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Fundamental Functor on Hypergroups

Behnam Afshar, Reza Ameri

Abstract

For a hypergroup $(H,\circ)$ we consider $γ^{\ast}$, as the smallest equivalence relation on $H$ such that the quotion $(H/γ^{\ast},\tiny{\otimes})$ is an abelian group. We study some more properties of $γ^{\ast}$. Initially, it is investigated which subhypergroup the congruence relation modulo is strongly regular on, and its quotient results in an abelian group? This is directly related to the fundamental relation $γ^{\ast}$, since such subhypergroups must contain $S_γ$. Then, we examine the functor $γ^{\ast}$ from a categorical perspective and investigate properties such as continuity and cocontinuity concerning it using the decomposition $γ=δ\tiny{\ast}β$. For this purpose, we define the reduced words on strongly regular hypergroups. This has a direct application in studying how the functor $γ^{\ast}$ affects on the stalks of the sheaves of hypergroups.

Fundamental Functor on Hypergroups

Abstract

For a hypergroup we consider , as the smallest equivalence relation on such that the quotion is an abelian group. We study some more properties of . Initially, it is investigated which subhypergroup the congruence relation modulo is strongly regular on, and its quotient results in an abelian group? This is directly related to the fundamental relation , since such subhypergroups must contain . Then, we examine the functor from a categorical perspective and investigate properties such as continuity and cocontinuity concerning it using the decomposition . For this purpose, we define the reduced words on strongly regular hypergroups. This has a direct application in studying how the functor affects on the stalks of the sheaves of hypergroups.

Paper Structure

This paper contains 5 sections, 19 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Canonical quotient hypergroups and $\gamma^{\ast}$ relation
  4. Categorical properties of $\gamma^{*}$
  5. Conclusions

Key Result

Theorem 2.2

Survey2 Let $(H,\circ)$ be a semi-hypergroup and $\mathcal{R}$ be an equivalence relation on $H$; If $\mathcal{R}$ is regular, then $H/\mathcal{R}$ is a semi-hypergroup with respect to the hyperoperation $\mathcal{R}(x)\tiny{\otimes}\mathcal{R}(y)=\{{\mathcal{R}(z); z\in x\tiny{\circ}y}\}$; If the

Theorems & Definitions (33)

  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Definition 2.9
  • Lemma 3.1
  • ...and 23 more