Regularity and maximal subsemigroups in semigroups of full transformations with fixed permuting sets

Abstract

Let [Formula: see text] be a nonempty set and [Formula: see text] denote the full transformation semigroup on [Formula: see text]. For a fixed nonempty subset [Formula: see text] of [Formula: see text], let [Formula: see text] where [Formula: see text] is a permutation group on [Formula: see text]. Then [Formula: see text] is a regular submonoid of [Formula: see text]. In this paper, we describe all intra-regular and unit regular elements of [Formula: see text] and give necessary and sufficient conditions for [Formula: see text] to be intra-regular and unit regular. We also count the number of these elements when [Formula: see text] is a finite set. Moreover, we classify the maximal subsemigroups of [Formula: see text] and prove that these maximal subsemigroup coincide with the maximal regular subsemigroups of [Formula: see text] when [Formula: see text] is a finite set.