论文标题
内态的相对等级是有限的$ g $ set的
The relative rank of the endomorphism monoid of a finite $G$-set
论文作者
论文摘要
对于在$ x $上作用的组$ g $，令$ \ text {end} _g（x）$是所有$ g $ equivariant转换的单体，或$ g $ endomorphisms的$ x $，然后让$ \ text {aut} _g（x aut} _g（x）$是其一组units。在一般环境中讨论了很少的基本结果之后，我们重点介绍了$ g $和$ x $都是有限的，以确定一组$ w \ subseteq \ subseteq \ text {end} _g（x）$的最小基数，以便$ w \ cup \ cup \ cup \ text \ text {aut} _g（x）这在半群理论中被称为$ \ text {end} _g（x）$ modulo $ \ text {aut} _g（x）$的相对等级。
For a group $G$ acting on a set $X$, let $\text{End}_G(X)$ be the monoid of all $G$-equivariant transformations, or $G$-endomorphisms, of $X$, and let $\text{Aut}_G(X)$ be its group of units. After discussing few basic results in a general setting, we focus on the case when $G$ and $X$ are both finite in order to determine the smallest cardinality of a set $W \subseteq \text{End}_G(X)$ such that $W \cup \text{Aut}_G(X)$ generates $\text{End}_G(X)$; this is known in semigroup theory as the relative rank of $\text{End}_G(X)$ modulo $\text{Aut}_G(X)$.
