GENERATING SETS, PRESENTATIONS, AND GROWTH OF TROPICAL MATRIX MONOIDS
Aird, Thomas · Research Explorer (The University of Manchester) · 2025
We construct minimal and irredundant generating sets for a family of submonoids of the monoid of n×n upper triangular matrices over a commutative semiring. We show that the monoid of n×n matrices over the tropical integers, Mₙ(ℤₘₐₓ), is finitely generated if and only if n ≤ 2, and finitely presented only if n = 1. We then construct a finite presentation for the monoid of n×n upper triangular matrices over the tropical integers, UTₙ(ℤₘₐₓ). Finally, we establish upper bounds on the polynomial degree of the growth function of finitely generated semigroups of bipotent matrices and show that these bounds are sharp for tropical matrices.