A study of graph closed subsemigroups of a full transformation semigroup

Let ${T_X}$ be the full transformation semigroup on the set X and let S be a subsemigroup of ${T_X}$. We may associate with S a digraph $g(S)$ with X as set of vertices as follows: $i \to j \in g(S)$ iff there exists $\alpha \in S$ such that $\alpha (i) = j$. Conversely, for a digraph G having certain properties we may assign a semigroup structure, $S(G)$, to the underlying set of G. We are thus able to establish a ``Galois correspondence'' between the subsemigroups of ${T_X}$ and a particular class of digraphs on X. In general, S is a proper subsemigroup of $S \cdot g(S)$.