Generating sets, presentations, and growth of tropical matrix monoids

TL;DR

This paper investigates the generation, presentation, and growth of tropical matrix monoids, providing explicit constructions and bounds, and characterizing when these monoids are finitely generated or presented.

Contribution

It constructs minimal generating sets, determines finite presentability conditions, and establishes growth bounds for tropical matrix monoids, advancing understanding of their algebraic structure.

Findings

The monoid of n×n matrices over tropical integers is finitely generated iff n ≤ 2.

It is finitely presented only when n=1.

Explicit minimal generating sets are constructed for n ≤ 3.

Abstract

We construct minimal and irredundant generating sets for a family of submonoids of the monoid of $n \times n$ upper triangular matrices over a commutative semiring. We show that the monoid of $n \times n$ matrices over the tropical integers, $M_n(\mathbb{Z}_\mathrm{max})$, is finitely generated if and only if $n \leq 2$, and finitely presented if and only if $n = 1$. Minimal and irredundant generating sets are explicitly constructed when $n \leq 3$. We then construct a presentation for the monoid of $n \times n$ upper triangular matrices over the tropical integers, $UT_n(\mathbb{Z}_\mathrm{max})$, demonstrating that it is finitely presented for all $n \in \mathbb{N}$. Finally, we establish upper bounds on the polynomial degree of the growth function of finitely generated subsemigroups of the monoid of $n \times n$ matrices over a bipotent semiring and show that these bounds are sharp for the tropical semiring.