Representations and identities of plactic-like monoids

TL;DR

This paper constructs faithful matrix representations for various plactic-like monoids over semirings, revealing their algebraic structures and the varieties they generate, including connections to natural numbers and finite monoids.

Contribution

It provides explicit faithful matrix representations of several plactic-like monoids over semirings and analyzes their generated varieties, linking them to natural numbers and finite monoids.

Findings

Faithful upper triangular matrix representations over semirings.

Varieties generated by hypoplactic, stalactic, and taiga monoids.

Connections to varieties generated by natural numbers and finite monoids.

Abstract

We exhibit faithful representations of the hypoplactic, stalactic, taiga, sylvester, Baxter and right patience sorting monoids of each finite rank as monoids of upper triangular matrices over any semiring from a large class including the tropical semiring and fields of characteristic $0$. By analysing the image of these representations, we show that the variety generated by a single hypoplactic (respectively, stalactic or taiga) monoid of rank at least $2$ coincides with the variety generated by the natural numbers together with a fixed finite monoid $\mathcal{H}$ (respectively, $\mathcal{F}$) forming a proper subvariety of the variety generated by the plactic monoid of rank $2$.