On certain semigroups of transformations whose restrictions belong to a given semigroup

Let T(X) (resp. L(V)) be the semigroup of all transformations (resp. linear transformations) of a set X (resp. vector space V). For a subset Y of X and a subsemigroup \(\mathbb {S}(Y)\) $\mathbb{S}(Y)$ of T(Y), consider the subsemigroup \(T_{\mathbb {S}(Y)}(X) = \{f\in T(X):f_{\upharpoonright _Y} \in \mathbb {S}(Y)\}\) $T_{\mathbb{S}(Y)}^{}(X)=\{f\in T(X):f_{{\upharpoonright }_Y}\in \mathbb{S}(Y)\}$ of T(X), where \(f_{\upharpoonright _Y}\in T(Y)\) $f_{{\upharpoonright }_Y}\in T(Y)$ agrees with f on Y. We give a new characterization for \(T_{\mathbb {S}(Y)}(X)\) $T_{\mathbb{S}(Y)}^{}(X)$ to be a regular semigroup [inverse semigroup]. For a subspace W of V and a subsemigroup \(\mathbb {S}(W)\) $\mathbb{S}(W)$ of L(W), we define an analogous subsemigroup \(L_{\mathbb {S}(W)}(V) = \{f\in L(V) :f_{\upharpoonright _W} \in \mathbb {S}(W)\}\) $L_{\mathbb{S}(W)}^{}(V)=\{f\in L(V):f_{{\upharpoonright }_W}\in \mathbb{S}(W)\}$ of L(V). We describe regular elements in \(L_{\mathbb {S}(W)}(V)\) $L_{\mathbb{S}(W)}^{}(V)$ and determine when \(L_{\mathbb {S}(W)}(V)\) $L_{\mathbb{S}(W)}^{}(V)$ is a regular semigroup [inverse semigroup, completely regular semigroup]. If \(\mathbb {S}(Y)\) $\mathbb{S}(Y)$ (resp. \(\mathbb {S}(W)\) $\mathbb{S}(W)$ ) contains the identity of T(Y) (resp. L(W)), we describe unit-regular elements in \(T_{\mathbb {S}(Y)}(X)\) $T_{\mathbb{S}(Y)}^{}(X)$ (resp. \(L_{\mathbb {S}(W)}(V)\) $L_{\mathbb{S}(W)}^{}(V)$ ) and determine when \(T_{\mathbb {S}(Y)}(X)\) $T_{\mathbb{S}(Y)}^{}(X)$ (resp. \(L_{\mathbb {S}(W)}(V)\) $L_{\mathbb{S}(W)}^{}(V)$ ) is a unit-regular semigroup.