Let X X be a non-empty set and Y Y a non-empty subset of X X . Denote the full transformation semigroup on X X by T ( X ) T\left(X) and write f ( X ) = { f ( x ) ∣ x ∈ X } f\left(X)=\{f\left(x)| x\in X\} for each f ∈ T ( X ) f\in T\left(X) . It is well known that T ( X , Y ) = { f ∈ T ( X ) ∣ f ( X ) ⊆ Y } T\left(X,Y)=\{f\in T\left(X)| f\left(X)\subseteq Y\} is a subsemigroup of T ( X ) T\left(X) and R T ( X , Y ) RT\left(X,Y) , the set of all regular elements of T ( X , Y ) T\left(X,Y) , also forms a subsemigroup of T ( X , Y ) T\left(X,Y) . Green’s ∗ \ast -relations and Green’s ∼ <mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> \sim </mml:mpadded>\hspace{0.08em} -relations (with respect to a non-empty subset U U of the set of idempotents) were introduced by Fountain in 1979 and Lawson in 1991, respectively. In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup T ( X , Y ) T\left(X,Y) . This investigation proves that the semigroup T ( X , Y ) T\left(X,Y) is always a left Ehresmann semigroup, and R T ( X , Y ) RT\left(X,Y) is orthodox (resp. completely regular) if and only if Y Y contains at most two elements.

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关于范围受限的变换的半群的一些结果

令XX为非空集，而YY为XX的非空子集。用T（X）T \ left（X）表示XX上的完整变换半群，并写f（X）= {f（x）∣ x∈X} f \ left（X）= \ {f \ left（x） | 对于每个f∈T（X）f \ in T \ left（X）。众所周知，T（X，Y）= {f∈T（X）∣ f（X）⊆Y} T \ left（X，Y）= \ {f \ in T \ left（X）| f \ left（X）\ subseteq Y \}是T（X）T \ left（X）和RT（X，Y）RT \ left（X，Y）的子半群，T（X（Y，Y））的所有常规元素的集合X，Y）T \ left（X，Y），也形成T（X，Y）T \ left（X，Y）的一个半子群。Green的* ast关系和Green的<mml：mpapped xmlns：ali =“ http://www.niso.org/schemas/ali/1.0/” xmlns：xsi =“ http://www.w3.org/ 2001 / XMLSchema-instance“> \ sim </ mml：mpadded> \ hspace {0。08em}-关系（关于幂等集的非空子集UU）分别由Fountain在1979年和Lawson在1991年引入。在本文中，我们打算介绍半群T（X，Y）T \ left（X，Y）的这两组格林关系的某些特征。该研究证明，半群T（X，Y）T \ left（X，Y）始终是左Ehresmann半群，而RT（X，Y）RT \ left（X，Y）则是正统的（分别是完全规则的）当且仅当YY最多包含两个元素。