Left (Right) Regular Elements of Some Transformation Semigroups

Abstract

For a nonempty set X, let T(X) be the total transformation semigroup on X. In this paper, we consider the subsemigroups of T(X) which are defined by T(X,Y) ={α∈T(X):Xα⊆Y}andS(X,Y)={α∈T(X):Yα⊆Y} where Y is a non-empty subset of X. We characterize the left regular and right regular elements of both T(X,Y) and S(X,Y). Moreover, necessary and sufficient conditions for T(X,Y) and S(X,Y) to be left regular and right regular are given. These results are then applied to determine the numbers of left and right regular elements in T(X,Y) for a finite set X.