Identities of the stylic monoid

Abstract

We observe that for each n ≥ 2, the identities of the stylic monoid with n generators coincide with the identities of n-generated monoids from other distinguished series of J -trivial monoids studied in the literature, e.g., Catalan monoids and Kiselman monoids. This solves the Finite Basis Problem for stylic monoids.A monoid identity is a pair of words, i.e., elements of the free monoid X * over an alphabet X, written as a formal equality. An identity w = w ′ with w, w ′ ∈ X * is said to hold in a monoid M if wϕ = w ′ ϕ for each homomorphism ϕ : X * → M; alternatively, we say that the monoid satisfies w = w ′ . Clearly, if M satisfies w = w ′ , then so does every homomorphic image of M.Given any set Σ of monoid identities, we say that an identity w = w ′ follows from Σ if every monoid satisfying all identities of Σ satisfies the identity w = w ′ as well. A subset ∆ ⊆ Σ is called a basis for Σ if each identity in Σ follows from ∆. The Finite Basis Problem for a monoid M is the question of whether or not the set of all identities that hold in M admits a finite basis.A monoid M is said to be J -trivial if every principal ideal of M has a unique generator, that is, MaM = MbM implies a = b for all a, b ∈ M. Finite J -trivial monoids attract much attention because of their distinguished role in algebraic theory of regular languages [14,15] and representation theory [7]. Several series of finite J -trivial monoids parameterized by positive integers appear in the literature, including Straubing monoids [17,18], Catalan monoids [16], double Catalan monoids [13], Kiselman monoids [9,11,12], and gossip monoids [5,8,10]. These monoids arise due to completely unrelated reasons and consist of elements of a very different nature. Surprisingly, studying identities of the listed monoids has revealed that the n-th monoids in each series satisfy exactly the same identities! Recently, a new family of finite J -trivial monoids, coined stylic monoids, have been introduced by Abram and Reutenauer [1], with motivation coming from combinatorics of Young tableaux. It is quite natural to ask whether the above phenomenon extends to this new family, i.e., whether the n-th stylic monoid again satisfies the same identities as do the n-th monoids in each aforementioned series. The present note aims to answers this question in the affirmative.The combinatorial definition of stylic monoids can be found in [1], but here we only need their presentation via generators and relations established in [1, Theorem 8.1(ii)]. Thus, let the stylic monoid Styl n be the monoid generated by a 1 , a 2 , . . . , a n subject to the

Remark on the identities of the grammic monoid with three generators

“…In this note, we meet another cluster grouping around the 3-generated plactic monoid; as follows from a recent result in [5], this cluster also hosts the monoid of all upper triangular 3 × 3-matrices over the tropical semiring. For an example from the realm of finite monoids, see the author's recent note [9]. It is very tempting to understand the intrinsic reasons of this clustering phenomenon.…”

Tropical representations and identities of the stylic monoid

“…The anonymous referee has also observed the following: In [63], it is shown that styl 3 is a homomorphic image of the Kiselman monoid Kis 3 and the Catalan monoid Cat 3 is a homomorphic image of styl 3 . It can be easily checked that these properties still hold when considering the mentioned monoids with involution.…”

Representations and identities of Baxter monoids with involution

“…Aird and Ribeiro have given a faithful representation of the stylic monoid and its involution case of each finite rank, and then solved the finite basis problems for them [5]. Also, Volkov has solved the finite basis problem for the stylic monoid by different mean [52]. The identity checking problems for the Baxter, sylvester and stylic monoids have been considered [5,14], which can be done in polynomial time.…”