Commuting Graphs of Semigroups

Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted $\cg(S)$, is the graph whose vertices are the non-central elements of $S$ and whose edges are the sets $\{a,b\}$ of vertices such that $a\ne b$ and $ab=ba$. I will discuss some recent results on commuting graphs of semigroups, obtained jointly with João Araújo (CAUL, Portugal) and Michael Kinyon (University of Denver). Denote by $T(X)$ the semigroup of full transformations on a finite set $X$. Let $J$ be any ideal of $T(X)$ such that $J$ is different from the ideal of constant transformations on $X$. We have proved that if $|X|\geq4$, then, with a few exceptions, the diameter of $\cg(J)$ is $5$. On the other hand, for every positive integer $n$, there exists a semigroup $S$ such that the diameter of $\cg(S)$ is $n$. We have also studied the left paths in $\cg(S)$, that is, paths $a_1-a_2-\cdots-a_m$ such that $a_1\ne a_m$ and$a_1a_i=a_ma_i$ for all $i\in \{1,\ldots ,m\}$. We have proved that for every positive integer $n\geq2$, except $n=3$, there exists a semigroup whose shortest left path has length~$n$. As a corollary, we have obtained a solution to a purely algebraic problem posed by B.M. Schein in 1978.