Orderings and valuations in hyperfields

TL;DR

This paper extends the theory of valuations and orderings from real fields to real hyperfields, introducing compatibility notions, exploring their relations, and generalizing key theorems like Baer-Krull to this broader context.

Contribution

It introduces the concept of compatibility between valuations and orderings in real hyperfields and generalizes the Baer-Krull theorem to this setting.

Findings

Compatibility notions between valuations and orderings are established.

Relations between valuations and orderings on hyperfield quotients are analyzed.

The Baer-Krull theorem is successfully generalized to real hyperfields.

Abstract

We introduce and study in detail the notion of compatibility between valuations and orderings in real hyperfields. We investigate their relation with valuations and orderings induced on factor and residue hyperfields. Much of the theory from real fields can be generalized to real hyperfields; we point out facts that cannot. We generalize the Baer-Krull theorem to real hyperfields.