The Sch\"utzenberger groups and maximal subgroups of tropical matrices

TL;DR

This paper classifies Schützenberger groups and maximal subgroups of tropical matrix monoids, revealing new group structures and correcting previous results in tropical algebra.

Contribution

It provides a comprehensive classification of Schützenberger groups over tropical matrices and identifies differences between these groups and maximal subgroups.

Findings

Classified Schützenberger groups for matrices over tropical semirings

Classified maximal subgroups of tropical matrix monoids

Discovered groups that are Schützenberger groups but not maximal subgroups

Abstract

We classify the Sch\"utzenberger groups of the category of matrices over the tropical semiring, $M(\mathbb{T})$, in doing so, we obtain a classification for the Sch\"utzenberger groups of the semigroupoid of matrices over the finitary tropical semiring, $M(\mathbb{F}\mathbb{T})$. We then classify the maximal subgroups of the monoid of $n \times n$ matrices over the tropical semiring, $M_n(\mathbb{T})$, for all $n \in \mathbb{N}$; generalising a result in the literature and correcting an erroneous proof. We proceed to show that for some $n \in \mathbb{N}$ there exists a group which appears as a Sch\"utzenberger group of $M_n(\mathbb{T})$ but does not appear as a maximal subgroup.