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Commutative Finite A-Hypergroups of Length Two
Annals of Discrete Mathematics 37 (1988) 147-156 0 Elsevier Science Publishers B.V. (North-Holland)
147
COMMUTATIVE FINITE A-HYPERGROUPS OF LENGTH TWO
Mario DE SALVO?:Via Palermo, 836 - S c a l a R i t i r o (HE), I t a l y
1. INTRODUCTION A h y p e r g r o u p o i d H = < H , o > i s a non-empty s e t H cquipped w i t h a hypcroo ( i . e . a mapping whose domain i s H X H and whose r a n g e i s t h e s e t of non-empty s u b s e t s of H ) . If A 5 H, B 2 H, t h e n A 0 B d e n o t e s u{a o b/a E A , b E n ] . A hypergroupoid H i s c a l l e d h y p e r g r o u p i f t h e h y p e r o p e r a t i o n s a t i s f i e s t h e f o l l o w i n g axioms : ( i ) ( x o y ) ~z = x . ( y o z ) , V ( X , Y , Z )E H3 i ( i i ) x o H = H n x = H, ~ E H . We 'say t h a t a hypergroupoid H h a s l e n g t h n , i f F ( x , y ) E H2, 1x0 yI = 11. I n [ 6 ] , T. Vougiouklis i n t r o d u c e d a new class of hypergroups, c o n s t r u c t c d from o r d i n a r y g r o u p s as f o l l o w s : ~ hyl e t C = < C,*> be a group and AC G, A f flj wc s h a l l c a l l A - I i y p e r g r o u ~tlic p e r g r o u p GA=, equipped w i t h t h e f o l l o w i n g h y p e r o p e r a t i o n
ration
k
: (x,y)-+
x . A-y.
S i n c e V ( s , y ) E G X G, Ix t\ yI = 1A1, t h c s c hypcrgroiips have l e n g t h I h l . Throughout t h e paper, we c o n s i d e r A-hypergroups of l e n g t h 2, c o n s t r u c t c d from a b e l i a n g r o u p s . These hypergroups a r e s t u d i e d and t h e i r number i s d e t e r mined when G i s c y c l i c or / G I < 1 2 .
2. NOTATIONS AND RECALLS We r e c a l l some d e f i n i t i o n s and n o t a t i o n s t y p i c a l of hypergroups. L e t A be a non-empty s u b s e t o f a hypergroup H; A i s c o m p l e t e i f V n E Nit and V(x,, x , ) E H" such t h a t O& x j n A 4 0 then -u; L A .
...,
OA 1-1
b8J
We d c n o t e by CH(A) t h e complete c l o s u r e of A, i . e . t h e i n t e r s e c t i o n of all s u b s e t s of H which are complete and c o n t a i n A , L e t R$ d e n o t e t h e t r a n s i t i v e c l o s u g e of t h e b i n a r y r e l a t i o n DH, d e f i n e d i n
t h e f o l l o w i n g way: xRHy i f f t h e r e e x i s t nEN'r and
$6
(21,
Research supported by the G.N.S.A.G.A.
..., z , ) E H "
of the C.N.R.
such t h a t { x , y } ~o~
(Italy).
zi.
M.De Salvo
I48
It is known that the relation x$ y iff xt$(ty)) is an equivnlrncc and moreover C,, = DA ( [ S I). Let 0 : H-+H/Da denote the canonical projection. If H is a hypergroup, then H/D$ is a group. Hence, we define the core of H, w(H), as the kernel of $.
H is said to be a n-hypergroupiff Vx€ H, IC, (x) 1
= n. H is said to be completeiff b'(x,y)~H', x o y =C,(xoy). Let H = , HI = < H~,o> be two hypergroups and f be a mapping of H into HI ; f is said to be a homomorphism(good homomorphism)if f(x y) 5 f (x) 0 f(y) respectively) b'(x,y)EH X H. (f(x0y) = f(x)of(y), We say that a homomorphism f is anisomorphism iff f is bijective and good. HI is a homomorphism (isomorphism) then b'x€ H, f(CH (x))C. If f : Hcaf(x)) (f(CH(x)) = C,,(f(x)), respectively). I I is said to beregular iff E(H) = { e E H / Y x E H , X E C o x n x - e ) i. fl and Y x E H , i(x) = IxVEH/3eeE(H) such that e E x oxlnxl. x) 4 8 . ( [ l ] , ) . 0
3. A-HYPERGROUPS CONSTRUCTED FROM ABJXCAN GROUPS In the following let G be an abelian group. We shall u s e the additive notation; s o , + will denote the operation in the group, 0 will be the unit element and -a the inverse element of a. Clearly, if G is abelian, then GA is commutative. It is known (see [ 61) that GA is regular and E ( G A ) = -A = [-a/aeA REMARK 3.1. The hyperproducts of two elements in G A may be defined as fo lows: O b x = A + X; VXEG, a '? b = 0 $! (a+b). Y (a,b) E C X G ,
We begin with the following result.
PROPOSITION 3.1. GA is a complete hypergroup iff mily of cosets of A .
G
is partitioned by the fa-
PROOF. If { A + is a partition of G, then by REMARK 3.1, the hyperproducts of two elements are pair-wise disjoint sets, thus GA is complete. Similariy, we may show the converse. As
an immediate consequence, we deduce
COROLLARY 3.1.
If Gis a finite g r o u p and GAis complete, then
1Al I I G I .
We can characterize the complete commutative A-hypergroups of length 2, with the
PROPOSITION 3.2.
If A = {a,bl, then Gp, is complete iff
2 a = 2 b
.
Commutative Finite A-Hypergroups of Length Two
149
PROOF. Suppose 2a = 2b. We show that G is partitioned by the family IA+g} gEG' In fact, g p g', such that {A + g}nIA + g') 7 $ ! l there , are two possib'(g,g')EG X G, (i) a + g = b + g'; bilities : (ii) b + g = a + g'. In the first case, a+g = b+g'+ 2a+g = a+b+g' +2b+g = a+b+g' + b+g = a+g' and then A+g = A+g'. The other case brings the same result. By PROPOSITION 3.1, Gfiis complete. Conversely, let G, be complete. From PROPOSITION 3.1, {A+g}9EG is a partition of G. Consider the cosets A+a = {2a, a+b}, A+b = ta+b, Zb}. Since {A + a}n{:A + b} $ Q , it follows that 2a = 2b. Observe that if G is a finite group, then, by COROLLARY 3.1, I G I must be even.
REMARK 3.2. bhEN+t-{l}, V(xl, In fact we have
n
4-q1-J
=
..., xn)€G",
x, + A + x2 +
...
3 g e G such that8Tqxi 1- I
+ xn-l + A + + )
...
+ xn). vity) = (n-l)A + (xl + In the rest of the paper, consider A-hypergroups of length It is easy to see that VnENI', we obtain nA = {pa + ub/O
5 p(n,
0 (v5nJ
p
+v
=
n
=
(n-l)A+g.
=
(by commutati-
2.
Put A =(a,b}.
>.
The number of distinct elements of nA depends on the order of (a-b). In fact, there i s the following result, where O(a-b) denotes the order of (a-b), and n
PROPOSITION 3.3.
If O(a-b)
=
n, then (mAl = m+l Ym
PROOF. If O(a-b) = n, then na= nb, and so InAlin. Now, we prove that for every {x,y}c nA - {na}, such that x = ua + vb, y = ra + sb, u f r, we have x f y. If u . > r then u-r = s-v = k
LEMHA 3.1. If Gf(a-b) PROOF. Let m
=
=
n a n d nA = (n-l)A, then mA
=
nA vm >n.
n + q. The result may be obtained by induction on q .
LEMHA 3.2. If O(a-b)
=
n a n d nA # (n-l)A, then nA n(n-1)A
=
@.
PROOF. From PROPOSITION 3.3., we have lnAl = I(n-1)Al = n. If xEnAn(n-l)A, such that x = ua+vb = ra+sb and l r, then set u = r+q. We have r+q+v = r+s+l, hence v = s+l-q and ra+qa+sb+(l-q)b = ra+sb, that is
M.De Salvo
150
(q-1)b. Note that ,n'q'l since q = u-r. If q = 1 then a = 0 and nA = (n-l)A. If q = n then na = (n-l)b, hence, since na = nb, we have b = 0 and nA = (n-l)A. Let q & 1 and q f n. We have: ( u ) qa = (q-1)b. Set t = n-q and add to (a) all the elements pa + vb such that P + U = t. After (t+l) additions, we have: (1) na = ta+(q-1)b (2) (n-l)a+b = (t-l)a+qb
qa
=
=
................
(t+l) qa+tb = (n-1)b. From (l), since na = nb, we have: ( Y ) (n-q+l)b = (n-q)a. Now, add to (Y) all the elements pa + 6b such that After (q-1) additions, obtain: (1)' (n-q+l)b+(q-l)a = (n-1)a (2)' (n-q+Z)b+(q-2)a = (n-2)a+b
p+d =
q-1 and n-q+l+p
................
(q-1)' (n-l)b+a = (n-q+l)a+(q-2)b. From (l), (2), (t+l), (l)',
....
3.3. If Q(a-b) (g,g')sG2, g
=
+ g',and
(2)',
.... (q-l)', nA
n,then (mA+g)n(mA+g') for e v e r y m,n-1.
f 0
-+
=
mA+g
(n-l)A
=
mA+g
, for
every
Let m = n+d (dENu(-ll). We have YgEG, :mA + g =(ma+g, (m-l)a+b+g, (d+Z)a+(n-2)b+g, (d+l)a+(n-l)b+gl. Suppose aE(mA+g)n (mA+g'). There are integers r,s,v,w such that a = ra+sb+g = va+wb+g', r+s = v+w = m, rE(d+l, d+2, m)3v, s E { O , 1, n-113~. Let r > v and r = V+E. Then s = m-r = m-v- - E = W-C, hence va+ca+wb-Eb+g = va+wb+g', that is: (1) ca+g = Eb+g'. Therefore ma+g = [E+(m--E)] a+g = ~a+(m-~)a+g= ~b+(m-~)a+g'= (m-E)a+Eb+g'. Similarly, using ( E ) , we obtain the following, where k = n-E: (m-l)a+b+g = (m-e-l)a+(E +l)b+g'
PROOF.
....
....
......
.............
[ m-(k-l)] a+(k-l)b+g (m-k)a+kb+g = ma+g'
=
(d+l)a+(n-l)b+g'
.............
[m-k-k -l)] a+ [k+(~-l)] b+g Whence, mA+g = mA+g'.
=
(m-E+l)a+(E-l)b+g'
Now we come the main theorem of this section.
T H E O W 3.1.
If A
=
(a,bl a n d O(a-b)
=
n, then Gn is a n-hypergroup.
PROOF. From PROPOSITION 3.3 we have Vh < n-1, [hAl = h+l and Ek, n-I, IkA( = n. The following cases are possible: (i) nAn(n-l)A 9; (ii) nAn(n-l)A = 9. In the first case, from LEMMAS 3.1, 3.2, we have mA = (n-l)A m,n-1. So, for every hyperproduct P of k elements with k l n , there exists g € G such that But, from LEMMA 3.3, the familyF= {(n-1)A+gIgEG is a partition P = (n-l)A+g.
+
Commutative Finite A-Hypergroups of Length Two
151
of G . S i n c e GA i s r e g u l a r , f o r e v e r y h y p e r p r o d u c t Q of s e l e m e n t s w i t h s < n, 3 g ' E G s u c h t h a t Q C ( n - l ) A + g ' , whence t h e s e t s of , i a r e t h e c l a s s e s mod. * Thus, s i n c e b k E G , I(n-I)A+gl = n, we deduce t h a t GA i s a n-hypergroup. I n t h e second c a s e , set O ( a ) = q . C l e a r l y , q 1, s i n c e nA $ (n-1)A. We have (n-l)A = (n-l+q)Aj s o , e v e r y h y p e r p r o d u c t of k e l e m e n t s w i t h k F n i s of t h e form hA+g where g € G and n - l L h l n - Z + q . But, f o r r e g u l a r i t y , a l l a f o r e s a i d g€G. A s r e g a r d s t h e h y p e r p r o d u c t s c a n be c o n s i d e r e d of t h e form (n-Z+q)A+g, r e m a i n i n g h y p e r p r o d u c t s , e a c h of them i s c o n t a i n e d i n (n-Z+q)A + g ' , f o r some g ' . By LEEINA 3.3, we can s a y t h a t G i s p a r t i t i o n e d i n t h e c l a s s e d mod. Ri' by t h e f a m i l y {(n-2+q)A+g} S i n c e 1 (n-Z+q)A+gl = n p g e G, t h e theorem i s gEG proved.
'FA
+
.
The theorem l e a d s t o t h e f o l l o w i n g c o r o l l a r y .
COROLLARY 3.1. -b).
I f G i s a f i n i t e g r o u p and A
4. FINITE A-HYPERGROUPS FROM crcm
=
Ia,bl,
t h e n IGA/R*
I
=
lGl,b(a-
GROUPS OR GROUPS OF ORDER LESS THAN
iz.
Now, we f i n d ( t o w i t h i n a n isomorphism) a l l A-hypergroups of l e n g t h 2, cons t r u c t e d from f i n i t e a b e l i a n g r o u p s of o r d e r t , when t h e g r o u p i s c y c l i c o r t < 1 2 . I n [61 t h e f o l l o w i n g s t a t e m e n t s a r e proved: ( 4 . 1 ) Let a : G --GI b e a n isomorphism o f groups arid A a non-empty s u b s e t of G . I f A ' = a (A), t h e n GA = C A I . In particular: (4.2)
I f a i s a n automorphism o f t h e g r o u p G , t h e n G A " G , ( A ) for e v e r y A € P ( G ) -
4).
F o r l a t e r u s e , we r e c a l l , from [ l ] , : ( 4 . 3 ) I f F i s t h e f a m i l y o f complete n-hypergroups, o f order nh, t h e n t h e r e are ( t o w i t h i n a n isomorphism) a s m a n y h y p e r g r o u p s i n p , a s the p a i r - w i s e non-isomorphic g r o u p s o f order h . REMARK 4.1. I n o r d e r t o g e n e r a t e ( t o w i t h i n a n isomorphism) a l l A-hypcrgroups of f i n i t e o r d e r t , by ( 4 . 1 ) , i t i s s u f f i c i e n t t o c o n s i d e r t h e A-hypergroups c o n s t r u c t e d froin t h e r e p r e s e n t a t i v e s of t h e c l a s s e s o f isomorphism of groups of o r d e r t . We b e g i n w i t h a n u s e f u l lemma!
LEMMA 4.1. I f A = {a,b}, A ' = {c,dlare s u b s e t s o f a f i n i t e g r o u p G such t h a t G A B, t h e n Gda-b) = O ( c - d ) .
GA
152
M.De Salvo
PROOF.
C
If Q(a-b) f Q(c-d), t h e n from COROLLARY 3.1, i t f o l l o w s t h a t l C f i / ~ * GAIlOx , a c o n t r a d i c t i o n . I . But GA ^. G i m p l i e s GA/D'
1 d
lGAl/Rx
Now we come t o t h e i m p o r t a n t theorem:
THEORM 4.1. Let A = I a , b l , A ' = I c , d l b e two subsets o f a f i n i t e g r o u p G such 3 ( x , y ) € G X G such that A + x = A ' + y , then CA = GAt t h a t Q(a-b) = O(c-d).
.
rf
PROOF. Suppose O(a-b) = O ( c - d ) = n. Let n = 2. From THEOREM 3.1 and PROPOSITION 3.2, i t f o l l o w s t h a t GA and GAI a r e complete, 2-hypergroups. By COROLLARY 3.1, I G J must be a n even number3 s e t IGI = 2k and d e n o t e GA = { a l , ak 9 h+ bk}, G A I = IC,, c k , d, , dk} w i t h ai W b i , ciD"di ViE11, 2,
...,
...,
..., k i .
...,
. ..,
S i n c e 03s i s a s t r o n g l y r e g u l a r e q u i v a l e n c e , we o b t a i n 0 d ai = 0 4 bi and k}. ci = 0 d ' d i P i c { 1 , 2 , IA+ai} 9 G A I = ' i E { , , k],' { A ' i q } where Then we may w r i t e : GA = . i f f i = j j a d a n a l o g o u s l y A ' + ci = A ' + c j i f f 1 = j . A + ai = A + a . I vai€{al., ak>, 3 w € G such t h a t ai = w + x, and t h u s A + a i = A + x + w = A ' + y + w, whence 3 c p € { c 1 , ck} such t h a t A ' + cp = A ' + y + w, and cp i s c l e a r l y unique. I n t h e same way, VciEic, , ck} t h e r e e x i s t s a n unique element a , , d a l , %} such t h a t A ' + c i =. A + 4 So, we may d e f i n e a b i j e c t i o n a1 : G A - + G A I i n t h e f o l l o w i n g way: a , ( a i ) = c j and u l ( b i ) = d j i f A + a i = A ' + c j . L a t e r , we ViiE {l, k} s h a l l show t h a t a1 i s a n isomorphism. Now, l e t n > 2. Then GA and G A I a r e n-hypergroups. We have GA = {A + X I and G A I = x&'G{At + x}. C o n s i d e r A + z C G A . Then 3 t ~ Gsuch t h a t z = x + t , whence A + z = A + x + t = A ' + y + t C G A 1 . Note t h a t y + t = z ' i s t h e o n l y element such t h a t A + z = A ' + z ' j i n f a c t , i f t h e r e e x i s t s w # z ' such t h a t A + z = A ' + w, t h e n i t f o l l o w s t h a t A ' + z ' = A ' + w, t h a t i s I c + z ' , d + z ' } = { c + w,d + w l , whence c + z ' = d + w and d + z ' = c + w . Hence z ' = d + w - c , d + d + w - c = c + w, and so, 2d = 2c, a c o n t r a d i c t i o n , s i n c e O(c-d) > 2 . I n a similar way, Y{A' + v } c G# t h e r e e x i s t s a n o n l y s such t h a t A + s = such t h a t V X E G A , = A ' + v . T h e r e f o r e , we may d e f i n e a b i j e c t i o n a 2 : GA -->GAI L a t e r on, t h e b i j e c t i o n s u 1 and a2 w i l l be deaz (x) = y i f f A + x = A ' + y . noted by t h e same l e t t e r a Now, we prove t h a t a, and a2 s a t i s f y t h e f o l l o w i n g p r o p e r t y : ( A ) V(X,Y)E G X G , a ( x + y ) = a ( x ) + y = x + a ( y ) . I f a ( x ) = x' and a ( y ) = y ' , t h e n we have A + x = A ' + x' and A + y = A ' + + y ' , whence A + x + y = A ' + x ' + y = A ' + x + y ' . So, u ( x + y ) = x ' + y = a ( x ) + y and u ( x + y ) = x + y ' = x + a ( y ) . A t l a s t , u s i n g ( A ) , we prove t h a t a1 and uZ a r e isomorphisms: we have V ( z , v ) e G X G w i t h a ( z ) = w, a(v) = u, a(z)bu(v) = A ' + u ( z ) + u ( v ) = A ' + w + u = A + z + u and a ( z 4 v ) = a (A + v + + z ) ; l e t y E u ( A + v + z ) , t h e n 3pEA such t h a t y = a ( p + v + z ) = p + z + a ( v ) = = p + z + uEA + z + uj 1 e t p e . A + z + u , t h e n 3pEA such t h a t p = p + z + u = = p + z + u ( v ) = cx(p + z + v ) ~ a ( A+ z + v ) . T h i s completes t h e p r o o f .
0
...,
d
...,
...,
...,
...,
...,
.
.....,
.
As a consequence of t h e p r e c e d i n g theorem, w e o b t a i n t h e
COROLLARY 4.1.
I f G i s a f i n i t e gr'oup o f ordwm:andIAI = 2, then ( t o within
an
Commutative Finite A-Hypergroups of Length Two
153
isomorphisml t h e number of A-hypergroups. constructed f r o m G , i s s t r i c t l y l e s s than m. PROOF. From THEOREM 4.1, it is sufficient to consider the A-hypergroups with { O t . In fact, for each Ia,b)CG - to}, 3(x,y)EG X G such A = :O,x} and xEG that {a,b} = {O,x} + y, where x = b - a and y = a.
-
We require some lemmas.
The case d = m is trivial. Let x~(m/d)G, m # d j then 3 a ~ 5 s u c hthat whence dx = ma = 0 and O(x)ld. Suppose that O ( x ) = p d j then 3t such that d = pt and px = p(m/d)a = (d/t)(m/d)a = (m/t)a = 0, a contradiction Therefore xeI(d). Conversely, if xeI(d), then x gesince O(a) = m >(m/t). nerates the only subgroup of order d. So, 3as5 such that x = (m/d)a. This completes the proof. PROOF.
+
x = (m/d)a,
LEMMA 4.3. Let t h e n GA- = GA
. j
G - 2 , a n d denoteAi = IO,xi>~qxiEG-{OL.If
o(xi)
= O(Xj)
=
d
1 PROOF. From LEMMA 4 . 2 , it follows that 3 ( q , aj ) E G X G such that xi = (m/d)ai and xj = (m/d)aj. If a is the automorphism of G such that a (ai) = aj , then we = (m/d)a(ai) = (m/d)aj = xj and s o , a ( A i ) = Aj. Therehave a ( x i ) = a((m/d)ai) fore, using ( 4 . 2 ) , the lemma is proved.
By means of LEMMAS 4.1, theorem
4.2,
4.3, and COROLLARY 4.1, we have the following
THEOREM 4.2. If G i s a f i n i t e c y c l i c g r o u p o f order m . t h e n t h e r e a r e ( t o w i t h i n an isomorphisml a s m a n y A-hypergroups of l e n g t h 2 , constructed f r o m G , as d i v i s o r s o f m. e x c e p t 1. As an immediate consequence, we state:
COROLLARY 4.2. If G is a f i n i t e g r o u p of p r i m e o r d e r , t h e n t h e r e e x i s t s ( t o w i t h i n a n isomorphisml e x a c t l y one .A-hypergroup o f l e n g t h 2 . constructed from G. Now, we are going to find the number of A-hypergroups of length 2, constructed from abelian groups of order t less than 1 2 . The cases when t is prime or tE (6,101 lead respectively to COROLLARY 4 . 2 and THEOREM 4.2. The cases t = 4, t = 8, t = 9, remain to be studied. We begin with a lemma, where mZ, denotes the direct sum Z, e Z, e Z,, with m summands.
...
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154
LEPW 4.4- I f G e m Z ,
.
t h e n t h e r e e x i s t s ( t o w i t h i n an Isomorphisml e x a c t l y One ~ - ) j y p e r g t a o u p o f l e n g t h 2. constructed from G , G A . Moreover, GA i s a complete h y p e r g r o u p and G A I B * = (m-l)z,. PROOF. C l e a r l y , V X E G - { O } , O(x) = Z j t h e r e f o r e , f o r e v e r y AEP(G) such t h a t I A l = 2, GA i s a complete 2-hypergroup. By COROLLARY 4.1, i t i s s u f f i c i e n t t o c o n s i d e r t h e A-hypergroups of t h e For each j € G - { O ] , d e n o t e by j t h e form G A i w i t h Ai = { O , i } and i E G - [ O } . c l a s s mod. of which j i s t h e r e p r e s e n t a t i v e . I f A = { O , i ) , t h e n we have 5 = A + j = { j , i + j l , whence 25 = A = 6 j t h e r e we havc t h a t G A / R X e ( m - I ) Z , . f o r e , t a k i n g i n t o account t h e COROLLARY 3.1, L a s t l y , by COROLLARY 2.1 of [4],, t h e s t a t e m e n t of t h e lemma. I n t h e r e s t , Vn€N+c-, we s h a l l d e n o t e Z,
= {O,
= 4. There a r e two c a s e s :
1,
...,
n-l}
. Now,
l e t IGI
=
( i ) G=Z,j ( i i ) G = 2, Q Z,. I f G=Z,, t h e n , u s i n g THEOREM 4 . 2 , t h e r e a r e two A-hypergroups G A , GAI, where we may choose A = { O , l } , A ' = {0,2}. Note t h a t , as a consequence of PROPOSITION 3.2, GAt i s complete. I n t h e c a s e ( i i ) , by (3.3) and LEMMA 4.4, we o b t a i n hypergroups which a r e i s o m o r p h i c t o G,, Now., l e t ( G I = 8. By c o m m u t a t i v i t y , t h e r e a r e t h r e e c a s e s : ( I ) G "Zsj ( 1 1 ) G=Z,eZ,cZ,; (111) c = z,.* z,
.
.
( I ) By THEOREM 4.2, w e o b t a i n t h r e e A-hypergroups: GA, , GA, , GA,, where, a c c o r d i n g t o LEMIYA 4.3 and COROLLARY 4.1, we may suppose A, = { O , l ) , A, = IO,2), A, = {0,41. Note t h a t GA = w(GA ), Gn, i s a 4-hypergroup and CA, i s a complete 2-llyz,. p e r g r o u p . Moreoier, i t i s easy t o s e e t h a t G~,/R*= ( 1 1 ) By LEMMA 4.4, t h i s c a s e l e a d s t o a n o n l y A-hypergroup, p l e t e and Gn/". "Z, Z,, whence GA f: Gq,
.
Q
(111) L e t G
[a,,
G,.
G i i s com-
..., a,}
where V i e Z , , a ; = ( 0 , i ) and sics = ( 1 , i ) . If we c o n s i d e r t h e sets {a,, a k l w i t h O ( a k ) = 2, t h e n we o b t a i n h y p e r g r o u p s , e a c h of which, by ( 4 . 3 ) , i s i s o m o r p h i c t o G i o r t o GA,. T h e r e f o r e , u s i n g COROLLARY 4.1, t h e two f o l l o w i n g hypergroups remain t o be c o n s i d e r e d : G4 , G C , , where C, = {a, , a , } , C, = {a, , a s } .
V a i E G - {a,}
=
#fa;)
a,,
€{Z,41.
But t h e b i j e c t i o n ad
G,, ^. have a, ,a, then
a3
a2
a,
'
a,
a, a, a, a,
i s as isomorphism and s o ,
GC,. S i n c e G C , and Gn, a r e 4-liypergroups, t h e y could be i s o m o r p h i c . Ye C, = { a ,a,> , A, = t0,21 E(G,-,) = t% ,a,>, E ( G k , ) = t 0 , 6 1 , w ( G C I ) = [a,, ,a,} , w(GA, ) = t0,2,4,61. If t h e r e e x i s t s a n isomorphism 6 : Gc, 3 Gh 6 ( E ( G c ) ) = E(GA ), whence we have two p o s s i b i l i t i e s :
(j) 6 ( a , ) = 0,
&(a,1
= 6;
Commutative Finite A-Hypergroups of Length Two
155
I n t h e f i r s t c a s e , 6( a, *C ' a o ) = 6 ( {a, ,a,} ) = IO, 6(a,.) 1 and 6(a, ) *4 6((a0 ) = = 0 $ 0 = (0,2}, whence &(a,) = 2 and s i n c e 6 ( u ( G c 1 ) ) = i t follows = {0,2) t h a t d a , ) = 4. Then &(a, 5 1 a , ) = 6 ( I a , , a , } ) = { 0 , 2 } . But 67:j)i45(a4)
i f f 6 ( % ) € { 0 , 4 , = 6(Ia,,a2} ), which i s a c o n t r a d i c t i o n . I n a similar way, a l s o t h e second c a s e l e a d s t o a c o n t r a d i c t i o n , hence we niay s a y t h a t t h e r e e x i s t ( t o w i t h i n a n isomorphism) e x a c t l y 5 commutative A-hypergroups o f o r d e r 8 and l e n g h t 2 : GA, , G A J ~ G A ~ ,Gii, G q
-
F i n a l l y , we examine t h e c a s e [ G I or (1) G=Z, (2) G = Z , s 2,.
=
9. We have:
( 1 ) Reasoning a s i n t h e p r e c e d i n g c a s e s , we o b t a i n two hypergroups Cs, where we may choose S, = t O , l } and S, = { O , 3 } . ( 2 ) L e t G = {x,, x , , i+s = ( 2 , i ) . We have
vi
, Cs,
...,
xs}, where Y i E Z , , xi = (O,i), x i + , = ( l , i ) , k 0, O ( x i ) = 3. For COROLLARY 3 . 1 , i t i s s u f f i c i e n t t o
c o n s i d e r two hypergroups: GT, GV w i t h T = {so,%}, V = (s ,q}. It i s e a s y t o s e e t h a t u(GT) = {x, ,x,,x,} , C G , ( x , ) = is, , x b , x 5 1 , CG T (x,) = { x s , x 7,x81 and w ( G , ) = { \ ,ss, x 7 1 , C,,(x,) = {xi,x, , x s l CGV(sz) = {q ,x,,s& . I f t h e r e e x i s t s a n isomorphism f : GT -->Gv, t h e n V x E G , f ( C G , ( s ) ) = cG,,(f(x)) and f(w(GT ) ) = = w(Gy). T h e r e f o r e , we must have f ( { x , ,x,+,&} ) = t s , , s , , x , } o r f({x,,x,,x,} ) = But, i n e v e r y c a s e , w e o b t a i n f ( x , s,) t f ( s , ) ?1 f ( x , 1 . Thus, = {x,,xb,xs}. GT i s n o t i s o m o r p h i c t o G v . S i n c e Gs, i s a 3-hypergroup, i t c o u l d be isouiorp h i c t o G T o r t o G,. I f g : Gs, -> GT i s a n isomorphism, t h e n , r e a s o n i n g as b e f o r e , we always f i n d g ( 1 $2 1 ) f g ( 1 ) 1 g ( 1 ) o r g ( 2 $22) f g ( 2 ) g(2). So, G s , i s n o t i s o m o r p h i c t o GT, and i n a similar way, i t i s p o s s i b l e t o s e e t h a t GS2 i s n o t i s o m o r p h i c t o C,.
,
I n t h e f o l l o w i n g t a b l e , we l i s t a l l d i f f e r e n t commutative A-liypergrou&w o f l e n g t h 2, of e a c h o r d e r from 2 t o 11:
[ I ] P. CORSINI: [ * I , I p e r g r u p p i serniregolari e r e g o l a r i , Rend. Sem. Mat. Univ. P o l i t e c . T o r i n o , v o l . 40, (19821, 35-40; [ * I 2 R e c e n t i r i s u l t a t i i n teoria d e g l i ipergruppi, B.U.M.I., 2-A, (19831, 133-138; [*I, Prolegogorneni a l l a t e o r i a d e g l i i p e r g r u p p i , Q u a d e r n i d e l l ' I s t i t u t o d i Matematica, I n f o r m a t i c a e S i s t e m i s t i c a d e l l ' U n i v e r s i t 8 d i Udine ( 1 9 8 6 ) . [ Z ] P. CORSINI and G. ROMEO, Hypergroups completes et 7 - g r o u p o i d e s , A t t i Convegno s u S i s t e m i B i n a r i e l o r o A p p l i c a z i o n i , Taormina I t a l y (1978), 129-146. [3] F. DE MARIA, G r u p p i s e l e z i o n i d i i p e r g r u p p i , A t t i Sem. Mat. F i s . Univ. Mo-
M.De Salvo
156
dena, XXX, (1981), 76-82. [ 4 ] M. DE SALVO:[.],Su l e p o t e n z e ad esponente i n t e r o in u n i p e r g r u p p o e g l i r-ipergruppi, Riv. Mat. Univ. Parma ( 4 ) 11 (19851, 409-421;['], Nuovi riXXXIV, s u l t a t i s u g l i (H,G)- i p e r g r u p p i , A t t i Sem. Mat. F i s . Univ. Modena; (1985-86), 1-14; [ - ] , I p e r g r u p p i f i n i t i di I u n g h e z z a c o s t a n t e , Rend. 1st. Cornbardo Accad. S c i . L e t t . Sez. A, 120 (1986), 41-56. [ 51 M. KOSKAS, Groupoides, demi-hypergroupes et h y p e r g r o u p e s , J . Math. P u r e s e t Appl. 49 (1970), 155-192. [ 6 ] T . VOUGIOUKLIS, G e n e r a l i z a t i o n of P- h y p e r g r o u p s , t o a p p e a r i n Rend. Circol o Mat. d i Palermo.
147
COMMUTATIVE FINITE A-HYPERGROUPS OF LENGTH TWO
Mario DE SALVO?:Via Palermo, 836 - S c a l a R i t i r o (HE), I t a l y
1. INTRODUCTION A h y p e r g r o u p o i d H = < H , o > i s a non-empty s e t H cquipped w i t h a hypcroo ( i . e . a mapping whose domain i s H X H and whose r a n g e i s t h e s e t of non-empty s u b s e t s of H ) . If A 5 H, B 2 H, t h e n A 0 B d e n o t e s u{a o b/a E A , b E n ] . A hypergroupoid H i s c a l l e d h y p e r g r o u p i f t h e h y p e r o p e r a t i o n s a t i s f i e s t h e f o l l o w i n g axioms : ( i ) ( x o y ) ~z = x . ( y o z ) , V ( X , Y , Z )E H3 i ( i i ) x o H = H n x = H, ~ E H . We 'say t h a t a hypergroupoid H h a s l e n g t h n , i f F ( x , y ) E H2, 1x0 yI = 11. I n [ 6 ] , T. Vougiouklis i n t r o d u c e d a new class of hypergroups, c o n s t r u c t c d from o r d i n a r y g r o u p s as f o l l o w s : ~ hyl e t C = < C,*> be a group and AC G, A f flj wc s h a l l c a l l A - I i y p e r g r o u ~tlic p e r g r o u p GA=
ration
k
: (x,y)-+
x . A-y.
S i n c e V ( s , y ) E G X G, Ix t\ yI = 1A1, t h c s c hypcrgroiips have l e n g t h I h l . Throughout t h e paper, we c o n s i d e r A-hypergroups of l e n g t h 2, c o n s t r u c t c d from a b e l i a n g r o u p s . These hypergroups a r e s t u d i e d and t h e i r number i s d e t e r mined when G i s c y c l i c or / G I < 1 2 .
2. NOTATIONS AND RECALLS We r e c a l l some d e f i n i t i o n s and n o t a t i o n s t y p i c a l of hypergroups. L e t A be a non-empty s u b s e t o f a hypergroup H; A i s c o m p l e t e i f V n E Nit and V(x,, x , ) E H" such t h a t O& x j n A 4 0 then -u; L A .
...,
OA 1-1
b8J
We d c n o t e by CH(A) t h e complete c l o s u r e of A, i . e . t h e i n t e r s e c t i o n of all s u b s e t s of H which are complete and c o n t a i n A , L e t R$ d e n o t e t h e t r a n s i t i v e c l o s u g e of t h e b i n a r y r e l a t i o n DH, d e f i n e d i n
t h e f o l l o w i n g way: xRHy i f f t h e r e e x i s t nEN'r and
$6
(21,
Research supported by the G.N.S.A.G.A.
..., z , ) E H "
of the C.N.R.
such t h a t { x , y } ~o~
(Italy).
zi.
M.De Salvo
I48
It is known that the relation x$ y iff xt$(ty)) is an equivnlrncc and moreover C,, = DA ( [ S I). Let 0 : H-+H/Da denote the canonical projection. If H is a hypergroup, then H/D$ is a group. Hence, we define the core of H, w(H), as the kernel of $.
H is said to be a n-hypergroupiff Vx€ H, IC, (x) 1
= n. H is said to be completeiff b'(x,y)~H', x o y =C,(xoy). Let H =
3. A-HYPERGROUPS CONSTRUCTED FROM ABJXCAN GROUPS In the following let G be an abelian group. We shall u s e the additive notation; s o , + will denote the operation in the group, 0 will be the unit element and -a the inverse element of a. Clearly, if G is abelian, then GA is commutative. It is known (see [ 61) that GA is regular and E ( G A ) = -A = [-a/aeA REMARK 3.1. The hyperproducts of two elements in G A may be defined as fo lows: O b x = A + X; VXEG, a '? b = 0 $! (a+b). Y (a,b) E C X G ,
We begin with the following result.
PROPOSITION 3.1. GA is a complete hypergroup iff mily of cosets of A .
G
is partitioned by the fa-
PROOF. If { A + is a partition of G, then by REMARK 3.1, the hyperproducts of two elements are pair-wise disjoint sets, thus GA is complete. Similariy, we may show the converse. As
an immediate consequence, we deduce
COROLLARY 3.1.
If Gis a finite g r o u p and GAis complete, then
1Al I I G I .
We can characterize the complete commutative A-hypergroups of length 2, with the
PROPOSITION 3.2.
If A = {a,bl, then Gp, is complete iff
2 a = 2 b
.
Commutative Finite A-Hypergroups of Length Two
149
PROOF. Suppose 2a = 2b. We show that G is partitioned by the family IA+g} gEG' In fact, g p g', such that {A + g}nIA + g') 7 $ ! l there , are two possib'(g,g')EG X G, (i) a + g = b + g'; bilities : (ii) b + g = a + g'. In the first case, a+g = b+g'+ 2a+g = a+b+g' +2b+g = a+b+g' + b+g = a+g' and then A+g = A+g'. The other case brings the same result. By PROPOSITION 3.1, Gfiis complete. Conversely, let G, be complete. From PROPOSITION 3.1, {A+g}9EG is a partition of G. Consider the cosets A+a = {2a, a+b}, A+b = ta+b, Zb}. Since {A + a}n{:A + b} $ Q , it follows that 2a = 2b. Observe that if G is a finite group, then, by COROLLARY 3.1, I G I must be even.
REMARK 3.2. bhEN+t-{l}, V(xl, In fact we have
n
4-q1-J
=
..., xn)€G",
x, + A + x2 +
...
3 g e G such that8Tqxi 1- I
+ xn-l + A + + )
...
+ xn). vity) = (n-l)A + (xl + In the rest of the paper, consider A-hypergroups of length It is easy to see that VnENI', we obtain nA = {pa + ub/O
5 p(n,
0 (v5nJ
p
+v
=
n
=
(n-l)A+g.
=
(by commutati-
2.
Put A =(a,b}.
>.
The number of distinct elements of nA depends on the order of (a-b). In fact, there i s the following result, where O(a-b) denotes the order of (a-b), and n
PROPOSITION 3.3.
If O(a-b)
=
n, then (mAl = m+l Ym
PROOF. If O(a-b) = n, then na= nb, and so InAlin. Now, we prove that for every {x,y}c nA - {na}, such that x = ua + vb, y = ra + sb, u f r, we have x f y. If u . > r then u-r = s-v = k
LEMHA 3.1. If Gf(a-b) PROOF. Let m
=
=
n a n d nA = (n-l)A, then mA
=
nA vm >n.
n + q. The result may be obtained by induction on q .
LEMHA 3.2. If O(a-b)
=
n a n d nA # (n-l)A, then nA n(n-1)A
=
@.
PROOF. From PROPOSITION 3.3., we have lnAl = I(n-1)Al = n. If xEnAn(n-l)A, such that x = ua+vb = ra+sb and l
M.De Salvo
150
(q-1)b. Note that ,n'q'l since q = u-r. If q = 1 then a = 0 and nA = (n-l)A. If q = n then na = (n-l)b, hence, since na = nb, we have b = 0 and nA = (n-l)A. Let q & 1 and q f n. We have: ( u ) qa = (q-1)b. Set t = n-q and add to (a) all the elements pa + vb such that P + U = t. After (t+l) additions, we have: (1) na = ta+(q-1)b (2) (n-l)a+b = (t-l)a+qb
qa
=
=
................
(t+l) qa+tb = (n-1)b. From (l), since na = nb, we have: ( Y ) (n-q+l)b = (n-q)a. Now, add to (Y) all the elements pa + 6b such that After (q-1) additions, obtain: (1)' (n-q+l)b+(q-l)a = (n-1)a (2)' (n-q+Z)b+(q-2)a = (n-2)a+b
p+d =
q-1 and n-q+l+p
................
(q-1)' (n-l)b+a = (n-q+l)a+(q-2)b. From (l), (2), (t+l), (l)',
....
3.3. If Q(a-b) (g,g')sG2, g
=
+ g',and
(2)',
.... (q-l)', nA
n,then (mA+g)n(mA+g') for e v e r y m,n-1.
f 0
-+
=
mA+g
(n-l)A
=
mA+g
, for
every
Let m = n+d (dENu(-ll). We have YgEG, :mA + g =(ma+g, (m-l)a+b+g, (d+Z)a+(n-2)b+g, (d+l)a+(n-l)b+gl. Suppose aE(mA+g)n (mA+g'). There are integers r,s,v,w such that a = ra+sb+g = va+wb+g', r+s = v+w = m, rE(d+l, d+2, m)3v, s E { O , 1, n-113~. Let r > v and r = V+E. Then s = m-r = m-v- - E = W-C, hence va+ca+wb-Eb+g = va+wb+g', that is: (1) ca+g = Eb+g'. Therefore ma+g = [E+(m--E)] a+g = ~a+(m-~)a+g= ~b+(m-~)a+g'= (m-E)a+Eb+g'. Similarly, using ( E ) , we obtain the following, where k = n-E: (m-l)a+b+g = (m-e-l)a+(E +l)b+g'
PROOF.
....
....
......
.............
[ m-(k-l)] a+(k-l)b+g (m-k)a+kb+g = ma+g'
=
(d+l)a+(n-l)b+g'
.............
[m-k-k -l)] a+ [k+(~-l)] b+g Whence, mA+g = mA+g'.
=
(m-E+l)a+(E-l)b+g'
Now we come the main theorem of this section.
T H E O W 3.1.
If A
=
(a,bl a n d O(a-b)
=
n, then Gn is a n-hypergroup.
PROOF. From PROPOSITION 3.3 we have Vh < n-1, [hAl = h+l and Ek, n-I, IkA( = n. The following cases are possible: (i) nAn(n-l)A 9; (ii) nAn(n-l)A = 9. In the first case, from LEMMAS 3.1, 3.2, we have mA = (n-l)A m,n-1. So, for every hyperproduct P of k elements with k l n , there exists g € G such that But, from LEMMA 3.3, the familyF= {(n-1)A+gIgEG is a partition P = (n-l)A+g.
+
Commutative Finite A-Hypergroups of Length Two
151
of G . S i n c e GA i s r e g u l a r , f o r e v e r y h y p e r p r o d u c t Q of s e l e m e n t s w i t h s < n, 3 g ' E G s u c h t h a t Q C ( n - l ) A + g ' , whence t h e s e t s of , i a r e t h e c l a s s e s mod. * Thus, s i n c e b k E G , I(n-I)A+gl = n, we deduce t h a t GA i s a n-hypergroup. I n t h e second c a s e , set O ( a ) = q . C l e a r l y , q 1, s i n c e nA $ (n-1)A. We have (n-l)A = (n-l+q)Aj s o , e v e r y h y p e r p r o d u c t of k e l e m e n t s w i t h k F n i s of t h e form hA+g where g € G and n - l L h l n - Z + q . But, f o r r e g u l a r i t y , a l l a f o r e s a i d g€G. A s r e g a r d s t h e h y p e r p r o d u c t s c a n be c o n s i d e r e d of t h e form (n-Z+q)A+g, r e m a i n i n g h y p e r p r o d u c t s , e a c h of them i s c o n t a i n e d i n (n-Z+q)A + g ' , f o r some g ' . By LEEINA 3.3, we can s a y t h a t G i s p a r t i t i o n e d i n t h e c l a s s e d mod. Ri' by t h e f a m i l y {(n-2+q)A+g} S i n c e 1 (n-Z+q)A+gl = n p g e G, t h e theorem i s gEG proved.
'FA
+
.
The theorem l e a d s t o t h e f o l l o w i n g c o r o l l a r y .
COROLLARY 3.1. -b).
I f G i s a f i n i t e g r o u p and A
4. FINITE A-HYPERGROUPS FROM crcm
=
Ia,bl,
t h e n IGA/R*
I
=
lGl,b(a-
GROUPS OR GROUPS OF ORDER LESS THAN
iz.
Now, we f i n d ( t o w i t h i n a n isomorphism) a l l A-hypergroups of l e n g t h 2, cons t r u c t e d from f i n i t e a b e l i a n g r o u p s of o r d e r t , when t h e g r o u p i s c y c l i c o r t < 1 2 . I n [61 t h e f o l l o w i n g s t a t e m e n t s a r e proved: ( 4 . 1 ) Let a : G --GI b e a n isomorphism o f groups arid A a non-empty s u b s e t of G . I f A ' = a (A), t h e n GA = C A I . In particular: (4.2)
I f a i s a n automorphism o f t h e g r o u p G , t h e n G A " G , ( A ) for e v e r y A € P ( G ) -
4).
F o r l a t e r u s e , we r e c a l l , from [ l ] , : ( 4 . 3 ) I f F i s t h e f a m i l y o f complete n-hypergroups, o f order nh, t h e n t h e r e are ( t o w i t h i n a n isomorphism) a s m a n y h y p e r g r o u p s i n p , a s the p a i r - w i s e non-isomorphic g r o u p s o f order h . REMARK 4.1. I n o r d e r t o g e n e r a t e ( t o w i t h i n a n isomorphism) a l l A-hypcrgroups of f i n i t e o r d e r t , by ( 4 . 1 ) , i t i s s u f f i c i e n t t o c o n s i d e r t h e A-hypergroups c o n s t r u c t e d froin t h e r e p r e s e n t a t i v e s of t h e c l a s s e s o f isomorphism of groups of o r d e r t . We b e g i n w i t h a n u s e f u l lemma!
LEMMA 4.1. I f A = {a,b}, A ' = {c,dlare s u b s e t s o f a f i n i t e g r o u p G such t h a t G A B, t h e n Gda-b) = O ( c - d ) .
GA
152
M.De Salvo
PROOF.
C
If Q(a-b) f Q(c-d), t h e n from COROLLARY 3.1, i t f o l l o w s t h a t l C f i / ~ * GAIlOx , a c o n t r a d i c t i o n . I . But GA ^. G i m p l i e s GA/D'
1 d
lGAl/Rx
Now we come t o t h e i m p o r t a n t theorem:
THEORM 4.1. Let A = I a , b l , A ' = I c , d l b e two subsets o f a f i n i t e g r o u p G such 3 ( x , y ) € G X G such that A + x = A ' + y , then CA = GAt t h a t Q(a-b) = O(c-d).
.
rf
PROOF. Suppose O(a-b) = O ( c - d ) = n. Let n = 2. From THEOREM 3.1 and PROPOSITION 3.2, i t f o l l o w s t h a t GA and GAI a r e complete, 2-hypergroups. By COROLLARY 3.1, I G J must be a n even number3 s e t IGI = 2k and d e n o t e GA = { a l , ak 9 h+ bk}, G A I = IC,, c k , d, , dk} w i t h ai W b i , ciD"di ViE11, 2,
...,
...,
..., k i .
...,
. ..,
S i n c e 03s i s a s t r o n g l y r e g u l a r e q u i v a l e n c e , we o b t a i n 0 d ai = 0 4 bi and k}. ci = 0 d ' d i P i c { 1 , 2 , IA+ai} 9 G A I = ' i E { , , k],' { A ' i q } where Then we may w r i t e : GA = . i f f i = j j a d a n a l o g o u s l y A ' + ci = A ' + c j i f f 1 = j . A + ai = A + a . I vai€{al., ak>, 3 w € G such t h a t ai = w + x, and t h u s A + a i = A + x + w = A ' + y + w, whence 3 c p € { c 1 , ck} such t h a t A ' + cp = A ' + y + w, and cp i s c l e a r l y unique. I n t h e same way, VciEic, , ck} t h e r e e x i s t s a n unique element a , , d a l , %} such t h a t A ' + c i =. A + 4 So, we may d e f i n e a b i j e c t i o n a1 : G A - + G A I i n t h e f o l l o w i n g way: a , ( a i ) = c j and u l ( b i ) = d j i f A + a i = A ' + c j . L a t e r , we ViiE {l, k} s h a l l show t h a t a1 i s a n isomorphism. Now, l e t n > 2. Then GA and G A I a r e n-hypergroups. We have GA = {A + X I and G A I = x&'G{At + x}. C o n s i d e r A + z C G A . Then 3 t ~ Gsuch t h a t z = x + t , whence A + z = A + x + t = A ' + y + t C G A 1 . Note t h a t y + t = z ' i s t h e o n l y element such t h a t A + z = A ' + z ' j i n f a c t , i f t h e r e e x i s t s w # z ' such t h a t A + z = A ' + w, t h e n i t f o l l o w s t h a t A ' + z ' = A ' + w, t h a t i s I c + z ' , d + z ' } = { c + w,d + w l , whence c + z ' = d + w and d + z ' = c + w . Hence z ' = d + w - c , d + d + w - c = c + w, and so, 2d = 2c, a c o n t r a d i c t i o n , s i n c e O(c-d) > 2 . I n a similar way, Y{A' + v } c G# t h e r e e x i s t s a n o n l y s such t h a t A + s = such t h a t V X E G A , = A ' + v . T h e r e f o r e , we may d e f i n e a b i j e c t i o n a 2 : GA -->GAI L a t e r on, t h e b i j e c t i o n s u 1 and a2 w i l l be deaz (x) = y i f f A + x = A ' + y . noted by t h e same l e t t e r a Now, we prove t h a t a, and a2 s a t i s f y t h e f o l l o w i n g p r o p e r t y : ( A ) V(X,Y)E G X G , a ( x + y ) = a ( x ) + y = x + a ( y ) . I f a ( x ) = x' and a ( y ) = y ' , t h e n we have A + x = A ' + x' and A + y = A ' + + y ' , whence A + x + y = A ' + x ' + y = A ' + x + y ' . So, u ( x + y ) = x ' + y = a ( x ) + y and u ( x + y ) = x + y ' = x + a ( y ) . A t l a s t , u s i n g ( A ) , we prove t h a t a1 and uZ a r e isomorphisms: we have V ( z , v ) e G X G w i t h a ( z ) = w, a(v) = u, a(z)bu(v) = A ' + u ( z ) + u ( v ) = A ' + w + u = A + z + u and a ( z 4 v ) = a (A + v + + z ) ; l e t y E u ( A + v + z ) , t h e n 3pEA such t h a t y = a ( p + v + z ) = p + z + a ( v ) = = p + z + uEA + z + uj 1 e t p e . A + z + u , t h e n 3pEA such t h a t p = p + z + u = = p + z + u ( v ) = cx(p + z + v ) ~ a ( A+ z + v ) . T h i s completes t h e p r o o f .
0
...,
d
...,
...,
...,
...,
...,
.
.....,
.
As a consequence of t h e p r e c e d i n g theorem, w e o b t a i n t h e
COROLLARY 4.1.
I f G i s a f i n i t e gr'oup o f ordwm:andIAI = 2, then ( t o within
an
Commutative Finite A-Hypergroups of Length Two
153
isomorphisml t h e number of A-hypergroups. constructed f r o m G , i s s t r i c t l y l e s s than m. PROOF. From THEOREM 4.1, it is sufficient to consider the A-hypergroups with { O t . In fact, for each Ia,b)CG - to}, 3(x,y)EG X G such A = :O,x} and xEG that {a,b} = {O,x} + y, where x = b - a and y = a.
-
We require some lemmas.
The case d = m is trivial. Let x~(m/d)G, m # d j then 3 a ~ 5 s u c hthat whence dx = ma = 0 and O(x)ld. Suppose that O ( x ) = p d j then 3t such that d = pt and px = p(m/d)a = (d/t)(m/d)a = (m/t)a = 0, a contradiction Therefore xeI(d). Conversely, if xeI(d), then x gesince O(a) = m >(m/t). nerates the only subgroup of order d. So, 3as5 such that x = (m/d)a. This completes the proof. PROOF.
+
x = (m/d)a,
LEMMA 4.3. Let t h e n GA- = GA
. j
G - 2 , a n d denoteAi = IO,xi>~qxiEG-{OL.If
o(xi)
= O(Xj)
=
d
1 PROOF. From LEMMA 4 . 2 , it follows that 3 ( q , aj ) E G X G such that xi = (m/d)ai and xj = (m/d)aj. If a is the automorphism of G such that a (ai) = aj , then we = (m/d)a(ai) = (m/d)aj = xj and s o , a ( A i ) = Aj. Therehave a ( x i ) = a((m/d)ai) fore, using ( 4 . 2 ) , the lemma is proved.
By means of LEMMAS 4.1, theorem
4.2,
4.3, and COROLLARY 4.1, we have the following
THEOREM 4.2. If G i s a f i n i t e c y c l i c g r o u p o f order m . t h e n t h e r e a r e ( t o w i t h i n an isomorphisml a s m a n y A-hypergroups of l e n g t h 2 , constructed f r o m G , as d i v i s o r s o f m. e x c e p t 1. As an immediate consequence, we state:
COROLLARY 4.2. If G is a f i n i t e g r o u p of p r i m e o r d e r , t h e n t h e r e e x i s t s ( t o w i t h i n a n isomorphisml e x a c t l y one .A-hypergroup o f l e n g t h 2 . constructed from G. Now, we are going to find the number of A-hypergroups of length 2, constructed from abelian groups of order t less than 1 2 . The cases when t is prime or tE (6,101 lead respectively to COROLLARY 4 . 2 and THEOREM 4.2. The cases t = 4, t = 8, t = 9, remain to be studied. We begin with a lemma, where mZ, denotes the direct sum Z, e Z, e Z,, with m summands.
...
M. De Saivo
154
LEPW 4.4- I f G e m Z ,
.
t h e n t h e r e e x i s t s ( t o w i t h i n an Isomorphisml e x a c t l y One ~ - ) j y p e r g t a o u p o f l e n g t h 2. constructed from G , G A . Moreover, GA i s a complete h y p e r g r o u p and G A I B * = (m-l)z,. PROOF. C l e a r l y , V X E G - { O } , O(x) = Z j t h e r e f o r e , f o r e v e r y AEP(G) such t h a t I A l = 2, GA i s a complete 2-hypergroup. By COROLLARY 4.1, i t i s s u f f i c i e n t t o c o n s i d e r t h e A-hypergroups of t h e For each j € G - { O ] , d e n o t e by j t h e form G A i w i t h Ai = { O , i } and i E G - [ O } . c l a s s mod. of which j i s t h e r e p r e s e n t a t i v e . I f A = { O , i ) , t h e n we have 5 = A + j = { j , i + j l , whence 25 = A = 6 j t h e r e we havc t h a t G A / R X e ( m - I ) Z , . f o r e , t a k i n g i n t o account t h e COROLLARY 3.1, L a s t l y , by COROLLARY 2.1 of [4],, t h e s t a t e m e n t of t h e lemma. I n t h e r e s t , Vn€N+c-, we s h a l l d e n o t e Z,
= {O,
= 4. There a r e two c a s e s :
1,
...,
n-l}
. Now,
l e t IGI
=
( i ) G=Z,j ( i i ) G = 2, Q Z,. I f G=Z,, t h e n , u s i n g THEOREM 4 . 2 , t h e r e a r e two A-hypergroups G A , GAI, where we may choose A = { O , l } , A ' = {0,2}. Note t h a t , as a consequence of PROPOSITION 3.2, GAt i s complete. I n t h e c a s e ( i i ) , by (3.3) and LEMMA 4.4, we o b t a i n hypergroups which a r e i s o m o r p h i c t o G,, Now., l e t ( G I = 8. By c o m m u t a t i v i t y , t h e r e a r e t h r e e c a s e s : ( I ) G "Zsj ( 1 1 ) G=Z,eZ,cZ,; (111) c = z,.* z,
.
.
( I ) By THEOREM 4.2, w e o b t a i n t h r e e A-hypergroups: GA, , GA, , GA,, where, a c c o r d i n g t o LEMIYA 4.3 and COROLLARY 4.1, we may suppose A, = { O , l ) , A, = IO,2), A, = {0,41. Note t h a t GA = w(GA ), Gn, i s a 4-hypergroup and CA, i s a complete 2-llyz,. p e r g r o u p . Moreoier, i t i s easy t o s e e t h a t G~,/R*= ( 1 1 ) By LEMMA 4.4, t h i s c a s e l e a d s t o a n o n l y A-hypergroup, p l e t e and Gn/". "Z, Z,, whence GA f: Gq,
.
Q
(111) L e t G
[a,,
G,.
G i i s com-
..., a,}
where V i e Z , , a ; = ( 0 , i ) and sics = ( 1 , i ) . If we c o n s i d e r t h e sets {a,, a k l w i t h O ( a k ) = 2, t h e n we o b t a i n h y p e r g r o u p s , e a c h of which, by ( 4 . 3 ) , i s i s o m o r p h i c t o G i o r t o GA,. T h e r e f o r e , u s i n g COROLLARY 4.1, t h e two f o l l o w i n g hypergroups remain t o be c o n s i d e r e d : G4 , G C , , where C, = {a, , a , } , C, = {a, , a s } .
V a i E G - {a,}
=
#fa;)
a,,
€{Z,41.
But t h e b i j e c t i o n ad
G,, ^. have a, ,a, then
a3
a2
a,
'
a,
a, a, a, a,
i s as isomorphism and s o ,
GC,. S i n c e G C , and Gn, a r e 4-liypergroups, t h e y could be i s o m o r p h i c . Ye C, = { a ,a,> , A, = t0,21 E(G,-,) = t% ,a,>, E ( G k , ) = t 0 , 6 1 , w ( G C I ) = [a,, ,a,} , w(GA, ) = t0,2,4,61. If t h e r e e x i s t s a n isomorphism 6 : Gc, 3 Gh 6 ( E ( G c ) ) = E(GA ), whence we have two p o s s i b i l i t i e s :
(j) 6 ( a , ) = 0,
&(a,1
= 6;
Commutative Finite A-Hypergroups of Length Two
155
I n t h e f i r s t c a s e , 6( a, *C ' a o ) = 6 ( {a, ,a,} ) = IO, 6(a,.) 1 and 6(a, ) *4 6((a0 ) = = 0 $ 0 = (0,2}, whence &(a,) = 2 and s i n c e 6 ( u ( G c 1 ) ) = i t follows = {0,2) t h a t d a , ) = 4. Then &(a, 5 1 a , ) = 6 ( I a , , a , } ) = { 0 , 2 } . But 67:j)i45(a4)
i f f 6 ( % ) € { 0 , 4 , = 6(Ia,,a2} ), which i s a c o n t r a d i c t i o n . I n a similar way, a l s o t h e second c a s e l e a d s t o a c o n t r a d i c t i o n , hence we niay s a y t h a t t h e r e e x i s t ( t o w i t h i n a n isomorphism) e x a c t l y 5 commutative A-hypergroups o f o r d e r 8 and l e n g h t 2 : GA, , G A J ~ G A ~ ,Gii, G q
-
F i n a l l y , we examine t h e c a s e [ G I or (1) G=Z, (2) G = Z , s 2,.
=
9. We have:
( 1 ) Reasoning a s i n t h e p r e c e d i n g c a s e s , we o b t a i n two hypergroups Cs, where we may choose S, = t O , l } and S, = { O , 3 } . ( 2 ) L e t G = {x,, x , , i+s = ( 2 , i ) . We have
vi
, Cs,
...,
xs}, where Y i E Z , , xi = (O,i), x i + , = ( l , i ) , k 0, O ( x i ) = 3. For COROLLARY 3 . 1 , i t i s s u f f i c i e n t t o
c o n s i d e r two hypergroups: GT, GV w i t h T = {so,%}, V = (s ,q}. It i s e a s y t o s e e t h a t u(GT) = {x, ,x,,x,} , C G , ( x , ) = is, , x b , x 5 1 , CG T (x,) = { x s , x 7,x81 and w ( G , ) = { \ ,ss, x 7 1 , C,,(x,) = {xi,x, , x s l CGV(sz) = {q ,x,,s& . I f t h e r e e x i s t s a n isomorphism f : GT -->Gv, t h e n V x E G , f ( C G , ( s ) ) = cG,,(f(x)) and f(w(GT ) ) = = w(Gy). T h e r e f o r e , we must have f ( { x , ,x,+,&} ) = t s , , s , , x , } o r f({x,,x,,x,} ) = But, i n e v e r y c a s e , w e o b t a i n f ( x , s,) t f ( s , ) ?1 f ( x , 1 . Thus, = {x,,xb,xs}. GT i s n o t i s o m o r p h i c t o G v . S i n c e Gs, i s a 3-hypergroup, i t c o u l d be isouiorp h i c t o G T o r t o G,. I f g : Gs, -> GT i s a n isomorphism, t h e n , r e a s o n i n g as b e f o r e , we always f i n d g ( 1 $2 1 ) f g ( 1 ) 1 g ( 1 ) o r g ( 2 $22) f g ( 2 ) g(2). So, G s , i s n o t i s o m o r p h i c t o GT, and i n a similar way, i t i s p o s s i b l e t o s e e t h a t GS2 i s n o t i s o m o r p h i c t o C,.
,
I n t h e f o l l o w i n g t a b l e , we l i s t a l l d i f f e r e n t commutative A-liypergrou&w o f l e n g t h 2, of e a c h o r d e r from 2 t o 11:
[ I ] P. CORSINI: [ * I , I p e r g r u p p i serniregolari e r e g o l a r i , Rend. Sem. Mat. Univ. P o l i t e c . T o r i n o , v o l . 40, (19821, 35-40; [ * I 2 R e c e n t i r i s u l t a t i i n teoria d e g l i ipergruppi, B.U.M.I., 2-A, (19831, 133-138; [*I, Prolegogorneni a l l a t e o r i a d e g l i i p e r g r u p p i , Q u a d e r n i d e l l ' I s t i t u t o d i Matematica, I n f o r m a t i c a e S i s t e m i s t i c a d e l l ' U n i v e r s i t 8 d i Udine ( 1 9 8 6 ) . [ Z ] P. CORSINI and G. ROMEO, Hypergroups completes et 7 - g r o u p o i d e s , A t t i Convegno s u S i s t e m i B i n a r i e l o r o A p p l i c a z i o n i , Taormina I t a l y (1978), 129-146. [3] F. DE MARIA, G r u p p i s e l e z i o n i d i i p e r g r u p p i , A t t i Sem. Mat. F i s . Univ. Mo-
M.De Salvo
156
dena, XXX, (1981), 76-82. [ 4 ] M. DE SALVO:[.],Su l e p o t e n z e ad esponente i n t e r o in u n i p e r g r u p p o e g l i r-ipergruppi, Riv. Mat. Univ. Parma ( 4 ) 11 (19851, 409-421;['], Nuovi riXXXIV, s u l t a t i s u g l i (H,G)- i p e r g r u p p i , A t t i Sem. Mat. F i s . Univ. Modena; (1985-86), 1-14; [ - ] , I p e r g r u p p i f i n i t i di I u n g h e z z a c o s t a n t e , Rend. 1st. Cornbardo Accad. S c i . L e t t . Sez. A, 120 (1986), 41-56. [ 51 M. KOSKAS, Groupoides, demi-hypergroupes et h y p e r g r o u p e s , J . Math. P u r e s e t Appl. 49 (1970), 155-192. [ 6 ] T . VOUGIOUKLIS, G e n e r a l i z a t i o n of P- h y p e r g r o u p s , t o a p p e a r i n Rend. Circol o Mat. d i Palermo.