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Functors Associated to Relations on Hypergroups and Hypermodules

Behnam Afshar, Reza Ameri

TL;DR

This work shows a deep link between regular relations on strongly regular hypergroups and subhypergroups containing the canonical subset $0_{H}$, via a lattice isomorphism that pairs quotient-structure with congruence-like subobjects. Building on this, it introduces the functor of relations: hom-invariant sequences of strongly regular relations that assign to each hypergroup a quotient and to each morphism a compatible map, thereby producing a rich categorical toolbox. This framework yields a notion of free objects and a tensor product in the category of regular hypergroups, enabling abelian-group-like constructions (e.g., tensor products) to be realized from hypergroup data. Concrete calculations of beta-tensor products illustrate how to control quotients (cyclic, trivial, etc.) from hypergroup information, and the results pave the way for generalizing congruence-type and tensorial methods to broader hyperstructure settings, including Krasner 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.

Functors Associated to Relations on Hypergroups and Hypermodules

TL;DR

This work shows a deep link between regular relations on strongly regular hypergroups and subhypergroups containing the canonical subset , via a lattice isomorphism that pairs quotient-structure with congruence-like subobjects. Building on this, it introduces the functor of relations: hom-invariant sequences of strongly regular relations that assign to each hypergroup a quotient and to each morphism a compatible map, thereby producing a rich categorical toolbox. This framework yields a notion of free objects and a tensor product in the category of regular hypergroups, enabling abelian-group-like constructions (e.g., tensor products) to be realized from hypergroup data. Concrete calculations of beta-tensor products illustrate how to control quotients (cyclic, trivial, etc.) from hypergroup information, and the results pave the way for generalizing congruence-type and tensorial methods to broader hyperstructure settings, including Krasner 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 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.

Paper Structure

This paper contains 6 sections, 17 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regular relations and closed subhypergroups
  4. Functor of relations
  5. Applications
  6. Conclusion

Key Result

Corollary 2.1

If $(H,\circ)$ is a hypergroup and $\mathcal{R}$ be an equivalence relation on $H$, then $\mathcal{R}$ is regular (strongly regular) if and only if $(H/\mathcal{R},\otimes)$ is a hypergroup (group).

Figures (1)

  • Figure :

Theorems & Definitions (46)

  • Definition 2.1: Corsini
  • Corollary 2.1: Davvaz
  • Definition 2.2: Davvaz
  • Proposition 2.1: Jantosciak
  • Proposition 2.2: Jantosciak
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • ...and 36 more