Semigroup identities of tropical matrices through matrix ranks

TL;DR

This paper proves that the monoid of all tropical matrices of any size satisfies nontrivial semigroup identities by analyzing matrix ranks and their powers.

Contribution

It establishes the conjecture that tropical matrix monoids satisfy nontrivial identities, using properties of matrix ranks and powers.

Findings

Tropical matrix monoids satisfy nontrivial semigroup identities.

The factor rank of large powers of tropical matrices is bounded by the original tropical rank.

The proof confirms a longstanding conjecture in tropical algebra.

Abstract

We prove the conjecture that, for any $n$, the monoid of all $n \times n$ tropical matrices satisfies nontrivial semigroup identities. To this end, we prove that the factor rank of a large enough power of a tropical matrix does not exceed the tropical rank of the original matrix.