Fundamental Functor on Hypergroups

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.