Notes on valuation theory for Krasner hyperfields

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.