Representations and identities of involution Plactic-like monoids arising from the meet of the stalactic congruence and its dual

TL;DR

This paper studies a specific plactic-like monoid derived from stalactic congruences, revealing multiple involutions, their classifications, and providing combinatorial characterizations of identities, thus solving key algebraic problems.

Contribution

It introduces multiple involutions on the monoid, classifies them, and offers combinatorial characterizations of identities, addressing the finite basis and identity checking problems.

Findings

Multiple involutions are identified and classified into loor(n/2)+1 types.

Faithful representations of the monoid under each involution are constructed.

The finite basis problem and identity checking problem are solved for these monoids.

Abstract

Let $\mathsf{mSt}_n$ be the plactic-like monoid obtained by factoring the free monoid over a finite alphabet $\mathcal{A}_n$ by the meet of the stalactic congruence and its dual. In this paper, we prove that $\mathsf{mSt}_n$ can be equipped with multiple involutions, and divide these involutions into $\lfloor\frac{n}{2}\rfloor+1$ types. A faithful representation of $\mathsf{mSt}_n$ under each of these involutions is obtained. We give transparent combinatorial characterizations of identities for $\mathsf{mSt}_n$ under each involution, and so the finite basis problem and identity checking problem for them are solved.