Orderings and valuations in 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.

Generalisations of Tropical Geometry over Hyperfields

“…They will be recalled here. (Note that in the literature on ordered hyerpfields, such as [KLS21], the notion of quotient hyperfields is referred to as Factor hyperfields.) This is a strict partial order on H. It also has a corresponding non-strict partial order ď defined by…”

“…Unlike the analogous construction over fields, compatibility is particularly troublesome, as 𝑎 ' 𝑐 and 𝑏 ' 𝑐 may be sets and there is no canonical way to extend ď to sets. There is a discussion in Section 2 of [LS20] and Section 3 of [KLS21].…”

“…Definition 6.2.3. [AD19, Def.2.2] An ordering on a hyperfield pH, ', d, 𝟘, 𝟙q is a subsetH `Ă H satisfying: • H `' H `Ď H `, • H `d H `Ď H `,• H " H There are discussions of equivalent notions of orderings for hyperfields in[LS20] and[KLS21] and for multirings in[Mar06]. The intuition one should have is that the subset H Example 6.2.4.…”

“…KLS21, Theorem 3.4]. Explicitly, the set of orderings on H is denoted 𝜒pHq and the set of orderings on H containing a subset 𝑈 is denoted 𝜒pH | 𝑈q.…”

“…Explicitly, the set of orderings on H is denoted 𝜒pHq and the set of orderings on H containing a subset 𝑈 is denoted 𝜒pH | 𝑈q. Then,[KLS21, Theorem 3.4] shows that |𝜒p𝐾{𝑈q| " |𝜒p𝐾 | 𝑈q|. This relationship is used to demonstrate that hyperfields can have an uncountable number of orderings.…”