t g(M(J)) # M(K).
of I i n t o K .
P u t I(.)
= g [ J ] and denote by i
By 6.2, I(o) i s exchangeable.
the i d e n t i t y mapping
Moreover, M ( I ( 0 ) ) = g(M(J))
and i i s an i n j e c t i v e homomorphism of I ( 0 ) i n t o K.
We have m ( I ( 0 ) )
$ M(K)
< m(K), and
hence there a r e a primary partialgroupoid L and an i n j e c t i v e homomorphism h of L into I(0).
Now, i t s u f f i c e s t o put f = i h .
COROLLARY 6.6: Let K be a f i n i t e exchangeable partialgroupoid.
Then there e x i s t a
Lulance:i primary parsialgroupoid L and a immersion f o f L i n t o K.
PROOF: Apply 6.4 and 6.5. PROPOSITION 6.7: k t K be a f i n i t e exchangeable partialgroupoid. :brzlawed s t m n g Z y
Then there e x k t
primary purtialgroupoid L and a pseudoimmersion f o f L i n t o K.
PROOF: P u t m = card B ( K ) and n = card C(K).
I f I i s a reduced partialgroupoid and
g i s a pseudoimmersion of I i n t o K then I i s f i n i t e and card1 < m+n+mn.
Hence
there a r e a balanced primary partialgroupoid L and a pseudoimmersion f o f L i n t o K such t h a t card D(L) 2 card D(J) whenever J i s
a balanced primary p a r t i a l groupoid
with a pseudoimnersion h of J i n t o K (use 6 . 7 ) .
Obviously, L i s strongly primary.
LEMMA 6 . 8 : Let K,L E R and l e t f be a s u r j e c t i v e strong i m e r s i o n of L onto K. K i s exchangeable iresp. p r i m u r g , strongly p r i m a r y ) i f f L i s so.
.r?2;n -17
329
Group modifcations of some partial groupoids PROOF: I f L i s exchangeable t h e n K i s exchangeable by 6.2.
Conversely, i f K i s
exchangeable ( r e s p . p r i m a r y , s t r o n g l y p r i m a r y ) t h e n i t i s easy t o check t h a t L i s so ( s e e t h e p r o o f o f 6.1 and 6 . 3 ( i i ) ) .
NOW, suppose t h a t L i s ( s t r o n g l y ) p r i m a r y
and g i s a pseudoimmersion o f an exchangeable p a r t i a l g r o u p o i d I i n t o K .
Then K i s
exchangeable and t h e r e e x i s t a balanced exchangeable p a r t i a l g r o u p o i d J and a s u r j e c t i v e s t r o n g immersion h o f J o n t o I. By 2.4, gh = f k f o r a homomorphism k o f J i n t o L.
I t i s easy t o see t h a t k i s a pseudoimmersion, and hence k i s
s u r j e c t i v e and s t r o n g .
The r e s t i s c l e a r .
7. GROUP DISTANCES OF FINITE QUASIGROUPS L e t G be a g r o u p o i d a n d M a non-empty s u b s e t o f G C = {c; (b,c) E M } and D = {ab; (a,b) E M I .
K(0) = K[G,M ] a s f o l l o w s : K = B u C (a,b)
E
M; i n t h a t case, a
U
b = ab.
o
i t i s c a n c e l l a t i v e , p r o v i d e d G i s so.
PROPOSITION 7.1:
Let Q ( 0 ) and Q ( " ) be
and M = { ( a,b)
Q2; a
K[ Q ( * ) ,MI
E
0
b # a
9:
bt
2
.
Put B = {b; (b,c)
E
MI,
We d e f i n e a p a r t i a l g r o u p o i d
D and f o r a,b E K, a
0
b i s defined i f f
I t i s easy t o see t h a t K(0) i s reduced and
Moreover, M ( K ( 0 ) ) ) = M.
quasigroups w i t h the same underlying s e t Then the p a r t i a l groupoids I
have the same underlying set and form a couple of companions.
PROOF: Easy.
L e t G be a g r u p o i d , M a non-empty subset o f G
2
and K(0) = I
be a p a r t i a l g r o u p o i d w i t h t h e same u n d e r l y i n g s e t as K(0) and l e t M d e f i n e a g r o u p o i d G ( 2 t ) = G[M,K(r)] (a,b) 4 M; a
*
b = a
9~
as f o l l o w s : a
b f o r a l l a,b E G, (a,b)
1b
E
5
.
M(K(9:)).
Q(")
= Q(o)[ M,K(*)]
K(9c) = I
PROOF: Easy.
.
)
= K[ Q (0 ) ,MI
We
= ab f o r a l l a,b E G,
M.
PROPOSITION 7.2: Let Q ( 0 ) be a quasigroup and M a non-empty subset of Q
t h a t t h e p a r t i a l p u p o i d K( 0
L e t K(a)
2
.
Suppose
has a companion K(*) and p u t
Then Q(") is a quasigroup, M = { ( a , b ) ;
a
0
b # a;? b l a n d
A. Dripal and T. K e p h
330 PROPOSITION 7.3: 2nd K (
=
0 )
Let Q he a f i n i t e group, Q(") a quasigroup, M = t ( a , b ) ;
4 Q,M].
a b # a:'; b l
:'hen:
iil K ( 0 ) is a r e 9 Z a r exchangeable partialgroupoidand M ( K ( 0 ) ) )
= M.
/ - . Y k r e eriar; a regular balanced p r i m a r y p a r t i a l groupoid L and an i m e r ,:L)
siov f -f L h i t o
K(0).
(iii) There exi;; a pseudoregular balanced strongly primary p a r t i a l groupoid I
I i n t o K(
m.? ii pseucioimsrsion g sf
0 ) .
PROOF: Apply 3.6, 6.6 and 6.7.
LEMMA 7.4: Let Q
be a group
of a f i n i t e order n
A u ; gdist(n) = dist(Q,Q(")).
?ut M = {(a,b);
>
2 and Q ($2) a quasigroup such
ab # a
b}
f:
and K ( 0 ) = K[Q,M].
?nen K ( o ) i; a regular primary p m t i a l g r o u p o i d a n d g d i s t ( n ) = m ( K ( 0 ) ) .
PROOF: The r e s u l t i s an easy consequence o f 7.1, 7.2 and 6.2. where K runs through a l l
PROPOSITION 7.5: For auery n 2 2, g d i s t ( n ) = min m ( K )
, p s t d o , ' r e g u L a r balances' ( s t r o n g l y ) primary p a r t i a l groupoids such t h a t there e x i s t
p o u r Q of order n aizd a i p s e u d o l i m e r s i o n of K i n t o Q.
8:
PROOF: Apply 3.7,
7.3 and 7.4.
L e t Q be a group, u,v E Q, R a subgroup o f Q, R ( s ) a quasigroup and M = {(a,b) E RL; ab # a
x
0
a,b
-C
b).
Define a new o p e r a t i o n
y = xy i f x,y E Q and e i t h e r x E
R.
9 uR o r
y
on Q as f o l l o w s :
0
4 Rv; (ua)
0
( b v ) = u ( a s: b ) v f o r a l l
We s h a l l use t h e n o t a t i o n Q ( 0 ) = Q[ R(f:),u,v].
LEMMA 7.6: Q ( 0 ) is c: quasigroup, d i s t ( Q , Q ( o ) ) = dist(R,R(fg)) = c a r d M and, f o r x,y E Q, xy # x
0
y
i f f x = ua and y = bv f o r ( a , b )
E
M.
PROOF: Easy. L e t Q be a group,
B = { x ; (x,y) E 0 = {x
0
MI,
Q ( 0 )
a quasigroup, M = { ( x , Y )
C = ty; (x,y)
y; (x.y) E M I .
E
MI and 0
E
Q 2 ; XY #
= Ixy; (x,y) E M I .
X
Yl,
Then
Further, l e t (u,v) E M and l e t R be the subgroup of Q
Group modifcations o f some partial groupoids generated by a l l u-’x,
x
33 1
6, y E C. D e f i n e an o p e r a t i o n 9s on R as f o l l o w s : -1 -1 ) f o r a l l ( x , ~ ) E M; ( ~ - ’ X ) ~ ~ ( Y V - =’ ) R and (a,b) # ( u x,yv yv-’,
E
b = ab i f a,b E -1 = u-’(x y)v f o r a l l (x,y) E M. a
9:
0
LEMMA 7.7:
and Q( 0
)
R(9:)
=
i s a quasigroup,{(a,b)
E
R2; ab # a
9s
b } = { ( u -1 x,yv -1 ) ; ( x , ~ ) E M }
.
Q[ R ( 2 t ) ,u,v]
PROOF: Easy.
Further, l e t K
T and l e t f be a pseudoimmersion o f K i n t o Q such t h a t
C l e a r l y , m(K) = c a r d M.
f ( M ( K ) ) = M. f(a)
E
u and f ( b ) = v.
L e t x = (a,b) E M(K) be such t h a t
Defiine a mapping h o f K i n t o R by h ( c ) = u - l f ( c ) ,
h ( d ) = f ( d ) v - l and h ( c d ) = u - l f ( c ) f ( d ) v - ’
f o r e v e r y (c,d) E M(K).
pseudoimmersion and i t i s an i m n e r s i o n , p r o v i d e d f i s so.
Then h i s a
We have h ( a ) = 1 = h ( b )
and sx = k e r q
C k e r h. Hence h induces a pseudoimmersion k o f Kx i n t o R such x t h a t h = kqx. I f f i s an immersion t h e n k i s so. Moreover, t h e r e i s a u n i q u e
homomorphism g o f G(Kx) i n t o R such t h a t k = ggK
.
X
and S = Ker g i s a pseudoregular subgroup o f G(Kx). i s a r e g u l a r subgroup.
Obviously, g i s s u r j e c t i v e If
f i s an i m e r s i o n t h e n S
The group R i s i s o m o r p h i c t o t h e f a c t o r g r o u p G(Kx)/S.
Now, c o n s i d e r t h e isomorphism hx o f G(Kx) o n t o H(K) ( s e e 5 . 2 ) .
Then T = h x ( S ) i s
a normal subgroup o f H(K), T i s pseudoregular and i t i s r e g u l a r , p r o v i d e d f i s an imnersion.
Moreover, t h e groups R and H(K)/T a r e i s o m o r p h i c .
I f Q i s f i n i t e then
H(K)/T i s f i n i t e and c a r d H(K)/T d i v i d e s c a r d Q.
LEMMA 7.8: For every n
> 2 there e x i s t a f i n i t e (pseudoiregular ( s t r o n g l y ) primary
balanced p a r t i a l groupoid K and a lpseudolregular subgroup T of (H( K) such t h a t m(K) = g d i s t ( n ) , H(K)/T i s f i n i t e and card H(K)/T d i v i d e s n.
PROOF: The a s s e r t i o n i s an easy consequence o f 7.3,
7.4 and t h e p r e c e d i n g c o n s i d -
erations. LEMMA 7.9.:
Let n 2 2 be an i n t e g e r and Kanexchangeable balanced partia2 groupoid
such t h a t there e x i s t s a pseudoregular subgroup T of H(K) such t h a t H(K)/T is f i n i t e and card H(K)/T d i v i d e s n.
Then g d i s t ( n ) G m(K).
332
A . Drdpal and T,Kepku
PROOF: There i s a group Q o f order n such t h a t H(K)/T i s isomorphic t o a subgroup
R o f Q.
L e t x E M(K), q be t h e n a t u r a l homomorphism of H(K) o n t o H(K)/T and p an Put f = p q t g so t h a t f i s a pseudoimmersion o f K x K L e t L ( 0 ) = f[ K] = K[Q,MI, M = f ( M ( K ) ) . By 6.2, L ( 0 ) i s
isomorphism o f H(K)/T onto R. i n t o R (see 5.5).
exchangeable, and hence g d i s t ( n )
THEOREM 7.10: For euery n 2 2,
Q
m(L(0)) = m(K).
g d i s t ( n ) = min m(K) where K runs through a l l f i n i t e
Ipsezuio)regular balanced ( s t r o n g l y ) p r i m a r y p a r t i a l g r o u p o i d s s u c h t h a t H(K)/T i s f i n i z a cmd card H(K)/T d i v i d e s n f o r a Ipseudolregular subgroup T of H(K). PROOF: Apply 7.8 and 7.9.
BIBLIOGRAPHY
1.
A . Drhpal, Quasigroups r i c h i n a s s o c i a t i v e t r i p l e s ( t o appear)
MFF UK 186 00 PRAHA 8 Sokolovskd 83 Czechoslovakia
-
Kar1;n
Annals of Discrete Mathematics 18 (1983) 319-332 0 North-Holland Publishing Company
GROUP MODIFICATIONS OF SOME PARTIAL GROUPOIDS
A. Dr6pal and T. Kepka"
1. INTRODUCTION Let G
( 0 )
dist(G(O),G(n))
and G (") be twogroupoids w i t h t h e same u n d e r l y i n g s e t . =
c a r d {(a,b)
E
6';
a
0
b # a
A
b}.
For an i n t e g e r n
We p u t
> 2, l e t
g d i s t ( n ) = min d i s t ( Q , Q ( o ) ) where Q runs through a l l groups o f o r d e r n and through a l l quasigroups w i t h t h e same u n d e r l y i n g s e t such t h a t Q # Q ( 0 ) . d i v i d e s n then c l e a r l y g d i s t ( n )
Q
Q
m
I t seems t o be an open problem whether
2.
I n t h e present paper, a technique which m i g h t
be u s e f u l i n s o l v i n g t h i s problem i s developed. (i.e.
If 2
gdist(m), and t h e r e f o r e g d i s t ( n ) B g d i s t ( p ) , p
being t h e l e a s t prime number d i v i d i n g n. g d i s t ( n ) < g d i s t ( p ) f o r some n
Q(0)
I n p a r t i c u l a r , the modifications
r e l f e x i o n s ) o f c e r t a i n p a r t i a l g r o u p o i d s i n t o t h e category o f groups a r e i n -
vestigated.
L e t us n o t e here t h a t t h e numbers g d i s t ( n ) p l a y i m p o r t a n t r B l e i n t h e
enumeration questions concerning t h e number o f a s s o c i a t i v e t r i p l e s o f elements i n f i n i t e quasigroups (see [ 11). 2. PARTIAL GROUPOIDS By a p a r t i a l groupoid we mean a non-empty s e t t o g e t h e r w i t h a p a r t i a l b i n a r y o p e r a t i o n ( p o s s i b l y empty).
T h i s o p e r a t i o n w i l l be denoted m u l t i p l i c a t i v e l y i n
t h i s section. L e t K be a p a r t i a l groupoid. M(K,a) = {(b,c)
E
For every a E K we p u t
KL; a = bc}, so t h a t t h e o p e r a t i o n o f K i s d e f i n e d j u s t f o r o r -
dered p a i r s from t h e s e t M = M(K) =
U
M(K,a),
a
E
K.
Further, l e t
B(K) = { b E K; (b,c) E M I , C(K) = { c E K, (b,c) E M } , A(K) = B(K) D(K) = {bc; (b,c) E M I .
F i n a l l y , l e t m(K)
L e t K and L be p a r t i a l groupoids.
U
C ( K ) and
c a r d M.
By a homomorphism o f K i n t o L we mean any
mapping f o f K i n t o L s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n : ( 1 ) I f (a,b) E M(K) then ( f ( a ) , f ( b ) )
E M(L) and f ( a b ) = f ( a ) f ( b ) .
phism f i s s a i d t o be s t r o n g i f t h e f o l l o w i n g i s t r u e :
The homomor-
320
A . Drcipal and T. K e p h
( 2 ) I f (x,y) E M(L) and x,y
E
f ( K ) then t h e r e e x i s t s a p a i r (a,b) E M(K) such t h a t
f ( a ) = x and f ( b ) = y. P a r t i a l groupoids w i t h homomorphisms form a category which w i l l be denoted by
P. L e t f be a homomorphism o f a p a r t i a l groupoid K i n t o a p a r t i a l groupoid L. Put f(M(K)) = ( ( f ( a ) , f ( b ) ) ;
(a,b) E M(K)}
5M(L).
groupoid I ( 0 ) = F[ K ] a s f o l l o w s : I = f ( K ) and u i f f t h e r e a r e a,b
E
K such t h a t (a,b) E M(K),
(then (u,v) E M(L) and z = u v ) .
0
Further, d e f i n e a p a r t i a l v = z i s d e f i n e d f o r u,v,z E I
f ( a ) = u, f ( b ) = v and f ( a b ) = z
I t i s easy t o see t h a t f : K
homomorphism and t h e i d e n t i t y mapping i d : I(.)
-+
.+
I ( . ) i s a strong
L i s a homomorphism.
By a pseudoimmersion (resp. immersion) o f a p a r t i a l g r o u p o i d K i n t o a p a r t i a l
groupoidL we mean any homomorphism f o f K i n t o L such t h a t the r e s t r i c t i o n s f j
E(K) and
f
C(K) (resp. f
1 B(K),
L e t K be a p a r t i a l g r o u p o i d .
f
1 C(K)
and f
I D(K))
are i n j e c t i v e mappings.
By a congruence o f K we mean any equivalence
r e l a t i o n r d e f i n e d on K such t h a t :
( 3 ) I f (a,b),(c,d)
E
r and (a,c),(b,d)
E
M(K) then (ac,bd) E r.
I n t h i s case, we
can d e f i n e a s t r u c t u r e o f a p a r t i a l g r o u p o i d on the f a c t o r s e t K / r by xy = z f o r x,y,z
E K / r i f f t h e r e a r e a E x and b E y such t h a t (a,b) E M(K) and ab E z.
The
n a t u r a l p r o j e c t i o n o f K onto K / r i s then a s t r o n g homomorphism. An element e o f a p a r t i a l groupoid K i s s a i d t o be a p a r t i a l n e u t r a l element
of K i f the following three conditions are satisfied: ( 4 ) If (e,a) E M(K) then ea = a. ( 5 ) I f (a,e) E M ( K ) then ae = a. ( 6 ) (e,e)
E
WK).
A p a r t i a l g r o u p o i d K i s s a i d t o be reduced i f K = A(K)
u D(K).
R the f u l l subcategory o f P formed by reduced p a r t i a l groupoids.
We denote by
Obviously, every
p a r t i a l g r o u p o i d K can be expressed u n i q u e l y as t h e d i s j o i n t union K = L M(K)
5 LL
and
L
U
J where
i s reduced.
A p a r t i a l g r o u p o i d K i s s a i d t o be balanced i f t h e sets B ( K ) , C(K) and D(K) are p a i r - w i s e d i s j o i n t . A p a r t i a l g r o u p o i d K i s s a i d t o be c a n c e l l a t i v e i f t h e f o l l o w i n g two condi-
tions are satisfied: ( 7 ) I f (a,b),(a,c)
E
M(K) and ab = ac then b = c.
( 8 ) I f (b,a),(c,a)
E
M(K) and ba = ca then b = c.
A group i s a semigroup w i t h (unique) d i v i s i o n .
Thus a group i s a (reduced
32 1
Group modifcations of some partial groupoids
c a n c e l l a t i v e p a r t i a l ) g r o u p o i d and we denote by g t h e f u l l subcategory o f groups i n F u r t h e r , we denote by S ( T ) t h e f u l l subcategory o f R c o n s i s t i n g
t h e c a t e g o r y R.
o f reduced balanced (cancel l a t i v e ) p a r t i a l groupoids.
LEMMA 2.1:
of
Let K
E
R.
Then there e d s t L E S and a surjective strong immersion k
L onto K. Moreouer,L is canceZZative, provided K is
PROOF: L e t B,C,D h: D
+
D(K) b i j e c t i v e mappings.
f o r a l l b E 6, c
k = f
be p a i r - w i s e d i s j o i n t s e t s and f : B
U
g
U
E
+
SO.
B(K), g: C
C(K),
P u t L = B u C u D and d e f i n e bc = h - ' ( f ( b ) g ( c ) )
C such t h a t ( f ( b ) , g ( c ) ) E M(K).
Now, i t s u f f i c e s t o p u t
h.
L e t K be a p a r t i a l g r o u p o i d a n d x = (a,b) E M = M(K). t h e l e a s t congruence o f K such t h a t (a,b)
E
We denote by s = sx
s and a/s i s a p a r t i a l n e u t r a l element
o f K/s ( i t i s easy t o check t h a t t h e r e e x i s t s such a congruence). a r e l a t i o n s ' = s; ( 9 ) (c,c)
+
Further, define
on K as f o l l o w s :
E s ' f o r e v e r y c E K.
(10) I f (a,c) E M t h e n (ac,c),(c,ac)
E s'.
( 1 1 ) If (d,b) E M t h e n (db,d),(d,db)
E s'.
( 1 2 ) I f (a,c),(d,b)
E M and ac = db t h e n (c,d),(d,c)
LEMMA 2.2: Let K E T and x
E
Then sx =
M(K).
S' X
E
s'.
and the naturaZ homomorphism of
K onto Kfs is an immersion. PROOF: One may v e r i f y e a s i l y t h a t s;
i s a congruence o f K and t h e r e s t i s c l e a r .
LEMMA 2.3: Let h be a pseudoimmersion of a p a r t i a Z g r o u p o i d K into a canceZZatioe
partial groupoid L.
Then K is canceZlative.
PROOF: Obvious.
LEMMA 2.4: Let I,J,K be a p a r t i a l g r o u p o i & and 9: J
such that J E S , g(M(L))
5 f(M(1))
K, f: I + K homomovhisms
and f is an immersion.
homomorphism h of J into I such that f h = 9 . PROOF: Easy.
-t
Then there exists a
A , Drapal and T. Kepka
322
An element a o f a p a r t i a l g r o u p o i d K i s s a i d t o be idempotent i f (a,a) E M(K) and aa = a.
3. GROUP MODIFICATIONS OF REDUCED PARTIAL GROUPOIDS I n t h i s s e c t i o n , the operations o f general p a r t i a l g r o u p o i d s a r e denoted by
0
and the operations o f groups m u l t i p l i c a t i v e l y . Let K
E
R and l e t F(K) designate the f r e e group o f words over the s e t A(K). -ld-l
and L e t N(K) be the normal subgroup o f F(K) generated by a l l t h e words bce -1 where (b,c),(d,e) E M(K,a), a 6 D(K), and (u,v) E M(K,z), z E A(K) n D(K). uvz Put G(K) = F(K)/N(K) and d e f i n e a mapping g = gK o f K i n t o G(K) by g ( a ) = aN(K) and g(b
0
c ) = bcN(K) f o r a l l a E A(K) and (b,c) E M(K).
PROPOSITION 3.1: Lot K E R.
Then g = gK i s a we12 defined homomorphism of K i n t o
G{K) and
r pm&i g ( e ) = 1 f ~ every
Moreovoq
neutra2 element e E K.
PROOF: Using t h e d e f i n i t i o n o f N = N(K) and the f a c t t h a t K i s reduced, i t i s easy t o show t h a t g i s a homomorphism.
H.
a
Now, l e t h be a homomorphism o f K i n t o a group
There i s a homomorphism p o f F(K) i n t o H such t h a t p ( a ) = h ( a ) f o r every E
A = A(K).
Clearly, N
5 Ker
p, and hence we have a homomorphism f o f G = G(K)
i n t o H such t h a t f(aN) = h ( a ) f o r each a E A.
Then h = f g and the r e s t f o l l o w s
from the f a c t t h a t G i s generated by g(A). L e t K,L E R and l e t h be a homomorphism o f K i n t o L.
According t o 3.1,
t h e r e e x i s t s a unique homomorphism G(h) o f G(K) i n t o G(L) such t h a t G(h)gK = gLh. This homomorphism i s d e f i n e d by G(h)(aN(K)) = h(a)N(L) f o r every a E A(K). F u r t h e r , g K ( a ) g K ( b ) - l E Ker G(h) whenever a,b E K and h ( a ) = h ( b ) .
F i n a l l y , G(h)
i s s u r j e c t i v e , provided h i s so.
K,L E R an2 l e t h be u s u r j e c t i v e strong homomoqkism o f K onto L. -1 IIce,; Ker G(H) is j u s : tne tiormu2 s 5 g r m p of G(K) generated by a21 g ( a ) g ( b ) , K K a,b E K, h ( a ) = h ( b ) . LEMMA 3.2: L,t
PROOF: Denote by P t h a t normal subgroup.
c Ker G(h). Then P -
Put H = G(K)/P,
denote by p t h e n a t u r a l homomorphism o f G(K) onto H and d e f i n e a mapping f o f L i n t o H by f h ( a ) = pgK(a) f o r every a
K.
Then f i s a homomorphism, and hence
3 23
Group modifcations of some partial groupoids f = kg
f o r a homomorphism k o f G(L) i n t o H. L p = kG(h) and Ker G(h) 5 Ker p = P.
NOW, pgK = f h = kgLh = kG(h)gK,
L e t K E R , g = gK and x = (a,b) E M(K). G = G(K) generated by a l l g ( a ) - ’ g ( c ) ,
g(d)g(b)-’
Denote by H(K,x) t h e subgroup o f where (c,d) E M(K).
P(K,x) be t h e normal subgroup o f G generated by g ( a ) and g ( b ) .
Further, l e t
Denote by p x t h e
n a t u r a l homomorphism o f G o n t o G/P(K,x).
LEMMA 3.3:
Let K
E
R and x,y
px(H(K)) = G(K)/P(K,x).
E
M(K).
Then H(K,x) = H(K,y) = H(K) and
Moreover, i f K has a t l e a s t one idempotent element then
H(K) = G(K).
PROOF: Easy.
LEMMA 3.4: Let K,L E Rand l e t h be a homomorphism of K i n t o 1. G(h)(H(K))
5 H(L).
Then
Moreover, G ( h ) ( H ( K ) ) = H(L), provided h i s s u r j e c t i v e and
strong. PROOF: Easy.
Let K E R.
A normal subgroup R o f G(K) i s s a i d t o be pseudoregular ( r e s p .
r e g u l a r ) if t h e homomorphism f g K o f K i n t o G(K)/R, f b e i n g t h e n a t u r a l homomorphism o f G(K) o n t o G(K)/R,
i s a pseudoimnersion ( r e s p . an immersion).
The
g r o u p o i d K i s s a i d t o be pseudoregular ( r e s p . r e g u l a r ) i f t h e u n i t subgroup o f G(K) i s pseudoregular ( r e s p . r e g u l a r ) . LEMMA 3.5: Let K E R be pseudoregular.
Then K is c a n c e l l a t i v e .
PROOF: Every group i s c a n c e l l a t i v e and t h e r e s u l t f o l l o w s f r o m 2.3.
LEMMA 3.6: Let K E R.
Then K i s fpseudolregular i f f t h e r e e x i s t a group G and a
(pseudo)immersion of K i n t o G. PROOF: Obvious
LEMMA 3.7: Let K,L E R and l e t h be an (pseudolimmersion of K i n t o L.
If L is
324
A. Drdpaland T. Kepkn
:psel*dolregular then K i s so.
PROOF: Use 3.6.
4. GROUP MODIFICATIONS OF REDUCED BALANCED PARTIAL GROUPOIDS LEMMA 4.1 : Let K H( K) = 1
LP:G
6
S 3, such t h t curd K =3.
Then G( K) is a f r e e group of rank 2
.
PROOF: Obvious
Let K
LEMMA 4.2: k t K gK(K)
Denote by V(K) t h e normal subgroup o f G ( K ) g e n e r a t e d by H(K).
E S.
V(K) =
E S.
Then G(K)/V(K) is a f r e e group o f rank 2 , H(K)
5 V(K)
and
8.
PROOF: As one may see e a s i l y , t h e r e e x i s t L E S and a s u r j e c t i v e homomorphism h of The r e s t i s c l e a r : f r o m 3.2 and
K o n t o L such t h a t c a r d L = 3 and h i s s t r o n g .
4.1.
LEMMA 4.3: G(K)/P(K,x)
Then there e x i s t s an isomorphism f
Let K E S and x E M(K). onta H(K) snch t h a t f p x x
I H(K)
X
of
= id.
I H, G = G(K,x) and P = P(K,x). There i s a K’ p = pX s u r j e c t i v e homomorphism q o f F(K) o n t o H = H(K) such t h a t q ( c ) = g ( a ) - ’ g ( c ) and
PROOF: L e t x = (a,b),
g = g
I t i s easy t o check t h a t
q ( d ) = g ( d ) g ( b ) - l f o r each ( c , d ) E M = M(K).
N
= N(K) c Ker q and a,b E Ker q.
Hence P
homomorphism f = f x o f G/P o n t o H. and f p ( g ( d ) g ( b ) - l ) = g ( d ) g ( b ) - ’ ,
PROPOSITION 4.4: i ‘;
:f
L e t K E S,
q/H and q induces a s u r j e c t i v e ‘
F o r (c,d) E M, f p ( g ( a ) - ’ g ( c ) )
so t h a t f p = i d H .
x E M(K) a d U(K,x)
H(K) is i s o ~ o r p h i cto G(K)/P(K,x),
G(K) crnd G(K) is generated bg H(K) ( i G * lP(K,x)/U(K,x)
5 Ker
U
P(K,x).
is u f r e e group of r a n k
Similarly, p f = i d GfP‘
= P(K,x)
H(K)
2.
= g(a)-’g(c)
17 V(K).
P(K,x)
Then:
= 1, H(K) i s u r e t r a c t
Group modifcations of some partial groupoids
325
(iii)G( K)/U ( K,x) i s isomorphic t o the d i r e c t product H( K) x F where F i s
Q
free group of rank 2. PROOF: Apply 4.2 and 4.3.
L e t K E S, x E M(K) and t = tx = f x p x . phism o f G = G(K) o n t o H = H(K) and t and R = t - l ( S ) .
IH
= id.
By 4.3,
t i s a s u r j e c t i v e homomor-
L e t S be a normal subgroup o f
H
We s h a l l say t h a t S i s an x-(pseudo)regular subgroup o f H i f R i s
(pseudo)regular i n G.
Thus S. i s x-(pseudo)regular i f f q t g K i s a (pseudo)immer-
sion, q being t h e n a t u r a l homomorphism o f H onto H/S.
LEMMA 4.5: Let K E S and l e t S be a normal subgroup of H(K). x,y E M(K),S
Then, for a l l
i s x-(pseudolregular i f f it i s y-(pseudolregular.
PROOF: Suppose t h a t S i s x-pseudoregular.
L e t x = (a,b),
y = (c,d)
and l e t
u,v E B(K) be such t h a t q t g(u) = q t g ( v ) where g = gK and q i s t h e n a t u r a l Y Y E S, and hence homomorphism o f H = H(K) onto H/S. Then g ( c ) - ’ g ( u ) g ( v ) - ’ g ( c ) g ( a )-’g ( u ) g ( v )-’g ( a ) = g ( a I-’g ( c )g ( c I-’g ( u 1g ( v I-’g ( c 1g ( c I-’g ( a I E g(a)-’g(c)
E
H and S i s normal i n H.
S being x-pseudoregular.
s
s irice
Consequently, q t g(u) = q t x g ( v ) and u = v, X
S i m i l a r l y f o r u,v E C(K) and, provided S i s x - r e g u l a r ,
f o r u,v E D(K). L e t K E S and l e t S be a normal subgroup o f H(K).
We s h a l l say t h a t S i s
(pseudo)regular i f i t i s x-(pseudo)regular f o r some x E M ( K ) .
LEMMA 4.6: Let K
E
S be regular.
Then g i s i n j e c t i v e . K
PROOF: Easy.
5. GROUP HODIFICATIONS OF REDUCED BALANCED CANCELLATIVE PARTIAL GROUPOIDS L e t K E T and x E M(K).
Consider t h e congruence s = sx, p u t Kx = K/s and
denote by qx t h e n a t u r a l homomorphism o f K onto Kx.
Then Kx E R, Kx has a p a r t i a l
n e u t r a l element and qx i s a s t r o n g imnersion. LEMMA 5.1: Let K E T and x E M(K).
Then Ker G(qx) = P(K,X).
326
A . Drhpal and T. Kepka
PROOF: The a s s e r t i o n f o l l o w s e a s i l y f r o m 2 . 2 and 3 . 2 .
LEMMA 5.2:
H(K)
Ls.5 K E T
sh.2;~ thct
h x ( S ) c.:
an2 x
H(K)i;
Then there i s an isomorphism h of G(Kx) onto X
G(K ) i s (psedo)regular i f f the subgroup X
30.
is: K
t h e r e i s an isomorphism kx o f G(Kx) o n t o G(K)/P(K,x)
such
Now, i t s u f f i c e s t o p u t hx = f x k x .
= kxG(qx).
LEMMA 5.3:
M(K).
a tiormL aubgroup S of
PROOF: According t o 5.1, t h a t p,
E
T z d x,y
6
6
Then there e x i s t s an isomorphism h o f G( K )
M(K).
X
G ( K ) suc;. :;an a norm2 subgroup S of G(K ) i s (pseucb)reguZaz~i f f tize subY X ~ l l L r i i ) h(S) sf G ( K ) :‘s SO. Y -;::G
PROOF: T h i s i s an immediate consequence o f 5 . 2 .
PROOF: ( i ) i m p l i e s ( i i ) .
F o r , l e t x = (a,b)
(pseudo)regular.
P x g ( c ) = P x g ( d ) , 9 = gK. g(c)g(d)-’
We must show t h a t t h e subgroup P = P(K,x)
o f G(K) i s
and c,d E B(K) be such t h a t
Then f x p x ( g ( c ) g ( d ) - ’ )
= 19 g ( a ) - l g ( c ) g ( d ) - ’ g ( a )
= 1 , g ( c ) = g ( d ) and c = d, s i n c e K i s ( p s e u d o ) r e g u l a r .
= 1%
S i m i l a r l y the
rest. ( i i ) implies ( i i i ) . ( i i i ) implies ( i ) .
COROLLARY 5.5: r
ict K
T h i s i s an immediate consequence o f 5 . 3 . Use R . 7 and t h e f a c t t h a t q,
E
T and x E M(K).
:zLLc:
_6. COUPLES
.
i s an immersion.
?hen K i s lpseudolreguZar i f f t g i s a x K
OF COMPANIONS
I n t h i s s e c t i o n , we s h a l l use v a r i o u s symbols f o r p a r t i a l b i n a r y o p e r a t i o n s . Using a n o t h e r n o t a t i o n t h a n t h e m u l t i p l i c a t i v e one, we p u t t h e o p e r a t i o n symbol
327
Group modifcations of some partial groupoids i n t o t h e b r a c k e t s b e h i n d t h e symbol f o r t h e u n d e r l y i n g s e t . Let
K(Q) and K(a) be p a r t i a l g r o u p o i d s w i t h t h e same u n d e r l y i n g s e t .
s h a l l say t h a t t h e p a i r (K("),
K(n))
We
i s a c o u p l e o f companions i f b o t h t h e p a r t i a l
i g - o u p o i d s a r e reduced and c a n c e l l a t i v e , M ( K ( 0 ) ) = M(K(9:)) = M and f o r a l l (a,b) E M t h e r e e x i s t c,d,e,f (a,c),(d,b),(a,e),(f,b)
E
M such t h a t c
E
M, a
0
b = a
f:
c = d
# b # e, d # a # f, f:
b and a fi b = a
LEMMA 6.1 : Let ( K ( " ) ,K(fc)) be a couple of companions.
e = f
b.
0
Then there e x i s t a couple
and a s u r j e c t i v e mapping k of L onto K such t h a t both
o f companions ( L ( o ) , L ( a ) ) L(0)
0
and L ( n ) are balanced, k i s a strong i m e r s i o n o f L ( 0 ) onto K(0) and k i s a
strong immersion of L ( n ) onto K(f:). PROOF: Easy ( u s e t h e same t e c h n i q u e as i n t h e p r o o f o f 2 . 1 ) .
A partialgroupoid
K
i s s a i d t o be exchangeable i f i t i s reduced and
c a n c e l l a t i v e and i f i t possesses a t l e a s t one companion.
LEMMA 6.2: Let K be a p a r t i a l grupoid, L ( 0 ) an exchangeable p a r t i a l groupoid and f
a pseudoimersion of L ( o ) i n t o K.
Suppose t h a t e i t h e r f is an i m e r s i o n or
Then the partialgroupoid I(o) = f[ L(.)]
canceZZative.
PROOF: L e t L ( k ) be a companion o f L ( 0 ) .
I = f ( L ) by u
3:'
f ( b ) = v and f ( a
v = 9:
z
i s exchangeable.
Define a p a r t i a l binary operation
i f f t h e r e a r e a,b E L such t h a t (a,b) E M(L(f:)),
b) = z.
K is
sc
on
f(a) = u,
I t i s easy t o v e r i f y t h a t t h e p a i r (I(o),I(s'c))
is a
coup1 e o f companions.
A p a r t i a l g r o u p o i d K i s s a i d t o be ( s t r o n g l y ) p r i m a r y i f i t i s exchangeable and e v e r y p s e u d o i m e r s i o n o f an exchangeable p a r t i a l g r o u p o i d i n t o K i s a s u r j e c t i v e strong pseudoimersion (immersion).
LEMMA 6.3: Let f be a pseudoimmersion of an exchangeable p a r t i a l groupoid L i n t o a
p r i m a r y p a r t i a l groupoid K.
Then:
( i l f(M(L))
= A(K),
= M(K), f ( A ( L ) )
f ( B ( L ) ) = B(K), f ( C ( L ) ) = C(K) and
f ( D ( L ) ) = D(K).
( i f )The partiaZ groupoid L i s primary.
Moreover, L i s strongly primary Iresp.
balanced, provided K i s so.
(iii)I f K i s balanced and f is an immersion then f is an isomoprhism.
328
A . Drhpal and T. Kepko
PROOF: Easy. LEMMA 6.4: L,et K be a ( s t r o n g l y ) p r i m a r y partialgroupoid.
Then there e x i s t a
balance*! ( s t r o n g l y ) p G m r u partialgroupoid L cozd a s u r j e c t i v e strong i m e r s i o n k o f L onto K.
PROOF: Use 6.1 and 6 . 3 ( i i ) .
LEMMA 6.5: Let K be a f i n i t e exchangeable parsialgroupoid.
Then there e x i s t a
L i n t o K.
;,rlmaq. parcinlgroupoid L and mi i n j e c t i v e homomorphism f o f
PROOF: W e s h a l l proceed by induction on m ( K ) .
Suppose t h a t K i s not primary.
Then there a r e an exchangeable p a r t i a l groupoid J and a pseudoimmersion g o f J i n t o K such t h a t g(M(J)) # M(K).
of I i n t o K .
P u t I(.)
= g [ J ] and denote by i
By 6.2, I(o) i s exchangeable.
the i d e n t i t y mapping
Moreover, M ( I ( 0 ) ) = g(M(J))
and i i s an i n j e c t i v e homomorphism of I ( 0 ) i n t o K.
We have m ( I ( 0 ) )
$ M(K)
< m(K), and
hence there a r e a primary partialgroupoid L and an i n j e c t i v e homomorphism h of L into I(0).
Now, i t s u f f i c e s t o put f = i h .
COROLLARY 6.6: Let K be a f i n i t e exchangeable partialgroupoid.
Then there e x i s t a
Lulance:i primary parsialgroupoid L and a immersion f o f L i n t o K.
PROOF: Apply 6.4 and 6.5. PROPOSITION 6.7: k t K be a f i n i t e exchangeable partialgroupoid. :brzlawed s t m n g Z y
Then there e x k t
primary purtialgroupoid L and a pseudoimmersion f o f L i n t o K.
PROOF: P u t m = card B ( K ) and n = card C(K).
I f I i s a reduced partialgroupoid and
g i s a pseudoimmersion of I i n t o K then I i s f i n i t e and card1 < m+n+mn.
Hence
there a r e a balanced primary partialgroupoid L and a pseudoimmersion f o f L i n t o K such t h a t card D(L) 2 card D(J) whenever J i s
a balanced primary p a r t i a l groupoid
with a pseudoimnersion h of J i n t o K (use 6 . 7 ) .
Obviously, L i s strongly primary.
LEMMA 6 . 8 : Let K,L E R and l e t f be a s u r j e c t i v e strong i m e r s i o n of L onto K. K i s exchangeable iresp. p r i m u r g , strongly p r i m a r y ) i f f L i s so.
.r?2;n -17
329
Group modifcations of some partial groupoids PROOF: I f L i s exchangeable t h e n K i s exchangeable by 6.2.
Conversely, i f K i s
exchangeable ( r e s p . p r i m a r y , s t r o n g l y p r i m a r y ) t h e n i t i s easy t o check t h a t L i s so ( s e e t h e p r o o f o f 6.1 and 6 . 3 ( i i ) ) .
NOW, suppose t h a t L i s ( s t r o n g l y ) p r i m a r y
and g i s a pseudoimmersion o f an exchangeable p a r t i a l g r o u p o i d I i n t o K .
Then K i s
exchangeable and t h e r e e x i s t a balanced exchangeable p a r t i a l g r o u p o i d J and a s u r j e c t i v e s t r o n g immersion h o f J o n t o I. By 2.4, gh = f k f o r a homomorphism k o f J i n t o L.
I t i s easy t o see t h a t k i s a pseudoimmersion, and hence k i s
s u r j e c t i v e and s t r o n g .
The r e s t i s c l e a r .
7. GROUP DISTANCES OF FINITE QUASIGROUPS L e t G be a g r o u p o i d a n d M a non-empty s u b s e t o f G C = {c; (b,c) E M } and D = {ab; (a,b) E M I .
K(0) = K[G,M ] a s f o l l o w s : K = B u C (a,b)
E
M; i n t h a t case, a
U
b = ab.
o
i t i s c a n c e l l a t i v e , p r o v i d e d G i s so.
PROPOSITION 7.1:
Let Q ( 0 ) and Q ( " ) be
and M = { ( a,b)
Q2; a
K[ Q ( * ) ,MI
E
0
b # a
9:
bt
2
.
Put B = {b; (b,c)
E
MI,
We d e f i n e a p a r t i a l g r o u p o i d
D and f o r a,b E K, a
0
b i s defined i f f
I t i s easy t o see t h a t K(0) i s reduced and
Moreover, M ( K ( 0 ) ) ) = M.
quasigroups w i t h the same underlying s e t Then the p a r t i a l groupoids I
have the same underlying set and form a couple of companions.
PROOF: Easy.
L e t G be a g r u p o i d , M a non-empty subset o f G
2
and K(0) = I
be a p a r t i a l g r o u p o i d w i t h t h e same u n d e r l y i n g s e t as K(0) and l e t M d e f i n e a g r o u p o i d G ( 2 t ) = G[M,K(r)] (a,b) 4 M; a
*
b = a
9~
as f o l l o w s : a
b f o r a l l a,b E G, (a,b)
1b
E
5
.
M(K(9:)).
Q(")
= Q(o)[ M,K(*)]
K(9c) = I
PROOF: Easy.
.
)
= K[ Q (0 ) ,MI
We
= ab f o r a l l a,b E G,
M.
PROPOSITION 7.2: Let Q ( 0 ) be a quasigroup and M a non-empty subset of Q
t h a t t h e p a r t i a l p u p o i d K( 0
L e t K(a)
2
.
Suppose
has a companion K(*) and p u t
Then Q(") is a quasigroup, M = { ( a , b ) ;
a
0
b # a;? b l a n d
A. Dripal and T. K e p h
330 PROPOSITION 7.3: 2nd K (
=
0 )
Let Q he a f i n i t e group, Q(") a quasigroup, M = t ( a , b ) ;
4 Q,M].
a b # a:'; b l
:'hen:
iil K ( 0 ) is a r e 9 Z a r exchangeable partialgroupoidand M ( K ( 0 ) ) )
= M.
/ - . Y k r e eriar; a regular balanced p r i m a r y p a r t i a l groupoid L and an i m e r ,:L)
siov f -f L h i t o
K(0).
(iii) There exi;; a pseudoregular balanced strongly primary p a r t i a l groupoid I
I i n t o K(
m.? ii pseucioimsrsion g sf
0 ) .
PROOF: Apply 3.6, 6.6 and 6.7.
LEMMA 7.4: Let Q
be a group
of a f i n i t e order n
A u ; gdist(n) = dist(Q,Q(")).
?ut M = {(a,b);
>
2 and Q ($2) a quasigroup such
ab # a
b}
f:
and K ( 0 ) = K[Q,M].
?nen K ( o ) i; a regular primary p m t i a l g r o u p o i d a n d g d i s t ( n ) = m ( K ( 0 ) ) .
PROOF: The r e s u l t i s an easy consequence o f 7.1, 7.2 and 6.2. where K runs through a l l
PROPOSITION 7.5: For auery n 2 2, g d i s t ( n ) = min m ( K )
, p s t d o , ' r e g u L a r balances' ( s t r o n g l y ) primary p a r t i a l groupoids such t h a t there e x i s t
p o u r Q of order n aizd a i p s e u d o l i m e r s i o n of K i n t o Q.
8:
PROOF: Apply 3.7,
7.3 and 7.4.
L e t Q be a group, u,v E Q, R a subgroup o f Q, R ( s ) a quasigroup and M = {(a,b) E RL; ab # a
x
0
a,b
-C
b).
Define a new o p e r a t i o n
y = xy i f x,y E Q and e i t h e r x E
R.
9 uR o r
y
on Q as f o l l o w s :
0
4 Rv; (ua)
0
( b v ) = u ( a s: b ) v f o r a l l
We s h a l l use t h e n o t a t i o n Q ( 0 ) = Q[ R(f:),u,v].
LEMMA 7.6: Q ( 0 ) is c: quasigroup, d i s t ( Q , Q ( o ) ) = dist(R,R(fg)) = c a r d M and, f o r x,y E Q, xy # x
0
y
i f f x = ua and y = bv f o r ( a , b )
E
M.
PROOF: Easy. L e t Q be a group,
B = { x ; (x,y) E 0 = {x
0
MI,
Q ( 0 )
a quasigroup, M = { ( x , Y )
C = ty; (x,y)
y; (x.y) E M I .
E
MI and 0
E
Q 2 ; XY #
= Ixy; (x,y) E M I .
X
Yl,
Then
Further, l e t (u,v) E M and l e t R be the subgroup of Q
Group modifcations o f some partial groupoids generated by a l l u-’x,
x
33 1
6, y E C. D e f i n e an o p e r a t i o n 9s on R as f o l l o w s : -1 -1 ) f o r a l l ( x , ~ ) E M; ( ~ - ’ X ) ~ ~ ( Y V - =’ ) R and (a,b) # ( u x,yv yv-’,
E
b = ab i f a,b E -1 = u-’(x y)v f o r a l l (x,y) E M. a
9:
0
LEMMA 7.7:
and Q( 0
)
R(9:)
=
i s a quasigroup,{(a,b)
E
R2; ab # a
9s
b } = { ( u -1 x,yv -1 ) ; ( x , ~ ) E M }
.
Q[ R ( 2 t ) ,u,v]
PROOF: Easy.
Further, l e t K
T and l e t f be a pseudoimmersion o f K i n t o Q such t h a t
C l e a r l y , m(K) = c a r d M.
f ( M ( K ) ) = M. f(a)
E
u and f ( b ) = v.
L e t x = (a,b) E M(K) be such t h a t
Defiine a mapping h o f K i n t o R by h ( c ) = u - l f ( c ) ,
h ( d ) = f ( d ) v - l and h ( c d ) = u - l f ( c ) f ( d ) v - ’
f o r e v e r y (c,d) E M(K).
pseudoimmersion and i t i s an i m n e r s i o n , p r o v i d e d f i s so.
Then h i s a
We have h ( a ) = 1 = h ( b )
and sx = k e r q
C k e r h. Hence h induces a pseudoimmersion k o f Kx i n t o R such x t h a t h = kqx. I f f i s an immersion t h e n k i s so. Moreover, t h e r e i s a u n i q u e
homomorphism g o f G(Kx) i n t o R such t h a t k = ggK
.
X
and S = Ker g i s a pseudoregular subgroup o f G(Kx). i s a r e g u l a r subgroup.
Obviously, g i s s u r j e c t i v e If
f i s an i m e r s i o n t h e n S
The group R i s i s o m o r p h i c t o t h e f a c t o r g r o u p G(Kx)/S.
Now, c o n s i d e r t h e isomorphism hx o f G(Kx) o n t o H(K) ( s e e 5 . 2 ) .
Then T = h x ( S ) i s
a normal subgroup o f H(K), T i s pseudoregular and i t i s r e g u l a r , p r o v i d e d f i s an imnersion.
Moreover, t h e groups R and H(K)/T a r e i s o m o r p h i c .
I f Q i s f i n i t e then
H(K)/T i s f i n i t e and c a r d H(K)/T d i v i d e s c a r d Q.
LEMMA 7.8: For every n
> 2 there e x i s t a f i n i t e (pseudoiregular ( s t r o n g l y ) primary
balanced p a r t i a l groupoid K and a lpseudolregular subgroup T of (H( K) such t h a t m(K) = g d i s t ( n ) , H(K)/T i s f i n i t e and card H(K)/T d i v i d e s n.
PROOF: The a s s e r t i o n i s an easy consequence o f 7.3,
7.4 and t h e p r e c e d i n g c o n s i d -
erations. LEMMA 7.9.:
Let n 2 2 be an i n t e g e r and Kanexchangeable balanced partia2 groupoid
such t h a t there e x i s t s a pseudoregular subgroup T of H(K) such t h a t H(K)/T is f i n i t e and card H(K)/T d i v i d e s n.
Then g d i s t ( n ) G m(K).
332
A . Drdpal and T,Kepku
PROOF: There i s a group Q o f order n such t h a t H(K)/T i s isomorphic t o a subgroup
R o f Q.
L e t x E M(K), q be t h e n a t u r a l homomorphism of H(K) o n t o H(K)/T and p an Put f = p q t g so t h a t f i s a pseudoimmersion o f K x K L e t L ( 0 ) = f[ K] = K[Q,MI, M = f ( M ( K ) ) . By 6.2, L ( 0 ) i s
isomorphism o f H(K)/T onto R. i n t o R (see 5.5).
exchangeable, and hence g d i s t ( n )
THEOREM 7.10: For euery n 2 2,
Q
m(L(0)) = m(K).
g d i s t ( n ) = min m(K) where K runs through a l l f i n i t e
Ipsezuio)regular balanced ( s t r o n g l y ) p r i m a r y p a r t i a l g r o u p o i d s s u c h t h a t H(K)/T i s f i n i z a cmd card H(K)/T d i v i d e s n f o r a Ipseudolregular subgroup T of H(K). PROOF: Apply 7.8 and 7.9.
BIBLIOGRAPHY
1.
A . Drhpal, Quasigroups r i c h i n a s s o c i a t i v e t r i p l e s ( t o appear)
MFF UK 186 00 PRAHA 8 Sokolovskd 83 Czechoslovakia
-
Kar1;n