The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Portugaliae Mathematica

Full-Text PDF (99 KB) | Metadata | Table of Contents | PM summary
Online access to the full text of Portugaliae Mathematica is restricted to the subscribers of the journal, who are encouraged to communicate their IP-address(es) to their agent or directly to the publisher at
Volume 74, Issue 3, 2017, pp. 243–255
DOI: 10.4171/PM/2005

Published online: 2018-02-08

Commutativity theorems for groups and semigroups

Francisco Araújo[1] and Michael Kinyon[2]

(1) Colégio Planalto, Lisboa, Portugal
(2) University of Denver, USA and Universidade de Lisboa, Portugal

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.

Keywords: Separative semigroup, inverse semigroup, completely regular, commutativity theorems

Araújo Francisco, Kinyon Michael: Commutativity theorems for groups and semigroups. Port. Math. 74 (2017), 243-255. doi: 10.4171/PM/2005