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