(untitled) (European Journal of Pure and Applied Mathematics, 2025)
This study develops a novel framework for bipolar-valued intuitionistic fuzzy positive implicative ideals (BPVIFPIIs) in BCK-algebras by integrating bipolar-valued intuitionistic fuzzy set theory with algebraic structures. The primary objective is to define and explore the properties of BPVIFPIIs in BCK-algebras, providing rigorous theoretical foundations supported by illustra- tive examples. Key conditions under which a bipolar-valued intuitionistic fuzzy set qualifies as a BPVIFPII are established. The findings reveal significant connections between BPVIFPIIs and other fuzzy ideals, highlighting their role in advancing the understanding of uncertainty and algebraic reasoning. This research opens avenues for further exploration of bipolar fuzzy structures in algebra and their practical implications in decision-making processes involving uncertain data.