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On the hyperfields associated to valued fields
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abstract
One can associate to a valued field an inverse system of valued hyperfields $(\mathcal{H}_i)_{i \in I}$ in a natural way. We investigate when, conversely, such a system arise from a valued field. First, we extend a result of Krasner by showing that the inverse limit of certain systems are stringent valued hyperfields. Secondly, we describe a Hahn-like construction which yields a henselian valued field from a stringent valued hyperfield. In addition, we provide an axiomatisation of the theory of stringent valued hyperfields in a language consisting of two binary function symbols $\oplus$ and $\cdot$ and two constant symbols $\textbf{0}$ and $\textbf{1}$.
Forward citations
Cited by 2 Pith papers
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Algorithmic Construction of Real Hyperfields from Minimal Axioms
A complete enumeration algorithm for finite real hyperfields with cyclic positive cones, with classification data to order 15, C-characteristics to order 17, and many new non-Krasner quotient hyperfields.
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On the borderline of fields and hyperfields, part II -- Enumeration and classification of the hyperfields of order 7
There are exactly 277 seven-element hyperfields, all built on the cyclic multiplicative group of order six, and the paper classifies which of them arise as quotients of fields.
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