Primitive Permutation Groups and Strongly Factorizable Transformation Semigroups

Abstract

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 $⟨A⟩$ 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 $S_t:=⟨G,t⟩$ 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 $s^{\prime }\in S$ such that $s=s{s}^{\prime }s$.) The paper ends with a list of problems.