Regular and unit-regular elements in various monoids of transformations

Abstract

Let [Formula: see text] be the full transformation monoid on a nonempty set [Formula: see text]. An element [Formula: see text] of [Formula: see text] is said to be semi-balanced if the collapse of [Formula: see text] is equal to the defect of [Formula: see text]. In this paper, we prove that an element of [Formula: see text] is unit-regular if and only if it is semi-balanced. For a partition [Formula: see text] of [Formula: see text], we characterize unit-regular elements in the monoid [Formula: see text] under composition. We characterize regular elements in the submonoids [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the equivalence induced by [Formula: see text]. We also characterize unit-regular elements in [Formula: see text], [Formula: see text], and the other two known submonoids of [Formula: see text].