Let Y be a subset of X and T ( X , Y ) the set of all functions from X into Y . Then, under the operation of composition, T ( X , Y ) is a subsemigroup of the full transformation semigroup T ( X ). Let E be an equivalence on X . Define a subsemigroup $$T_E(X,Y)$$ T E ( X , Y ) of T ( X , Y ) by $$\begin{aligned} T_E(X,Y)=\{\alpha \in T(X,Y):\forall (x,y)\in E, (x\alpha ,y\alpha )\in E\}. \end{aligned}$$ T E ( X , Y ) = { α ∈ T ( X , Y ) : ∀ ( x , y ) ∈ E , ( x α , y α ) ∈ E } . Then $$T_E(X,Y)$$ T E ( X , Y ) is the semigroup of all continuous self-maps of the topological space X for which all E -classes form a basis carrying X into a subspace Y . In this paper, we give a necessary and sufficient condition for $$T_E(X,Y)$$ T E ( X , Y ) to be regular and characterize Green’s relations on $$T_E(X,Y)$$ T E ( X , Y ) . Our work extends previous results found in the literature.

中文翻译：

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保持等价的有限范围变换半群的正则性和格林关系

设 Y 是 X 的子集，T ( X , Y ) 是从 X 到 Y 的所有函数的集合。那么，在合成运算下，T(X,Y)是全变换半群T(X)的子半群。令 E 是 X 上的等价。通过 $$\begin{aligned} T_E(X,Y)=\{\alpha \in T(X) 定义 T ( X , Y ) 的子半群 $$T_E(X,Y)$$ TE ( X , Y ) ,Y):\forall (x,y)\in E, (x\alpha ,y\alpha )\in E\}。\end{aligned}$$ TE ( X , Y ) = { α ∈ T ( X , Y ) : ∀ ( x , y ) ∈ E , ( x α , y α ) ∈ E } 。那么 $$T_E(X,Y)$$ TE ( X , Y ) 是拓扑空间 X 的所有连续自映射的半群，其中所有 E 类形成一个基，将 X 带入子空间 Y 。在本文中，我们给出了 $$T_E(X,Y)$$ TE ( X , Y ) 为正则的充分必要条件，并刻画了 $$T_E(X,Y)$$ TE ( X , Y ,是）。