保留等价性的限定范围变换半群上的自然偏序

Pub Date : 2024-03-27 DOI:10.1007/s00233-024-10422-0

Kritsada Sangkhanan, Jintana Sanwong

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引用次数: 0

摘要

让 Y 是 X 的一个非空子集，T(X, Y) 是所有从 X 到 Y 的函数的集合。那么，T(X, Y) 的组成是完整变换半群 T(X) 的一个子半群。定义 T(X, Y) 的子半群 $T_E(X,Y)$ 为 $$\begin{aligned}.T_E(X,Y)={T(X,Y)中的(x,y):\forall (x,y)\in E, (x\alpha ,y\alpha )\in E\}.\end{aligned}$$我们用自然偏序来研究 $T_E(X,Y)$，并确定两个元素在此序下何时相关。我们还给出了 $T_E(X,Y)$上相容性的特征，然后描述了最大元素和最小元素。

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The natural partial order on semigroups of transformations with restricted range that preserve an equivalence

Let Y be a nonempty subset of X and T(X, Y) the set of all functions from X into Y. Then T(X, Y) with composition is a subsemigroup of the full transformation semigroup T(X). Let E be a nontrivial equivalence on X. Define a subsemigroup $T_E(X,Y)$ of T(X, Y) by

$$\begin{aligned} T_E(X,Y)=\{\alpha \in T(X,Y):\forall (x,y)\in E, (x\alpha ,y\alpha )\in E\}. \end{aligned}$$

We study $T_E(X,Y)$ with the natural partial order and determine when two elements are related under this order. We also give a characterization of compatibility on $T_E(X,Y)$ and then describe the maximal and minimal elements.

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