Special elements of semigroup of n-ary operations

Abstract

Let X be a finite set of m elements. Let On(X) be the set of all n-ary operations on X. On On(X), it is defined operation “+” with (f+g)(x¯)=f(g(x_1),g(x_2),…,g(x_n)), for all f, g ∈ On(X) and x¯=(x1,x2,…,xn) ∈ Xn, so that (On(X), +) is a semigroup. In this paper we give some properties of idempotent elements, regular elements, coregular elements, left zero elements and right identity elements on On(X). Moreover, from this properties, we provide some properties of nontrivial subsemigroups of the semigroup (On(X), +).Let X be a finite set of m elements. Let On(X) be the set of all n-ary operations on X. On On(X), it is defined operation “+” with (f+g)(x¯)=f(g(x_1),g(x_2),…,g(x_n)), for all f, g ∈ On(X) and x¯=(x1,x2,…,xn) ∈ Xn, so that (On(X), +) is a semigroup. In this paper we give some properties of idempotent elements, regular elements, coregular elements, left zero elements and right identity elements on On(X). Moreover, from this properties, we provide some properties of nontrivial subsemigroups of the semigroup (On(X), +).