Semigroups of transformations whose restrictions belong to a given semigroup

For a set X , denote by T ( X ) the semigroup of full transformations on X . For any subset Y of X and any subsemigroup S ( Y ) of T ( Y ), denote by T S ( Y ) ( X ) the semigroup of all transformations α ∈ T ( X ) such that α | Y ∈ S ( Y ) , where α | Y is the restriction of α to Y . In this paper, we describe the regular elements of T S ( Y ) ( X ) and determine when T S ( Y ) ( X ) is a regular semigroup [inverse semigroup, completely regular semigroup]. With the assumption that S ( Y ) contains the identity id Y , we describe Green’s relations in T S ( Y ) ( X ) in terms of the corresponding Green’s relations in S ( Y ) . We apply these general results to obtain more concrete results for the semigroup T Γ ( Y ) ( X ) , where Γ ( Y ) is the semigroup of full injective transformations on Y . We also discuss generalizations and extensions of the semigroup T S ( Y ) ( X ) .