SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET

Let T(X) denote the semigroup (under composition) of transformations from X into itself. 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 when S(X, Y) is isomorphic to T(Z) for some set Z and prove that every semigroup A can be embedded in S(A(1), A). Then we describe Green's relations for S(X, Y) and apply these results to obtain its group H-classes and ideals.