Idempotents and one-sided units in infinite partial Brauer monoids

TL;DR

This paper investigates the structure and properties of monoids generated by idempotents and units within infinite partial Brauer monoids, providing classifications, rank calculations, and properties related to semigroup theory.

Contribution

It introduces a comprehensive analysis of eight monoids generated by idempotents and units in infinite partial Brauer monoids, including rank calculations and property determinations.

Findings

Calculated relative ranks of the monoids

Determined the Sierpinski rank of each monoid

Identified which monoids have the semigroup Bergman property

Abstract

We study monoids generated by various combinations of idempotents and one- or two-sided units of an infinite partial Brauer monoid. This yields a total of eight such monoids, each with a natural characterisation in terms of relationships between parameters associated to Brauer graphs. We calculate the relative ranks of each monoid modulo any other such monoid it may contain, and then apply these results to determine the Sierpinski rank of each monoid, and ascertain which ones have the semigroup Bergman property. We also make some fundamental observations about idempotents and units in arbitrary monoids, and prove some general results about relative ranks for submonoids generated by these sets. Dedicated to Dr Des FitzGerald on the occasion of his 70th birthday.