JournalspmVol. 74, No. 3pp. 243–255

Commutativity theorems for groups and semigroups

  • Francisco Araújo

    Colégio Planalto, Lisboa, Portugal
  • Michael Kinyon

    University of Denver, USA and Universidade de Lisboa, Portugal
Commutativity theorems for groups and semigroups cover
Download PDF

A subscription is required to access this article.

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 SS we have xpyp=ypxpx^p y^p = y^p x^p and xqyq=yqxqx^q y^q = y^q x^q for all x,ySx,y\in S where pp and qq are relatively prime, then SS is commutative. In a separative or inverse semigroup SS, if there exist three consecutive integers ii such that (xy)i=xiyi(xy)^i = x^i y^i for all x,ySx,y\in S, then SS is commutative. Finally, if SS is a separative or inverse semigroup satisfying (xy)3=x3y3(xy)^3=x^3y^3 for all x,ySx,y\in S, and if the cubing map xx3x\mapsto x^3 is injective, then SS is commutative.

Cite this article

Francisco Araújo, Michael Kinyon, Commutativity theorems for groups and semigroups. Port. Math. 74 (2017), no. 3, pp. 243–255

DOI 10.4171/PM/2005