Regularity in semigroups of transformations with invariant sets

Abstract

Let T(X) be the semigroup of all transformations on a set X. For a fixed nonempty subset Y of X, let S(X, Y ) = {α ∈ T(X) : Y α ⊆ Y }. Then S(X, Y ) is a semigroup of total transformations on X which leave a subset Y of X invariant. In this paper, we characterize left regular, right regular and intra-regular elements of S(X, Y ) and consider the relationships between these elements. Moreover, we count the number of left regular elements of S(X, Y ) when X is a finite set. © 2013 Academic Publications, Ltd.