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 \(T_{{\mathbb {S}}(Y)}(X)\).

为一组X，表示由Ť（X）上满变换的半群 X。对于任意子集ÿ的 X和任何半群\（{\ mathbb {S}}（Y）\）的Ť（Ý），表示由\（T _ {{\ mathbb {S}}（Y）}（X）\ )所有变换的半群\(\alpha \in T(X)\)使得\(\alpha |_{Y}\in {\mathbb {S}}(Y)\)，其中\(\alpha | _Y\)是\(\alpha \)对Y 的限制。在本文中，我们描述了\(T_{{\mathbb {S}}(Y)}(X)\)并确定何时\(T_{{\mathbb {S}}(Y)}(X)\)是正则半群 [逆半群，完全正则半群]。假设\({\mathbb {S}}(Y)\)包含身份\({{\,\mathrm{id}\,}}_{{\tiny Y}}\)，我们描述格林的在关系\（T _ {{\ mathbb {S}}（Y）}（X）\）在对应的绿色的关系而言\（{\ mathbb {S}}（Y）\）。我们运用这些一般结果以获得关于半群更具体的结果\（T _ {\伽马（γ）}（X）\） ，其中\（\伽玛（Y）\）是满射变换的半群ÿ。我们还讨论了半群的推广和扩展\(T_{{\mathbb {S}}(Y)}(X)\)。