Regularity and Finiteness Conditions on Transformation Semigroups with Invariant Sets

Let X be a nonempty set and T(X) denote the semigroup of transformations from X to itself under the composition of functions. For a fixed nonempty subset Y of X, let S(X, Y ) = {alpha is an element of T(X) : Y alpha subset of Y }. Then S(X, Y ) is a semigroup of total transformations of X which leave a subset Y of X invariant. In this paper, we characterize coregular elements of S(X, Y ) and give necessary and sufficient conditions for S(X, Y ) to be coregular. Moreover, we study some properties of regularity on S(X, Y ) and give necessary and sufficient conditions for S(X, Y ) to be left regular, right regular, completely regular, intra-regular and directly finite.