Tropical one-relation monoids

TL;DR

This paper classifies all one-relation monoids that can be represented as semigroups of tropical matrices, extending previous results on the bicyclic monoid to a broader class.

Contribution

It provides a complete classification of one-relation monoids admitting tropical matrix representations, with only two exceptions.

Findings

Most one-relation monoids admit tropical matrix representations.

The classification includes all but two specific monoids.

Extends the known representation of the bicyclic monoid to a wider class.

Abstract

In 2010, Izhakian & Margolis proved that the bicyclic monoid admits a representation as a semigroup of upper triangular tropical matrices. We extend this result by classifying all one-relation monoids which admit such representations. We also classify, with two exceptions, all one-relation monoids which admit representations as semigroups of tropical matrices.