Characterizing the largest commutative (full and partial) transformation semigroups of certain types

TL;DR

This paper characterizes the largest commutative subsemigroups within transformation and partial transformation semigroups, providing insights into their structure, idempotent elements, and implications for the graphs formed by their commuting relations.

Contribution

It offers a comprehensive characterization of the largest commutative subsemigroups in transformation semigroups, including those with unique idempotents and nilpotent elements, advancing understanding of their algebraic structure.

Findings

Largest commutative subsemigroups characterized

Results on clique numbers and girths of commuting graphs

Alternative methods for nilpotent subsemigroup determination

Abstract

Let $X$ be a finite set. Let $\mathcal{T}(X)$ be the transformation semigroup on $X$ and let $\mathcal{P}(X)$ be the partial transformation semigroup on $X$. This paper is a contribution to the problem of characterizing the largest commutative subsemigroups of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$). In the process of looking for these semigroups, we also characterize the largest commutative subsemigroups of idempotents of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$); as well as the largest commutative subsemigroups of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$) that contain a unique idempotent. We also provide an alternative way to determine the largest commutative nilpotent subsemigroups of $\mathcal{T}(X)$ (which were previously characterized by Cain, Malheiro and the present author); and we describe the largest commutative nilpotent subsemigroups of $\mathcal{P}(X)$. These results allow us to make conclusions regarding the clique numbers of the commuting graphs of $\mathcal{T}(X)$ and of $\mathcal{P}(X)$. We also determine their girths and knit degrees.