The stylic monoid

Abram, Antoine; Reutenauer, Christophe

doi:10.1007/s00233-022-10285-3

Cited by 12 publications

(20 citation statements)

References 24 publications

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“…For a general background on the plactic monoid, see [47,Chapter 5]. For an in-depth look at the stylic monoid, see [1].…”

Section: The Stylic Monoidmentioning

confidence: 99%

“…The stylic monoid of rank n can be defined in two other ways, which will be the ones used in this work: It is defined by the presentation A n R styl [1,Theorem 8.1], where…”

Section: The Stylic Monoidmentioning

confidence: 99%

“…The defining relations are known as the stylic relations, and are the plactic relations together with a generator idempotent relation. As such, the stylic monoid of rank n can be viewed as a quotient of the plactic monoid [1, Proposition 5.1], and two words in the same stylic class have the same support [1,Lemma 5.3].…”

Section: The Stylic Monoidmentioning

confidence: 99%

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Tropical representations and identities of the stylic monoid

Aird

Ribeiro

2022

Semigroup Forum

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We exhibit a faithful representation of the stylic monoid of every finite rank as a monoid of upper unitriangular matrices over the tropical semiring. Thus, we show that the stylic monoid of finite rank n generates the pseudovariety $$\varvec{{\mathcal {J}}}_n$$ J n , which corresponds to the class of all piecewise testable languages of height n, in the framework of Eilenberg’s correspondence. From this, we obtain the equational theory of the stylic monoids of finite rank, show that they are finitely based if and only if $$n \le 3$$ n ≤ 3 , and that their identity checking problem is decidable in linearithmic time. We also establish connections between the stylic monoids and other plactic-like monoids, and solve the finite basis problem for the stylic monoid with involution.

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“…For a general background on the plactic monoid, see [47,Chapter 5]. For an in-depth look at the stylic monoid, see [1].…”

Section: The Stylic Monoidmentioning

confidence: 99%

“…The stylic monoid of rank n can be defined in two other ways, which will be the ones used in this work: It is defined by the presentation A n R styl [1,Theorem 8.1], where…”

Section: The Stylic Monoidmentioning

confidence: 99%

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Tropical representations and identities of the stylic monoid

Aird

Ribeiro

2022

Semigroup Forum

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“…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 , .…”

mentioning

confidence: 99%

“…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 , .…”

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confidence: 99%

Identities of the stylic monoid

Volkov

2022

Semigroup Forum

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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

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Grammic monoids with three generators

Choffrut

2022

Semigroup Forum

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The stylic monoid

Abstract: We construct a complete system of primitive orthogonal idempotents and give an explicit quiver presentation of the monoid algebra of the stylic monoid introduced by Abram and Reutenauer [AR22].

Cited by 12 publications

References 24 publications

Tropical representations and identities of the stylic monoid

Tropical representations and identities of the stylic monoid

Identities of the stylic monoid

Grammic monoids with three generators

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