Semigroups of transformations whose restrictions belong to a given semigroup

Abstract

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