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 some important representation theoretic results concerning the corresponding diagram algebras, the partial (or rook) Brauer algebra and Motzkin algebra

Presentations for rook partition monoids and algebras and their singular ideals

“…The previous result shows that RP n \S n is generated by its idempotents; this property is shared by many [7,21,24,56], but not all [13,14,17], diagram monoids. Under the above-mentioned embedding RP n → P n+1 , the generators o i ∈ RP n are mapped to t i,n+1 ∈ P n+1 .…”

“…Key to the approach used in [20] was the observation of Wilcox [75] (also implicit in [37]) that diagram algebras (including the partition algebras) arise as twisted semigroup algebras of corresponding diagram semigroups. This allows one to obtain information (concerning cellularity [17,24,31,75] or presentations [20,21], for example) about the algebras from corresponding information about the associated semigroups. Conversely, the theory of diagram algebras has led to a number of important families of semigroups and monoids that have been studied with increasing vigour in recent years; see for example [7, 13-17, 20, 21, 23, 24, 28, 49, 51, 56, 63, 64, 71], and especially the work of Auinger and his collaborators [1][2][3][4] on equational theories of involution semigroups.…”

Idempotents and one-sided units in infinite partial Brauer monoids

“…Note that 0 ≤ t(α), s(α), s * (α) ≤ |X| and that 0 ≤ h(α), h * (α) ≤ 1 2 |X|, with the " 1 2 " being unnecessary if X is infinite. We define the domain and codomain of α to be the sets Dom(α) = {x ∈ X : x belongs to a transversal of α}, Codom(α) = {x ∈ X : x ′ belongs to a transversal of α}, respectively, noting that |Dom(α)| = |Codom(α)| = t(α); elsewhere in the literature, the cardinal t(α) is sometimes called the rank or propagating number of α and denoted rank(α) or pn(α); see for example [9,28]. It is easy to see that…”

Maximal subsemigroups of finite transformation and diagram monoids

“…Proof. By [15], PB n is generated by its elements with rank at least n − 2, and any generating set contains elements of ranks n, n − 1, and n − 2. By Lemma 4.1, Green's J -relation in PB n is determined by rank.…”

“…([38],[45, Theorem 17],[37, Theorem 5], and[15, Theorem 2.4]). Let S ∈ {P n , PB n , B n , I * n , M n , J n } and let α, β ∈ S. Then:(a) αRβ if and only if dom(α) = dom(β) and ker(α) = ker(β); (b) αL β if and only if α * Rβ * , i.e.…”