Orderings and valuations in hyperfields

Convex geometry over ordered hyperfields

“…We use a definition from ordered field theory and real algebraic geometry rather than combinatorial notions from order theory; the reasons for this will become apparent in Section 3.3. Note that this definition has been used and studied in a number of places within the hyperfield literature [45,10,36].…”

On the hyperfields associated to valued fields

“…The interested reader can consult the introduction of [26] for a summary on earlier work of Krasner. For recent developments in the research on hyperfields see for instance [19,6]. We note that hyperfields have recently played a central role in an interesting paper of Junguk Lee on the model theory of valued fields (see [20,Theorem 5.8]).…”

“…Remark 1.9. In the literature (see for instance [19]) one may find a weaker notion of valuation on hyperfields. This is obtained by asking for axioms (V0), (V1) and (V2) only.…”

“…Let H and H ′ be valued hyperfields with the same value group Γ. 19. Notice that Krasner originally considered only valuations of rank one.…”

A Result of Krasner in Categorial Form

“…A hyperfield is a field-like structure where the latter property is relaxed for the additive operation. In the literature, such structures appear perhaps more than one would expect: hyperfields are of interest, e.g., in tropical geometry [1][2][3], symmetrization [4][5][6], projective geometry [7], valuation theory [8][9][10][11], and ordered algebra [12][13][14]. There are even reasons to believe that their theory generalizes field theory in ways that can be used to tackle deep problems such as the description of F 1 , the "field of characteristic one" (cf.…”