A Result of Krasner in Categorial Form

Abstract

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.

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.…”