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arXiv 0911.0445math.GR
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与任意变换一起生成正则半群或幂等生成半群的群

Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

Joao Araujo, J. D. Mitchell, Csaba Schneider

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摘要：

令$a$是有限集的不可逆变换，令$G$是同一集合上的置换群。则$\genset{G, a}\setminus G$是由$G$和$a$生成的半群中的所有不可逆变换构成的子半群。同样，$a$与元素$g\in G$的共轭$a^g=g^{-1}ag$生成记为$\genset{a^g | g\in G}$的半群。我们对有限集$X$上的有限置换群$G$进行分类，使得半群$\genset{G,a}$，$\genset{G, a}\setminus G$和$\genset{a^g | g\in G}$对$X$的所有变换都是正则的.我们还对有限集$X$上的置换群$G$进行了分类，使得对于$X$的所有不可逆变换，半群$\genset{G, a}\setminus G$和$\genset{a^g | g\in G}$都由它们的幂等元生成.

英文摘要：

Let $a$ be a non-invertible transformation of a finite set and let $G$ be a group of permutations on that same set. Then $\genset{G, a}\setminus G$ is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by $G$ and $a$. Likewise, the conjugates $a^g=g^{-1}ag$ of $a$ by elements $g\in G$ generate a semigroup denoted $\genset{a^g | g\in G}$. We classify the finite permutation groups $G$ on a finite set $X$ such that the semigroups $\genset{G,a}$, $\genset{G, a}\setminus G$, and $\genset{a^g | g\in G}$ are regular for all transformations of $X$. We also classify the permutation groups $G$ on a finite set $X$ such that the semigroups $\genset{G, a}\setminus G$ and $\genset{a^g | g\in G}$ are generated by their idempotents for all non-invertible transformations of $X$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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