Tropical representations and identities of the stylic monoid
Thomas J. Aird, Duarte Ribeiro · Semigroup Forum · 2022
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.