Primitive Permutation Groups and Strongly Factorizable Transformation Semigroups

TL;DR

This paper classifies permutation groups based on their ability to generate certain semigroups with specific factorizability properties and explores conditions for regularity in transformation semigroups.

Contribution

It provides an (almost) complete classification of permutation groups with a particular product decomposition property and characterizes regularity of semigroups formed with normalizers.

Findings

Classified permutation groups with specific semigroup generation properties

Proved equivalence of regularity between semigroups and their normalizers

Established conditions for regularity in 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 $\langle A\rangle$ 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.