ON ANTI-FUZZY IMPLICATIVE AND ANTI-FUZZY SUB-IMPLICATIVE IDEALS IN Z-ALGEBRAS

This paper presents a comprehensive examination of anti-fuzzy implicative and anti-fuzzy sub-implicative ideals in Z-algebras. Drawing from recent advances in algebraic fuzzy logic, we investigate the fundamental properties, structural relationships, and applications of these novel ideal concepts. We establish the interrelationships between fuzzy Z-ideals, fuzzy implicative ideals, and their anti-fuzzy counterparts, demonstrating that anti-fuzzy structures provide complementary frameworks for modelling uncertainty in non-associative algebraic systems. Key theoretical results include characterization theorems, preservation properties under homomorphisms, and conditions for equivalence between different ideal classes. Crucially, we provide formal derivations showing that the implicative condition implies the medial condition using Z-algebra axioms, and we verify the well-definedness of the supremum-based image construction under surjective homomorphisms. This research contributes to the broader understanding of how fuzzy and anti-fuzzy methodologies can coexist in abstract algebra to address limitations in classical ideal theory.