On the relation between hyperrings and fuzzy rings

Abstract

We construct a full embedding of the category of hyperfields into Dress’s category of fuzzy rings and explicitly characterize the essential image—it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker and Bowler’s theory of strong matroids over hyperfields with Dress’s theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker and Bowler’s partial demifields.