Minimal paths in the commuting graphs of semigroups

TL;DR

This paper investigates the structure of commuting graphs in finite semigroups, establishing bounds on their diameters, exploring left paths, and solving an old algebraic problem related to these graphs.

Contribution

It provides new bounds on the diameters of commuting graphs of certain semigroups and characterizes the possible lengths of left paths, addressing a longstanding algebraic question.

Findings

For ideals of transformation semigroups, the diameter is typically 5 for large sets.

Semigroups can have commuting graphs with arbitrarily large diameters.

The shortest left path lengths in semigroups can be any positive integer except 3.

Abstract

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 $ab=ba$. 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 prove that if $|X|\geq4$, then, with a few exceptions, the diameter of $\cg(J)$ is 5. On the other hand, we prove that for every positive integer $n$, there exists a semigroup $S$ such that the diameter of $\cg(S)$ is $n$. We also study the left paths in $\cg(S)$, that is, paths $a_1-a_2-...-a_m$ such that $a_1\ne a_m$ and $a_1a_i=a_ma_i$ for all $i\in \{1,\ldot, m\}$. We prove 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 use the previous results to solve a purely algebraic old problem posed by B.M. Schein.