We observe that for each $$n\ge 2$$ , the identities of the stylic monoid with n generators coincide with the identities of n-generated monoids from other distinguished series of $$\mathscr {J}$$ -trivial monoids studied in the literature, e.g., Catalan monoids and Kiselman monoids. This solves the Finite Basis Problem for stylic monoids. 

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Identities of the stylic monoid

Abstract 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 $$\varvec{{\mathcal {J}}}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub></mml:math> , 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 $$n \le 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math> , 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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Tropical representations and identities of the stylic monoid

Representations and identities of plactic-like monoids

We exhibit faithful representations of the hypoplactic, stalactic, taiga, sylvester, Baxter and right patience sorting monoids of each finite rank as monoids of upper triangular matrices over any semiring from a large class including the tropical semiring and fields of characteristic $0$. By analysing the image of these representations, we show that the variety generated by a single hypoplactic (respectively, stalactic or taiga) monoid of rank at least $2$ coincides with the variety generated by the natural numbers together with a fixed finite monoid $\mathcal{H}$ (respectively, $\mathcal{F}$) forming a proper subvariety of the variety generated by the plactic monoid of rank $2$.

In this paper, we show that all stalactic and taiga monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities, that all sylvester monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities and that all baxter monoids of rank greater than or equal to 2 are finitely based and satisfy the same identities. 

Finite basis problems for stalactic, taiga, sylvester and baxter monoids

This paper presents new results on the identities satisfied by the hypoplactic monoid. We show how to embed the hypoplactic monoid of any rank strictly greater than 2 (including infinite rank) into a direct product of copies of the hypoplactic monoid of rank 2. This confirms that all hypoplactic monoids of rank greater than or equal to 2 satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the variety generated by the hypoplactic monoid has finite axiomatic rank, by giving a finite basis for it.

Identities and bases in the hypoplactic monoid

This paper presents new results on the identities satisfied by the sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, and prove that the varieties generated by the sylvester and the Baxter monoids have finite axiomatic rank, by giving a finite basis for them.

Duarte Ribeiro

Papers

Identities and bases in the hypoplactic monoid

Tropical representations and identities of the stylic monoid

Identities and bases in the hypoplactic monoid

Identities and bases in the sylvester and Baxter monoids