Commutativity theorems for groups and semigroups (Q2413199)

Summary: 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 \(x^py^p=y^px^p\) and \(x^qy^q=y^qx^q\) 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 \((xy)^i=x^iy^i\) 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.