aa r X i v : . [ m a t h . R T ] M a r Finite Semihypergroups Built From Groups
Stan OnypchukSeptember 7, 2018
Abstract
Necessary and sufficient conditions for finite semihypergroups to be built fromgroups of the same order are established.
Introduction
The theory of hypergroups and semihypergroups was introduced by C.Dunkl [1], R. Jewett [2], and R. Spector [3] and is well developed now.Many examples of finite commutative semihypergroups and hypergroups can be foundin [4]. In [4] there is a precise physical definition of a finite semihypergroup - commuta-tive semihypergroup because this example describes a finite collection of particles whichinteract by colliding.Here is a more suitable example. Let us say we are observing a finite collection of events { e , . . . , e n } , which may be in a cause and effect relationship. To establish that wewill build a cube with frequencies (probabilities) a i,j ( k ) in which each event e k appears(be third) after any sequential pair of events e i and e j in our observed (infinitely long)sequence of events. So, we have a Markov chain of second order. Now assume thatconvolution - e i ∗ e j = P a i,j ( k ) e k - is associative and this semihypergroup (cube) canbe developed from a group. In terms of this example the main result of present articleshows that for such semihypergroups all observed events are not simple but rather area combination of some simple events, evolution of which can be described as a groupproduct for some appropriate group.Let H be a finite semihypergroup with n states e , e , . . . , e n and associative convolutionoperation defined by e i ∗ e j = X k a i,j ( k ) e k ( i, j, k = 1 , , . . . , n ) (1)where a i,j ( k ) > P nk =1 a i,j ( k ) = 1 for each i, j .Let us denotecolumns { a i,j (1) , . . . , a i,j ( n ) } by a i,j matrix with columns { a i, , . . . , a i,n } by A i (matrix of left regular representation)matrix with columns { a ,i , . . . , a n,i } by B i (matrix of right regular representation)and cube with matrices { A , . . . , A n } by C. efinition A semihypergroup H will be said to be derived from a group if any ofthe following two conditions is satisfied:(A) There are only n different rows in all A i and only n different rows in all B i and inany A i rows are linearly independent and in any B i rows are linearly independent.(B) There exists a group G of order n with elements { g , . . . , g n } and measure m on G that e i = m · g i where g i · g j is a group product and e i ∗ e j = ( m · g i ) · ( m · g j )Now let us establish some properties which are shared among all semihypergroups withcondition (A).In matrix terms the convolution operation (1) transforms to a i,j = A i e j (where e j isa column-vector with 1 on j -th position and 0 otherwise). The matrix P c i A i repre-sents a measure P c i e i so P nk =1 a i,j ( k ) A k represents e i ∗ e j . Now it is obvious that theconvolution operation (1) is associative if and only if A i A j = n X k =1 a i,j ( k ) A k (2) Theorem 1.
If a semihypergroup satisfies condition (A) then there exist n differentnon negative numbers among all a i,j ( k ) that P a k = 1 and for each a p from { a , . . . , a n } and each i, j = 1 , . . . , n there exist such k and l that a i,k ( j ) = a p and a l,i ( j ) = a p Let us denote as r i,j j -th row in A i . Then in A A the first row is P r , ( k ) r ,k andbecause of (2) it is P a , r k, . Now because r ,k ( k = 1 , · · · , n ) are linearly independentwe conclude that { r k, } contains the same set of linearly independent rows as { r ,k } andthe first column a , contains the same elements as the first row r , .The second row in A A is P r , ( k ) r ,k = P a , ( k ) r k, . Again we conclude that { r k, } contains the same set of linearly independent rows as { r ,k } and the row r , containsthe same elements as the column a , .Continuing this consideration for all A i A j , we conclude that each row in each A i andeach column in C contains the same elements. Now, if we repeat this process withmatrices B i , we will see that each row in each matrix B i contains the same elements aseach column in C .Now let us establish some properties which are shared among all semihypergroups derivedfrom a group. Corollary 1.