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arXiv · 2206.11725

Tropical Representations and Identities of the Stylic Monoid

Thomas Aird·Duarte Ribeiro
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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.

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Thomas Aird, Duarte Ribeiro. 2022-06-23. Tropical Representations and Identities of the Stylic Monoid. https://arxiv.org/abs/2206.11725

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