For a hypergroup (H, ) we consider ^, as the smallest equivalence relation on H such that the quotion (H/^, ) 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 S_. 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.
Afshar et al. (Wed,) studied this question.