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Primitive permutation groups and strongly factorizable transformation semigroups
Journal of Algebra ( IF 0.8 ) Pub Date : 2021-01-01 , DOI: 10.1016/j.jalgebra.2020.05.023
João Araújo , Wolfram Bentz , Peter J. Cameron

Let $\Omega$ be a finite set and $T(\Omega)$ be the full transformation monoid on $\Omega$. The rank of a transformation $t\in T(\Omega)$ is the natural number $|\Omega t|$. Given $A\subseteq T(\Omega)$, denote by $\langle A\rangle$ the semigroup generated by $A$. Let $k$ be a fixed natural number such that $2\le k\le |\Omega|$. In the first part of this paper we (almost) classify the permutation groups $G$ on $\Omega$ such that for all rank $k$ transformation $t\in T(\Omega)$, every element in $S_t:=\langle G,t\rangle$ can be written as a product $eg$, where $e^2=e\in S_t$ and $g\in G$. In the second part we prove, among other results, that if $S\le T(\Omega)$ and $G$ is the normalizer of $S$ in the symmetric group on $\Omega$, then the semigroup $SG$ is regular if and only if $S$ is regular. (Recall that a semigroup $S$ is regular if for all $s\in S$ there exists $s'\in S$ such that $s=ss's$.) The paper ends with a list of problems.

中文翻译:

原始置换群和强可分解变换半群

令 $\Omega$ 是一个有限集,$T(\Omega)$ 是 $\Omega$ 上的完整变换幺半群。变换$t\in T(\Omega)$ 的秩是自然数$|\Omega t|$。给定$A\subseteq T(\Omega)$,用$\langle A\rangle$ 表示由$A$ 生成的半群。令 $k$ 是一个固定的自然数,使得 $2\le k\le |\Omega|$。在本文的第一部分,我们(几乎)对 $\Omega$ 上的置换组 $G$ 进行分类,这样对于所有秩 $k$ 变换 $t\in T(\Omega)$,$S_t:= \langle G,t\rangle$ 可以写成一个积$eg$,其中$e^2=e\in S_t$ 和$g\in G$。在第二部分,我们证明,除其他结果外,如果 $S\le T(\Omega)$ 和 $G$ 是 $\Omega$ 上对称群中 $S$ 的归一化器,则半群 $SG$是正则的当且仅当 $S$ 是正则的。
更新日期:2021-01-01
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