Functors Associated to Relations on Hypergroups and Hypermodules

TL;DR

This paper explores the structure of regular relations on strongly regular hypergroups, establishing lattice isomorphisms, and introduces functors to define free objects and tensor products in the hypergroup category.

Contribution

It introduces a novel connection between regular relations and subhypergroups, and develops functors for free objects and tensor products in hypergroup theory.

Findings

Regular relations form a lattice isomorphic to subhypergroups.

Defined functors lead to new free objects in hypergroup categories.

Established tensor product constructions for hypermodules.

Abstract

If H is a strongly regular hypergroup, we show that the set of regular relations on H and the set of subhypergroups containing $0_{H}$ are two lattices that are isomorphic to each other. In the next step, we introduce and study the properties of functors that are constructed by a sequence of strongly regular relations. This helps us to define a specific type of free objects and tensor products on the category of regular hypergroups.