On the separator of subsets of semigroups

Abstract

By the separator $\operatorname{\mathit{Sep}}A$ of a subset A of a semigroup S we mean the set of all elements x of S which satisfy conditions xA⊆A, Ax⊆A, x(S−A)⊆(S−A), (S−A)x⊆(S−A). In this paper we deal with the separator of subsets of semigroups.

In Sect. 2, we investigate the separator of subsets of special types of semigroups. We prove that, in the multiplicative semigroup S of all n×n matrices over a field ${\mathbb{F}}$ and in the semigroup S of all transformations of a set X with |X|=n<∞, the separator of an ideal I≠S is the unit group of S. We show that the separator of a subset of a group G is a subgroup of G. Moreover, the separator of a subset A of a completely regular semigroup [a Clifford semigroup; a completely 0-simple semigroup] S is either empty or a completely regular semigroup [a Clifford semigroup, a completely simple semigroup (supposing ∅≠A≠S)] of S.

In Sect. 3 we characterize semigroups which satisfy certain conditions concerning the separator of their subsets. We prove that every subset A of a semigroup S with ∅⊂A⊂S has an empty separator if and only if S is an ideal extension of a rectangular band by a nil semigroup. We also prove that every subsemigroup of a semigroup S is the separator of some subset of S if and only if $\emptyset \subset \operatorname{\mathit{Sep}}A\subseteq A$ is satisfied for every subsemigroup A of S if and only if S is a periodic group if and only if $A=\operatorname{\mathit{Sep}}A$ for every subsemigroup A of S.

In Sect. 4 we apply the results of Sect. 3 for permutative semigroups. We show that every permutative semigroup without idempotent elements has a non-trivial group or a group with a zero homomorphic image. Moreover, if a finitely generated permutative semigroup S has no neither a non-trivial group homomorphic image nor a group with a zero homomorphic image then S is finite.