Conditions for the commutativity of semigroups

Abstract:

Let $S$ be a semigroup. Then by a theorem of Tully [7]: $S$ is a commutative semigroup iff $ab = {b^n}{a^m}$ for all $a,b \in S$ ($m,n \geqslant 1$, fixed). We prove the following: $S$ is a commutative semigroup iff $ab = {b^{n(a,b)}}{a^{m(a,b)}}$ for all $a,b \in S$, where one of the exponents $n(a,b)$ and $m(a,b)$ is constant and the other is independent of $a$ or $b$.