Tropical Representations and Identities of the Stylic Monoid

TL;DR

This paper provides a tropical matrix representation of stylic monoids of finite rank, linking them to piecewise testable languages, and explores their algebraic properties, including finite basis and decidability of identities.

Contribution

It introduces a faithful tropical matrix representation of stylic monoids, determines their equational theory, and addresses the finite basis problem and identity checking complexity.

Findings

Faithful tropical matrix representation of stylic monoids

Finite basis property holds iff rank ≤ 3

Identity checking is decidable in linearithmic time

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 $\boldsymbol{\mathcal{J}}_n$, 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 $n \leq 3$, 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.