半群$E\left(X,\sigma \right)$的扩张元和$F$、$G$富足性

Magnifying Elements and $F$、$G$ Abundance on Semigroup $E\left(X,\sigma \right)$

摘要: 设$T\left(X\right)$ 为非空集合$X$ 上的全变换半群，$\sigma$ 是$X$ 上的等价关系，定义$E\left(X,\sigma \right)=\left\{f\in T\left(X\right)|\sigma \subseteq \ker f\right\}$ ，则$E\left(X,\sigma \right)$ 是$T\left(X\right)$ 的子半群。本文刻画了$E\left(X,\sigma \right)$ 的左、右扩张元；描述了这类半群的$F$ 、$G$ 关系，并据此讨论了它的 $F$ 、$G$ 富足性。

Abstract: Let $T\left(X\right)$ be the full transformation semigroup on a nonempty set $X$ , and let $\sigma$ be an equivalence relation on $X$ . Define $E\left(X,\sigma \right)=\left\{f\in T\left(X\right)|\sigma \subseteq \ker f\right\}$ . Then $E\left(X,\sigma \right)$ is a subsemigroup of $T\left(X\right)$ . In this paper, we characterize the left and right magnifying elements of $E\left(X,\sigma \right)$ , describe the $F$ and $G$ relations on this semigroup, and based on this, discuss its $F$ abundance and $G$ abundance.