Motzkin monoids and partial Brauer monoids

TL;DR

This paper provides a comprehensive structural analysis of the Motzkin and partial Brauer monoids, including their ideals, ranks, and idempotent generation, and applies these findings to representation theory of related diagram algebras.

Contribution

It offers new structural insights and formulas for ranks and idempotent ranks of ideals in Motzkin and partial Brauer monoids, with applications to diagram algebra representations.

Findings

Derived formulas for the rank of each ideal.

Established conditions for ideals to be idempotent-generated.

Provided new proofs for representation theoretic results.

Abstract

We study the partial Brauer monoid and its planar submonoid, the Motzkin monoid. We conduct a thorough investigation of the structure of both monoids, providing information on normal forms, Green's relations, regularity, ideals, idempotent generation, minimal (idempotent) generating sets, and so on. We obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated. We find formulae for the rank (smallest size of a generating set) of each ideal, and for the idempotent rank (smallest size of an idempotent generating set) of the idempotent-generated subsemigroup of each ideal; in particular, when an ideal is idempotent-generated, the rank and idempotent rank are equal. Along the way, we obtain a number of results of independent interest, and we demonstrate the utility of the semigroup theoretic approach by applying our results to obtain new proofs of important representation theoretic results concerning the corresponding diagram algebras, the partial (or rook) Brauer algebra and Motzkin algebra.