Valuations of Semirings

TL;DR

This paper introduces a new valuation theory for semirings using hyperrings, classifies valuations on max-plus semifields, and explores their connection to tropical and Berkovich spaces.

Contribution

It develops hyperfield valuations for semirings, classifies valuations on $Q_{max}$ and $Q_{max}(T)$, and links these to tropical and analytic geometry.

Findings

Classified valuations on $Q_{max}$ and $Q_{max}(T)$.

Constructed the abstract curve associated to $Q_{max}(T)$.

Connected valuations to tropical and Berkovich spaces.

Abstract

We develop notions of valuations on a semiring, with a view toward extending the classical theory of abstract nonsingular curves and discrete valuation rings to this general algebraic setting; the novelty of our approach lies in the implementation of hyperrings to yield a new definition (\emph{hyperfield valuation}). In particular, we classify valuations on the semifield $\mathbb{Q}_{max}$ (the max-plus semifield of rational numbers) and also valuations on the `function field' $\mathbb{Q}_{max}(T)$ (the semifield of rational functions over $\mathbb{Q}_{max}$) which are trivial on $\mathbb{Q}_{max}$. We construct and study the abstract curve associated to $\mathbb{Q}_{max}(T)$ in relation to the projective line $\mathbb{P}^1_{\mathbb{F}_1}$ over the field with one element $\mathbb{F}_{1}$ and the tropical projective line. Finally, we discuss possible connections to tropical curves and Berkovich's theory of analytic spaces.