Fundamental Functor on Hypergroups

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