Regularity of some invariant transformation semigroups

Abstract

Let [Formula: see text] be a nonempty subset of [Formula: see text]. The invariant subsemigroup of the full transformation semigroup [Formula: see text] is denoted by [Formula: see text] In this paper, we characterize the regularity of the following subsemigroups: [Formula: see text] the complement sets of both [Formula: see text] and [Formula: see text]. These are generalizations of certain subsemigroups of [Formula: see text]. Furthermore, infinitely many regular and nonregular elements of each semigroup are given.