Finite Extensions of $\mathbb{Z}_\mathrm{max}$

TL;DR

This paper classifies all finite extensions of the max-plus semifield max, identifying the structure of semifields and division semirings that extend it finitely.

Contribution

It provides a complete classification of semifields and division semirings finitely generated over max, advancing the understanding of their algebraic extensions.

Findings

Classified all semifields containing max as finitely generated extensions.

Identified the structure of division semirings extending max.

Established the algebraic properties of these finite extensions.

Abstract

We classify the semifields and division semirings containing the max-plus semifield $\mathbb{Z}_\mathrm{max}$, which are finitely generated as $\mathbb{Z}_\mathrm{max}$-semimodules.