Generating sets, presentations, and growth of tropical matrix monoids

TL;DR

This work provides a comprehensive study of generating sets, presentations, and growth for tropical matrix monoids. It yields explicit minimal and irredundant generating sets for upper triangular and certain full matrix monoids, proves finite presentability of $UT_n(\mathbb{Z}_{\max})$ for all $n$, and shows finite presentability fails for $M_n(\mathbb{Z}_{\max})$ when $n\ge 2$. It also establishes sharp polynomial growth bounds for finitely generated subsemigroups over bipotent semirings, with precise dependence on matrix size and the rank of the generated abelian subgroup in the tropical setting. Collectively, these results illuminate the structure and complexity of tropical matrix semigroups, with implications for both algebraic theory and applications in tropical geometry and semiring theory.

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.

Theorems & Definitions (69)

• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Corollary 3.4
• Corollary 3.5