CREAT. MATH. INFORM. Volume 27 (2018), No. 1, Pages 79 - 88 Online version at https://creative-mathematics.cunbm.utcluj.ro/ Print Edition: ISSN 1584 - 286X; Online Edition: ISSN 1843 - 441X DOI: https://doi.org/10.37193/CMI.2018.01.11 A note on the morphism theorems for (n, m)−semirings ADINA POP and MONICA LAURAN ABSTRACT. In this paper, some properties of subtractive ideal of (n, m)-semirings are investigated. In ad- dition, we study the morphisms of (n, m)-semirings starting from the definitions given in the case of univer- sal algebras. We will present several theorems of correspondence for sub-(n, m)-semirings, ideals, subtractive ideals that represent the generalization of the morphism theorems of the binary case. 1. I NTRODUCTION Algebraic polyadic structures are applied in many disciplines such as theoretical physics, computer sciences, coding theory, automata theory and other. The concept of n−ary group was introduced by D¨ ornte [2] and developed by E. Post [14] , J. Timms [17] for commutative case. The m−semigroups are studied by F. M. Siosson [16], M. S. Pop [12], A. Pop [8]. I. Purdea [13] and G. Crombez [1] extended the usual ring concept to the case where the underlying group and semigroup is an commutative n−ary group and an m−ary semigroup, respectively. In some recently appeared papers, various authors continue the study of ordinary semigroups introduced by H. S. Vandiver [18] to the case where the underlying commutative additive semigroup and multiplicative semigroup are not binary but an n−ary and one m−ary respectively. The new obtained structures are called (n, m)−semirings [7], [9], [19]. We begin with some preliminaries about the m−semigroups, n−groups, (n, m)− semir- ings and (n, m)−rings. Traditionally in the theory of n−groups we use the following abbreviated notation: the sequence x i ,...,x j is denoted by x j i (for j<i this symbol is empty). If x i+1 = x i+2 = ... = x i+k = x, then instead of x i+k i+1 we write (k) x . The algebra (S, () + ) is called an n−semigroup if for any i ∈{2, 3,...,n} and all x 1 ,...,x 2n−1 ∈ S, the following asso- ciativity laws are true i.e ((x n 1 ) + ,x 2n−1 n+1 ) + =(x i−1 1 , (x i+n−1 i ) + ,x 2n−1 i+n ) + . An n−semigroup (S, () + ) is called n−group if for any i ∈{1, 2,...,n} and all a 1 ,...,a n ∈ S, the equation (a i−1 1 , x, a n i+1 ) + = a i has a unique solution in S. In some n-groups there is an element e ∈ S (called identity or neutral element) such that ( (i−1) e x (n−i) e ) + = x holds for all x ∈ S and for all i ∈{1,...,n}. It is interesting that there are n-groups with two or more neutral elements or which do not contain such elements [2],[14]. From the definition of the n-group (S, () + ) we can see that for every x ∈ S there is only one y ∈ S, satisfying the equation ( (n−1) x y) + = x. This element, denoted by x, so called querelement of x, defines the power x [−1] . W.D¨ ornte [2] proved that in any n-group for all a, x ∈ A; 2 ≤ i, j ≤ n, we have ( (i−2) x x (n−i) x a) + = a and (a (n−j) x x (j−2) x ) + = a. Received: 12.03.2018. In revised form: 13.03.2018. Accepted: 20.03.2018 2010 Mathematics Subject Classification. 20N15, 16Y60, 16Y99, 22A99, 16N99. Key words and phrases. n-group, n-semigroup, (n, m)-semirings, ideal and subtractive ideal of (n, m)-semiring, semiring morfism. 79