Characteristic, C-Characteristic and Positive Cones in Hyperfields

Abstract

We study the notions of the positive cone, characteristic and C-characteristic in (Krasner) hyperfields. We demonstrate how these interact in order to produce interesting results in the theory of hyperfields. For instance, we provide a criterion for deciding whether certain hyperfields cannot be obtained via Krasner’s quotient construction. We prove that any positive integer (larger than 1) can be realized as the characteristic of some infinite hyperfield and an analogous result for the C-characteristic. Finally, we study the (directed) graph associated with the strict partial order induced by a positive cone in a hyperfield in various examples.

Preface to the Special Issue “Algebraic Structures and Graph Theory”

“…For example, the hypergroups are a generalization of groups, while hyperfields are a generalization of the concept of fields. In [10] the authors work with the most well-known class of hyperfields, Krasner hyperfields, studying the notions of positive cone, characteristic and C-characteristic. Using these notions, they provide a criterion for deciding whether certain hyperfields cannot be obtained via Krasner's quotient construction.…”

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.…”

From HX-Groups to HX-Polygroups

“…Its commutative version, i.e., the canonical hypergroup, dates back to the beginning of 1970s, when Mittas [25] studied it as an independent structure in the framework of valuation theory, and not just as the additive structure of a hyperfield. In fact, this was the way that canonical hypergroups appeared in the first studies of Krasner [26] and have continued to be investigated as the additive structure of the Krasner hyperfields and the hypercompositional structure with the most applications in different areas, e.g., valuation theory [27][28][29], algebraic geometry [30], number theory, affine algebraic group schemes [31], matroids theory [32], tropical geometry [33], and hypermodules [34]. The state of the art in hyperfield theory was included in an article recently published by Ch.…”