Representations and identities of hypoplactic monoids with involution

TL;DR

This paper provides a faithful matrix representation of hypoplactic monoids with involution, characterizes their identities, and analyzes their finite basis properties and algorithmic identity checking.

Contribution

It introduces a matrix representation of hypoplactic monoids with involution and characterizes their identities and basis properties.

Findings

Faithful matrix representation over semirings including tropical semiring.

Characterization of word identities satisfied by the monoids.

Identification of cases where the monoid is non-finitely based and polynomial-time identity checking.

Abstract

Let $(\mathsf{hypo}_n,~^\sharp)$ be the hypoplactic monoid of finite rank $n$ with Sch\"{u}tzenberger's involution $^{\sharp}$. In this paper, we exhibit a faithful representation of $(\mathsf{hypo}_n,~^\sharp)$ as an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. We then give a transparent combinatorial characterization of the word identities satisfied by $(\mathsf{hypo}_n,~^\sharp)$. Further, we prove that $(\mathsf{hypo}_n,~^\sharp)$ is non-finitely based if and only if $n=2, 3$ and give a polynomial time algorithm to check whether a given word identity holds in $(\mathsf{hypo}_n,~^\sharp)$.