Identities and bases in the hypoplactic monoid

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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“…(hypo) is the varietal join V(styl 2 ) ∨ C, and is generated by styl 2 and the free commutative monoid.Proof A consequence of[19, Corollary 4.4].…”

mentioning

confidence: 93%

Tropical representations and identities of the stylic monoid

Aird

Ribeiro

2022

Semigroup Forum

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“…It is shown that all Baxter [resp. hypoplactic, sylvester, stalactic and taiga] monoids of rank greater than or equal to 2 generate the same variety and are finitely based [11,14,15,25]. 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].…”

Section: Introductionmentioning

confidence: 99%

Representations and identities of Baxter monoids with involution

Han¹,

Zhang²,

Luo³

et al. 2023

Preprint

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Let (baxtn, ♯ ) be the Baxter monoid of finite rank n with Schützenberger's involution ♯ . In this paper, it is shown that (baxtn, ♯ ) admits a faithful representation by an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. Then a transparent combinatorial characterization of the word identities satisfied by (baxtn, ♯ ) is given. Further, it is proved that (baxtn, ♯ ) is finitely based if and only if n = 3, and shown that the identity checking problem for (baxtn, ♯ ) can be done in polynomial time.

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“…These monoids satisfy non-trivial identities [16] and, except for the case of the right patience-sorting monoid, these identities are satisfied regardless of rank. The finite basis problem regarding the plactic-like monoids has also been studied by different authors (including the second author), using different techniques [17,19,12,26], and the identity checking problem has been studied in [19] by Cain, Malheiro and the second author.…”

Section: Introductionmentioning

confidence: 99%

Tropical Representations and Identities of the Stylic Monoid

Aird¹,

Ribeiro²

2022

Preprint

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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 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 ≤ 3, that the varieties they generate have uncountably many subvarieties for 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.

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Identities and bases in the hypoplactic monoid

Cited by 12 publications

References 30 publications

Tropical representations and identities of the stylic monoid

Tropical representations and identities of the stylic monoid

Representations and identities of Baxter monoids with involution

Tropical Representations and Identities of the Stylic Monoid

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