2022/01/28 by Aird, Thomas
#15A80 (Secondary) #20M10 (Primary) 16Y60 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2201.12166
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 if and only if n = 1. Minimal and irredundant generating sets are explicitly constructed when n ≤ 3. We then construct a presentation for the monoid of n × n upper triangular matrices over the tropical integers, UTn(ℤmax), demonstrating that it is finitely presented for all n ∈ ℕ. Finally, we establish upper bounds on the polynomial degree of the growth function of finitely generated subsemigroups of the monoid of n × n matrices over a bipotent semiring and show that these bounds are sharp for the tropical semiring.