Twisted Brauer monoids

Abstract

We investigate the structure of the twisted Brauer monoid ${\mathcal{B}}_n^{\tau }$, comparing and contrasting it to the structure of the (untwisted) Brauer monoid ${\mathcal{B}}_n$. We characterise Green's relations and pre-orders on ${\mathcal{B}}_n^{\tau }$, describe the lattice of ideals, and give necessary and sufficient conditions for an ideal to be idempotent-generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent-generated, its rank and idempotent rank are equal. As an application of our results, we also describe the idempotent-generated subsemigroup of ${\mathcal{B}}_n^{\tau }$ (which is not an ideal) as well as the singular ideal of ${\mathcal{B}}_n^{\tau }$ (which is neither principal nor idempotent-generated), and we deduce a result of Maltcev and Mazorchuk that the singular part of the Brauer monoid ${\mathcal{B}}_n$ is idempotent-generated.

v2 - 16 pages, 5 figures (post refereeing, to appear in PRSE). v1 - 17 pages, 7 figures

References in corpus (2)

• Motzkin monoids and partial Brauer monoids
• Enumeration of idempotents in diagram semigroups and algebras

Cited by in corpus (1)

• The idempotent generated subsemigroup of the Kauffman monoid