Abelian groups derived from Hv-groups

This paper explores new aspects of regular and fundamental relations on [Formula: see text]-groups, focusing on the introduction and properties of specific equivalence relations. We define the [Formula: see text] and [Formula: see text] relations on [Formula: see text]-groups, respectively, and demonstrate that these structures yield to significant results when applied to the quotient of an [Formula: see text]-group. Our main findings show that the fundamental relation [Formula: see text] enables the quotient set of an [Formula: see text]-group to form an abelian group. Through rigorous proofs, we establish conditions under which these relations maintain strongly regularity and transitivity. Additionally, we provide a geometric interpretation of these relations, expanding on their utility in broader algebraic contexts. This study generalizes earlier work and contributes to understanding hyperstructure allow for the application of traditional group theory techniques to more complex algebraic structures.