Abelian groups derived from H v -groups

Abstract

This paper explores new aspects of regular and fundamental relations on Hv-groups, focusing on the introduction and properties of specific equivalence relations. We define the α and α⋆ relations on Hv-groups, respectively, and demonstrate that these structures yield to significant results when applied to the quotient of an Hv-group. Our main findings show that the fundamental relation α⋆ enables the quotient set of an Hv-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.

Keywords: Abelian group, fundamental relation, H v -group, quotient group, regular equivalence relation