The hyperfield came into being due to a mathematical necessity that appeared during the study of valuation theory fields by M. Krasner, who also defined hyperring, which is related in same way as ring field. and hyperfields, well rings hyperrings, border on each other, it natural problems open questions arise their boundary areas. This paper presents such occasions, more specifically, introduce...

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

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

In this paper we introduce the complete parts on hyperrings and study the complete closure on hyperrings. Also, we consider the fundamental relation Γ on hyperrings and we prove that the relation Γ is transitive on hyperfields.

Baker and Bowler defined a category of algebraic objects called tracts which generalize both partial fields hyperfields. They also notion weak strong matroids over tract F , proved that if is perfect, meaning -vectors -covectors are orthogonal for every matroid then the notions -matroids coincide. We define class strongly fused prove such perfect. show stringent hyperfields fused; in this way, ...