Generating sets, presentations, and growth of tropical matrix monoids

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, $M_n({\mathbb{Z}}_{\max })$, is finitely generated if and only if $n\le 2$, and finitely presented if and only if $n=1$. Minimal and irredundant generating sets are explicitly constructed when $n\le 3$. We then construct a presentation for the monoid of $n\times n$ upper triangular matrices over the tropical integers, $U{T}_n({\mathbb{Z}}_{\max })$, 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.

arXiv comment27 pages. Reformatted and added new results