Notes on valuation theory for Krasner hyperfields

TL;DR

This paper develops valuation theory for Krasner hyperfields, generalizing classical valuation concepts from fields, and explores the structure and properties of valuations on these hyperfields, including specific cases and examples.

Contribution

It introduces a valuation framework for Krasner hyperfields, extending classical valuation theory, and analyzes the structure of valued hyperfields including their additive properties.

Findings

Valuation rings can be generalized to hyperfields.

Krasner's valued hyperfields have specific additive structures.

The induced valuation may differ from the original valuation.

Abstract

The main aim of this article is to study and develop valuation theory for Krasner hyperfields. In analogy with classical valuation theory for fields, we generalise the formalism of valuation rings to describe equivalence of valuations on hyperfields. After proving basic results and discussing several examples, we focus on the valued hyperfields that Krasner originally defined in 1957. We find that these must have a particular additive structure which in turns implies the existence of a valuation a'la Krasner. We note that given such a valued hyperfield $(F,v)$, the valuation induced by its additive structure does not have to be equivalent to $v$. We discuss the cases in which it does.