Fuzzy Hom-Groups: A New Perspective on Algebraic Generalization

Fuzzy algebraic structures extend classical algebra to model uncertainty, while Hom-groups introduce a twisting map α that modifies associativity and identity conditions. This paper unifies these concepts by introducing Fuzzy Hom-Groups, a generalization of fuzzy groups within the Hom-group framework. We define fuzzy Hom-subgroups and fuzzy Hom-normal subgroups, establishing their fundamental properties. A key result shows that each fuzzy Hom-subgroup induces an upper-level set forming a classical Hom-subgroup, bridging fuzzy group theory and Hom-algebra. We further analyze the structural relationships between fuzzy Hom-subgroups and Hom-subgroups. Illustrative examples highlight how the twisting map influences fuzzy Hom-structures. This study extends fuzzy algebra and Hom-group theory, with potential applications in decision-making, fuzzy control, and uncertainty modeling.