In this paper we investigate the generalized Hyers-Ulam stability of mappings of m semigroups m 2 N; m 2 into Banach spaces. For m D 3 the results can be found in Amyari and Moslehian [Approximate homomorphisms of ternary semigroups, Lett. Math. Phys. 77 (2006), 1-9] with the mention that they are true in the class of normal m semigroups which is larger than the class of commutative m semigroups. For m D 2 we find certain results of Hyers [On the stability of the linear functional equation,

In this paper, we investigate some properties of the additive and multiplicative idempotents of an (n, m)−semiring. In particular, the research is focused on differences between (n, m)−semirings, n, m ≥ 3 and usual semirings.

In this paper, some properties of subtractive ideal of (n,m)-semirings are investigated. In addition, we study the morphisms of (n, m)-semirings starting from the definitions given in the case of universal 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.

The aim of this paper is to investigate some properties for an ideal radical and an ideal radical boundary in a commutative Hausdorff topological (n, m)−semiring. We are going to give some generalizations of results due to Shum [Shum, K. P, On the boundary of algebraic radicals in topological semigroups, Acta Mathematica Acad. Scient. Hung., 25 (1974), No. (1-2), 15–19], Chow [Chow, H. L., Remarks on boundaries in semigroups , Period. Math. Hungar., 7 (1976), No. 2, 137–139] relative to semigroups and due Maria S. Pop [Pop, M. S., On boundary in topological n-semigroups, Mathematica, 22 (45) (1980), No. 1, 127–130] relative to n-semigroups.

In the paper [Marichal, J.-L. and Mathonet, P., A description of n-ary semigroups polynomial-derived from integral domains, Semigroup Forum, 83 (2011), No. 2, 241–249] the authors provide a complete classification of the nsemigroups, defined by polynomial functions over infinite commutative integral domains with identity (i.e., the n-semigroup structures polynomial derived from integral domains). We remark that some results from that paper can be extended for the n-semigroups polynomial-derived defined over infinite semidomains with zerosumfree element. Furthermore, in this note we give a description of (n, m)- semiring structures defined by polynomial functions over such semidomains.