Mathematical Foundations, Structural Embeddings, and Complement Operations in Hyperbolic Fuzzy Set Theory
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This paper presents a comprehensive study of Hyperbolic Fuzzy Sets (HyFS), a generalized fuzzy framework characterized by independent optimistic and pessimistic membership degrees constrained by a hyperbolic relation. We explore the embedding of q-Rung Orthopair Fuzzy Sets, (n,m)-Rung Orthopair Fuzzy Sets, and (n,m)-Power Root Fuzzy Sets into HyFS, thereby establishing its flexibility and generality. A comparative analysis is carried out based on computational complexity, runtime, and scalability, where HyFS demonstrates superior performance due to its simple structure. The study further investigates complement operations in HyFS, including classical and mixed-type complements using different fuzzy negation functions. Theoretical properties such as De Morgan’s laws, distributivity, and difference operations are examined. Our results highlight the robustness, efficiency, and logical consistency of HyFS, making it a promising tool for uncertainty modeling and decision-making applications.
