Fundamental Functor on Hypergroups

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 $S_{\hat{I}}^3$. Then, we examine the functor $γ^{\ast}$ from a categorical perspective and investigate properties such as continuity and cocontinuity concerning it using the decomposition ${\hat{I}}^3=\hat{I}´\ast {\hat{I}}^2$. 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.

19 pages