On certain semigroups of transformations with an invariant set

Abstract

Let [Formula: see text] be a nonempty set and let [Formula: see text] be the full transformation semigroup on [Formula: see text]. The main objective of this paper is to study the subsemigroup [Formula: see text] of [Formula: see text] defined by [Formula: see text] where [Formula: see text] is a fixed nonempty subset of [Formula: see text]. We describe regular elements in [Formula: see text], and show that [Formula: see text] is regular if and only if [Formula: see text] is finite. We characterize unit-regular elements in [Formula: see text], and prove that [Formula: see text] is unit-regular if and only if [Formula: see text] is finite. We characterize Green’s relations on [Formula: see text], and prove that [Formula: see text] on [Formula: see text] if and only if [Formula: see text] is finite. We also determine ideals of [Formula: see text] and investigate its kernel. This paper extends several results that have appeared in the literature.