Fundamental Functor on Hypergroups

TL;DR

This paper investigates the fundamental functor on hypergroups, focusing on the properties of the smallest equivalence relation that yields an abelian group quotient, from both algebraic and categorical perspectives.

Contribution

It introduces and analyzes the functor ^{*} on hypergroups, exploring its properties, regularity conditions, and effects on sheaves, providing new insights into hypergroup structure and categorical behavior.

Findings

Characterization of subhypergroups where the quotient is an abelian group

Properties of the functor ^{*} regarding continuity and cocontinuity

Application to sheaves of hypergroups and their stalks

Abstract

For a hypergroup $(H,\circ)$ we consider $\gamma^{\ast}$, as the smallest equivalence relation on $H$ such that the quotion $(H/\gamma^{\ast},\tiny{\otimes})$ is an abelian group. We study some more properties of $\gamma^{\ast}$. 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 $\gamma^{\ast}$, since such subhypergroups must contain $S_{\gamma}$. Then, we examine the functor $\gamma^{\ast}$ from a categorical perspective and investigate properties such as continuity and cocontinuity concerning it using the decomposition $\gamma=\delta\tiny{\ast}\beta$. For this purpose, we define the reduced words on strongly regular hypergroups. This has a direct application in studying how the functor $\gamma^{\ast}$ affects on the stalks of the sheaves of hypergroups.