Motzkin monoids and partial Brauer monoids

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.

34 pages, 13 figures, 5 tables. v2: post refereeing, to appear in J Algebra

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• Hecke algebras of finite type are cellular
• Enumeration of idempotents in diagram semigroups and algebras
• Variants of finite full transformation semigroups
• Idempotent generation in the endomorphism monoid of a uniform partition
• Idempotent rank in the endomorphism monoid of a non-uniform partition
• Presentation of the Motzkin Monoid
• Cell algebra structures on monoid and twisted monoid algebras

Cited by in corpus (7)

• The idempotent generated subsemigroup of the Kauffman monoid
• Maximal subsemigroups of finite transformation and diagram monoids
• Monoidal categories, representation gap and cryptography
• Sandwich semigroups in diagram categories
• Twisted Brauer monoids
• Cellular subalgebras of the partition algebra
• Classification of variants of partial Brauer monoids