On certain semigroups of transformations whose restrictions belong to a given semigroup (Q6582351)

Let \(T(X)\) be the semigroup of all transformations of a set \(X\). For a subset \(Y\) of \(X\) and a subsemigroup \(\mathbb{S}(Y)\) of \(T(Y)\), consider the subsemigroup \(T_{\mathbb{S}(Y)}(X)=\{f\in T(X):f_{\upharpoonright_Y}\in\mathbb{S}(Y)\}\) of \(T(X)\), where \(f_{\upharpoonright_Y}\in T(Y)\) agrees with \(f\) on \(Y\). In Section 3, the authors give a characterization for the regularity of \(T_{\mathbb{S}(Y)}(X)\) (Theorem 3.3) and for \(T_{\mathbb{S}(Y)}(X)\) to be an inverse semigroup (Theorem 3.4). If \(\mathbb{S}(Y )\) contains the identity of \(T (Y )\), then the authors describe unit-regular elements in \(T_{\mathbb{S}(Y)}(X)\) (Theorem 3.6) and determine when \(T_{\mathbb{S}(Y)}(X)\) is a unit-regular semigroup (Theorem 3.8).\N\NNext, let \(L(V)\) be the semigroup of all linear transformations of a vector space \(V\). For a subspace \(W\) of \(V\) and a subsemigroup \(\mathbb{S}(W)\) of \(L(W)\), consider the subsemigroup \(L_{\mathbb{S}(W)}(V)=\{f\in L(V):f_{\upharpoonright_W}\in\mathbb{S}(W)\}\) of \(L(V)\). In Section 4, the authors characterize regular elements in \(L_{\mathbb{S}(W)}(V)\) (Theorem 4.2) and determine when \(L_{\mathbb{S}(W)}(V)\) is a regular semigroup (Theorem 4.4). If \(\mathbb{S}(W)\) contains the identity of \(L(W)\), then the authors characterize unit-regular elements in \(L_{\mathbb{S}(W)}(V)\) (Theorem 4.6) and determine when \(L_{\mathbb{S}(W)}(V)\) is a unit-regular semigroup (Theorem 4.8). The authors also determine when \(L_{\mathbb{S}(W)}(V)\) is an inverse semigroup (Theorem 4.9) and when \(L_{\mathbb{S}(W)}(V)\) is a completely regular semigroup (Theorem 4.11).