Identities of tropical matrices and plactic monoids

TL;DR

This paper investigates the algebraic identities of tropical matrix semigroups and the plactic monoid, revealing dimension-dependent variety generation and introducing shorter identities for the rank 4 plactic monoid.

Contribution

It establishes that tropical matrix semigroups generate distinct varieties based on dimension and provides new, shorter identities for the rank 4 plactic monoid.

Findings

Different dimensions generate different semigroup varieties.

Shorter identities are found for the plactic monoid of rank 4.

The variety generated by the rank 4 plactic monoid is contained within that of 5x5 tropical matrices.

Abstract

We study semigroup varieties generated by full and upper triangular tropical matrix semigroups and the plactic monoid of rank 4. We prove that the upper triangular tropical matrix semigroup $UT_n(\mathbb{T})$ generates a different semigroup variety for each dimension $n$. We show a weaker version of this fact for the full matrix semigroup: full tropical matrix semigroups of different prime dimensions generate different semigroup varieties. For the plactic monoid of rank 4, $\mathbb{P}_4$, we find a new set of identities satisfied by $\mathbb{P}_4$ shorter than those previously known, and show that the semigroup variety generated by $\mathbb{P}_4$ is strictly contained in the variety generated by $UT_5(\mathbb{T})$.