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

## TL;DR

This paper classifies finite permutation groups based on whether their generated semigroups, with any transformation, are regular or idempotent-generated, revealing structural properties of these algebraic systems.

## Contribution

It provides a complete classification of permutation groups that produce regular or idempotent-generated semigroups with any non-invertible transformation.

## Key findings

- Identifies permutation groups yielding regular semigroups with any transformation.
- Classifies groups where generated semigroups are idempotent-generated.
- Provides structural insights into transformation semigroups and their generators.

## Abstract

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 $X$.