Semigroup identities in the monoid of two-by-two tropical matrices

TL;DR

This paper proves that the monoid of all 2x2 tropical matrices satisfies a non-trivial semigroup identity, highlighting a key difference from matrices over infinite fields, using tropical polynomial matrices and connections to the bicyclic monoid.

Contribution

It establishes the existence of a non-trivial semigroup identity in the monoid of 2x2 tropical matrices, linking it to the bicyclic monoid and employing tropical polynomial matrices.

Findings

The monoid of 2x2 tropical matrices satisfies a non-trivial semigroup identity.

A connection between tropical matrices and the bicyclic monoid is demonstrated.

A variant of Adian's identity holds in the tropical matrix monoid.

Abstract

The main result of the paper proves that the monoid of all 2 x 2 tropical matrices satisfies a non-trivial semigroup identity, unlike the case of matrices over infinite fields. We exploit a strong connection between the monoid of all 2 x 2 tropical matrices and the bicyclic monoid to show that a variant of Adian's identity for the bicyclic monoid holds in our case.We use generic matrices of tropical polynomials to prove our result.