In 1957, M. Krasner described a complete valued field (K,v) as the inverse limit of a system of certain structures, called hyperfields, associated with (K,v). We put this result in purely ...category-theoretic terms by translating it into a limit construction in certain slice categories of the category of valued hyperfields and their homomorphisms. We replace the original metric-dependent arguments employed by Krasner with a clean and elegant transition to certain slice categories.
With the aim of developing the recent theory of dependence relations, we elaborate a procedure to measure the strength of the influence of an element on another with respect to a given dependence ...relation on a finite set. We call this measure the degree of influence. Its definition is based on a partial hyperoperation and a directed graph which we associate with any dependence relation. We compute the degree of influence in various examples and prove some general properties. Among these properties, we find symmetries that have the potential to be applied in the realization of effective algorithms for the computations.
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.
Valuation theory is an important area of investigation in algebra, with applications in algebraic geometry and number theory. In 1957, M. Krasner introduced hyperfields, which are field-like objects ...with a multivalued addition, to describe some structures arising naturally from valued fields. We wish to discuss the possibility of generalising the notion of valuation to the multivalued setting and the potential that this higher point of view has in the understanding of classical valuation theory. We will see that a valuation on a field K is nothing but a homomorphism of hyperfields from K onto a special type of hyperfield, which we call a (generalised) tropical hyperfield.
In 1957 M.\ Krasner described a complete valued field \((K,v)\) via the projective limit of a system of certain structures, called hyperfields, associated to \((K,v)\). We put this result in purely ...category-theoretic terms by translating into a limit construction in certain slice categories of the category of valued hyperfields and their homomorphisms.
The main aim of this article is to study and develop valuation theory for Krasner hyperfields. In analogy with classical valuation theory for fields, we generalise the formalism of valuation rings to ...describe equivalence of valuations on hyperfields. After proving basic results and discussing several examples, we focus on the valued hyperfields that Krasner originally defined in 1957. We find that these must have a particular additive structure which in turns implies the existence of a valuation a'la Krasner. We note that given such a valued hyperfield \((F,v)\), the valuation induced by its additive structure does not have to be equivalent to \(v\). We discuss the cases in which it does.
Based on the theory of ball spaces introduced by Kuhlmann and Kuhlmann, we introduce and study Caristi–Kirk and Oettli–Théra ball spaces. We show that if the underlying metric space is complete, then ...these have a very strong property: every ball contains a singleton ball. This fact provides quick proofs for several results which are equivalent to the Caristi–Kirk fixed point theorem, namely Ekeland’s variational principles, the Oettli–Théra theorem, Takahashi’s theorem and the flower petal theorem.
Let \(\mathcal{P}\) be the set of points of a finite-dimensional projective space over a local field \(F\), endowed with the topology \(\tau\) naturally induced from the canonical topology of \(F\). ...Intuitively, continuous incidence abelian group structures on \(\mathcal{P}\) are abelian group structures on \(\mathcal{P}\) preserving both the topology \(\tau\) and the incidence of lines with points. We show that the real projective line is the only finite-dimensional projective space over an Archimedean local field which admits a continuous incidence abelian group structure. The latter is unique up to isomorphism of topological groups. In contrast, in the non-Archimedean case we construct continuous incidence abelian group structures in any dimension \(n \in \mathbb{N}\). We show that if \(n>1\) and the characteristic of \(F\) does not divide \(n+1\), then there are finitely many possibilities up to topological isomorphism and, in any case, countably many.
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 \(\textbf{1}\).
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