Tropical representations and 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 $$\varvec{{\mathcal {J}}}_n$$ 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.

Representations and identities of Baxter monoids with involution

“…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]. Also, Volkov has solved the finite basis problem for the stylic monoid by different mean [52].…”

“…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. By the characterization of the identities satisfied by the hypoplactic, stalactic and taiga monoids [11,14,25], it is easy to see that the identity checking problem for each of them is in the complexity class P.…”