Bipolar Fuzzy Commutative Hyper BCK-ideals in Hyper BCK-algebras

Abstract

This study introduces the concept of bipolar fuzzy commutative hyper BCK-ideals (BF-CHBCKIs) within the algebraic framework of hyper BCK-algebras, offering a novel approach to modeling dual uncertainty through bipolar fuzzy sets. By defining and classifying BF-CHBCKIs across multiple types and examining their structural relationships with reflexive, strong, and weak hyper BCK-ideals, we establish a comprehensive theoretical foundation supported by formal theorems and illustrative examples. These findings extend current understandings in hyperstructure theory and fuzzy algebra, contributing to the broader landscape of abstract mathematical reasoning. Importantly, this research aligns with the goals of Sustainable Development Goal 4 (SDG-4) by promoting inclusive and equitable quality education. The formalization of BF-CHBCKIs fosters advanced mathematical thinking and provides meaningful tools for enhancing learning environments, particularly in schools and institutions that emphasize research-oriented instruction. By integrating abstract algebraic structures with uncertainty modeling, this work supports the cultivation of analytical skills, mathematical creativity, and deeper engagement with formal logic among students and emerging researchers.