# Commutativity theorems for groups and semigroups

## TL;DR

This paper establishes new commutativity theorems for classes of semigroups, showing under what algebraic conditions such structures must be commutative, thus advancing the understanding of semigroup algebraic properties.

## Contribution

It introduces novel conditions involving powers and identities that guarantee semigroup commutativity, extending previous results in algebraic semigroup theory.

## Key findings

- Semigroups with certain power commutativity are necessarily commutative.
- Three consecutive power identities imply semigroup commutativity.
- Injectivity of the cubing map combined with a power identity ensures commutativity.

## Abstract

In this note we prove a selection of commutativity theorems for various classes of semigroups. For instance, if in a separative or completely regular semigroup $S$ we have $x^p y^p = y^p x^p$ and $x^q y^q = y^q x^q$ for all $x,y\in S$ where $p$ and $q$ are relatively prime, then $S$ is commutative. In a separative or inverse semigroup $S$, if there exist three consecutive integers $i$ such that $(xy)^i = x^i y^i$ for all $x,y\in S$, then $S$ is commutative. Finally, if $S$ is a separative or inverse semigroup satisfying $(xy)^3=x^3y^3$ for all $x,y\in S$, and if the cubing map $x\mapsto x^3$ is injective, then $S$ is commutative.