Generating sets, presentations, and growth of tropical matrix monoids

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.