Generating the monoid of $2 \times 2$ matrices over max-plus and min-plus semirings

TL;DR

This paper identifies generating sets for the monoids of all 2x2 matrices over specific finite tropical semirings, advancing understanding of their algebraic structure.

Contribution

It provides explicit generating sets for monoids of 2x2 matrices over finite tropical semirings, a novel contribution to tropical algebra.

Findings

Explicit generating sets for matrices over max-plus semiring

Explicit generating sets for matrices over min-plus semiring

Enhanced understanding of algebraic structure of tropical matrix monoids

Abstract

In this short note, we describe generating sets for the monoids of consisting of all $2 \times 2$ matrices over certain finite tropical semirings.