Jaiung Jun

On the relation between hyperrings and fuzzy rings

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.

Association schemes and hypergroups

ABSTRACT In this paper, we investigate hypergroups which arise from association schemes in a canonical way; this class of hypergroups is called realizable. We first study basic algebraic properties of realizable hypergroups. Then we prove that two interesting classes of hypergroups (partition hypergroups and linearly ordered hypergroups) are realizable. Along the way, we prove that a certain class of projective geometries is equipped with a canonical association scheme structure which allows us to link three objects; association schemes, hypergroups, and projective geometries (see, Section 1.2 for details).