Classification of variants of partial Brauer monoids

TL;DR

This paper classifies variants of the partial Brauer monoid formed by sandwiching with elements of nonzero rank, expanding understanding of their algebraic structure.

Contribution

It provides a classification of variants of the partial Brauer monoid based on the rank of the sandwich element, a novel structural analysis.

Findings

Classified variants of the partial Brauer monoid for nonzero rank sandwich elements

Identified structural differences based on the rank of the sandwich element

Extended the theory of semigroup variants in the context of partial Brauer monoids

Abstract

A variant of a semigroup $S$ with respect to an element $a\in S$ is the semigroup $S^{a}=(S,\star_{a})$, where $x\star_{a}y=xay$ for any $x,y\in S$. Here, $a$ is the sandwich element of $S^{a}$. In this article, we study variants of the partial Brauer monoid $\mathcal{PB}_{n}$ for $n\in\mathbb{N}$. We give the classification of these variants in the case when the rank of the sandwich element is nonzero.