On the relation between hyperrings and fuzzy rings

TL;DR

This paper establishes a categorical embedding linking hyperfields and fuzzy rings, connecting matroid theories over these structures, and explores the extension to hyperrings and their relation to partial demifields.

Contribution

It constructs a full embedding of hyperfields into fuzzy rings, clarifies the connection between matroid theories, and analyzes the extension to hyperrings and partial demifields.

Findings

Embedding of hyperfields into fuzzy rings is essentially full.

Matroid theories over hyperfields and fuzzy rings are equivalent within the image.

Extension from hyperfields to hyperrings is explicitly characterized.

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's theory of 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's partial demifields.