ON THE NILPOTENCY IN SEMIGROUPS

This paper is an extension of the results of papers [3] and [5]. The first three theorems of this paper are extensions of the first three theorems of the paper [3] and the fourth theorem of this paper is an extension of the theorem of paper [5]. We introduce three kinds of nilpotency and consider instead of ideals (which are as a rule used to define various kinds of radicals) more gene rally subsets or subsemigroups of a given semigroup S. Throughout the paper the empty subset 0 is considered as a subsemigroup of the semigroup S. By an ideal we mean a two-sided ideal. If we consider a left ideal or a right ideal we shall indicate it explicitely. The n-semilattice of all left (right) [two-sided] ideals of the semigroup S contains always the empty set 0. We now introduce three kinds of nilpotency and shall study the relations among them. Definition 1. Let S be a semigroup and Ma subset of S. a) Let x be such an element of S for which there exists a positive integer N(x