Semigroups of transformations with invariant sets

Let $T(X)$ denote the semigroup (under composition) of transformations from $X$ into itself. For a fixed nonempty subset $Y$ of $X$, let $$ S(X, Y) = \{\alpha\in T(X) : Y\alpha \subseteq Y\}.$$ Then $S(X, Y)$ is a semigroup of total transformations of $X$ which leave a subset $Y$ of $X$ invariant. In this talk, 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 $\mathcal{H}$-classes and ideals. (Joint work with Jintana Sanwong)