Diameters of commuting graphs of partial transformation semigroups

TL;DR

This paper calculates the diameter of the commuting graph of the partial transformation semigroup on a finite set and shows it matches that of the full transformation semigroup, revealing new structural insights.

Contribution

It determines the diameter of the commuting graph of the partial transformation semigroup and demonstrates its equality with that of the full transformation semigroup, highlighting new semigroup properties.

Findings

Diameter of the commuting graph of $ ext{P}(X)$ is determined.

The diameter matches that of the transformation semigroup $ ext{T}(X)$.

Existence of semigroup $S$ and subsemigroup $T$ with equal commuting graph diameters.

Abstract

Let $X$ be a finite set. We determine the diameter of the commuting graph of the partial transformation semigroup $\mathcal{P}(X)$ on $X$ and show that it coincides with the diameter of the commuting graph of the transformation semigroup $\mathcal{T}(X)$ on $X$, which was previously determined by Ara\'ujo, Kinyon and Konieczny. This proves the existence of a semigroup $S$ and of a proper subsemigroups $T$ of $S$ such that the diameters of the commuting graphs of $S$ and $T$ are equal.