Association schemes and Hypergroups

TL;DR

This paper explores the connection between association schemes and hypergroups, introducing the class of realizable hypergroups and demonstrating their properties and examples, including projective geometries.

Contribution

It defines realizable hypergroups derived from association schemes and proves that certain classes like partition and linearly ordered hypergroups are realizable.

Findings

Partition hypergroups are realizable.

Linearly ordered hypergroups are realizable.

Certain projective geometries have canonical association scheme structures.

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.