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Semigroups of transformations whose restrictions belong to a given semigroup
Semigroup Forum ( IF 0.7 ) Pub Date : 2021-09-22 , DOI: 10.1007/s00233-021-10227-5
Janusz Konieczny 1
Affiliation  

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



中文翻译:

限制属于给定半群的变换半群

为一组X,表示由ŤX)上满变换的半群 X。对于任意子集ÿ的 X和任何半群\({\ mathbb {S}}(Y)\)ŤÝ),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\(\alpha | _Y\)\(\alpha \)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\(T_{{\mathbb {S}}(Y)}(X)\)

更新日期:2021-09-23
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