Permutability of Matrices over Bipotent Semirings

TL;DR

This paper investigates the permutability properties of matrix semigroups over bipotent semirings, clarifies previous results, and classifies certain semirings and isomorphisms, with implications for tropical algebra.

Contribution

It proves that all such matrix semigroups are weakly permutable and characterizes conditions for strong permutability based on the semiring structure.

Findings

Every matrix semigroup over a commutative bipotent semiring is weakly permutable.

Conditions for strong permutability depend on the specific semiring.

Classification of monogenic bipotent semirings and isomorphisms of truncated tropical semirings.

Abstract

We study permutability properties of matrix semigroups over commutative bipotent semirings (of which the best-known example is the tropical semiring). We prove that every such semigroup is weakly permutable (a result previous stated in the literature, but with an erroneous proof) and then proceed to study in depth the question of when they are strongly permutable (which turns out to depend heavily on the semiring). Along the way we classify monogenic bipotent semirings and describe all isomorphisms between truncated tropical semirings.