Pseudovarieties generated by Brauer type monoids

Abstract

It is proved that the series of all Brauer monoids B n generates the pseudovariety of all finite monoids while the series of their aperiodic analogues, the Jones monoids J n (also called Temperly-Lieb monoids), generates the pseudovariety of all finite aperiodic monoids. The proof is based on the analysis of wreath product decomposition and Krohn-Rhodes theory. The fact that the Jones monoids J n form a generating series for the pseudovariety of all finite aperiodic monoids can be viewed as solution of an old problem popularized by J.-É. Pin. For the latter, the relationship between the Jones monoids J n and the monoids O n of order preserving mappings of a chain of length n is investigated.

Motzkin monoids and partial Brauer monoids

“…When R = C is the field of complex numbers, it is known that C τ [PBn] is generically semisimple [60, Theorem 1.1 and Corollary 3.6]; i.e., C τ [PBn] is semisimple for all but a finite number of choices of x, y 6. In [9, Theorem 5.14], semisimplicity was characterised in the case x = y in terms of certain expressions involving Chebyshev polynomials.…”

Infinite partition monoids

“…But then, by Lemma 9, there exists δ ∈ S X , απα ⊆ L X with d * (δ, 4) ≥ k * (απα, 4) = ℵ 0 . This completes the proof of the claim, since d * (δ, 3 Our goal will be to construct a permutation σ ∈ S X such that γ = δσβδ ∈ L X and k * (γ, ℵ 0 ) = ℵ 0 . The proof will then be complete since we will also have d…”

“…If one of x or y belongs to X\dom(α), then so too does the other, and (x, y) ∈ ker(α). 3 ) ∈ coker(α) and so on. But, since ker(β) ⊆ coker(α), it follows that (x 0 , x 1 ), (x 1 , x 2 ), .…”

“…Write α = (A x | B x ) and β = (C x | D x ) * . We first claim that there exists δ ∈ S X , α such that δ ∈ L X and d * (δ, 3…”

“…Partition algebras may be thought of as twisted semigroup algebras of partition monoids, and many properties of the partition algebras may be deduced from corresponding properties of the associated monoids [8,9,19,39]. Recent studies have also recognized partition monoids and some of their submonoids as key objects in the pseudovarieties of finite aperiodic monoids and semigroups with involution [2][3][4].…”