Generators of matrix monoids over commutative semirings (2201.12166v1)

Abstract: We determine minimal generating sets for matrix monoids over semirings and semifields. In particular, we produce minimal generating sets for the monoids of upper triangular and unitriangular matrices over a commutative semiring. We construct minimal and irredundant generating sets for the monoids of $2 \times 2$ and $3 \times 3$ matrices over the tropical integers, showing that $M_3({\mathbb{Z}}_\mathrm{max})$ is not finitely generated.