Complex Hyperbolic Fuzzy Sets: Theory and Construction

Abstract

Uncertainty modeling has evolved from intuitionistic fuzzy sets to q-rung orthopair fuzzy sets, with successive extensions seeking to enlarge the admissible space for membership and non-membership evaluations. However, these conventional orthopair models remain bounded by restrictive linear or polynomial constraints, which may limit their ability to represent scenarios involving simultaneously high membership and non-membership degrees. Furthermore, standard orthopair models do not inherently represent periodicity, seasonality, or two-dimensional directional information. To address these limitations, this paper introduces the Complex Hyperbolic Fuzzy Set (Complex-HyFS), a novel framework that combines the algebraic flexibility of hyperbolic geometry with the vector-based representation of complex fuzzy sets. The Complex-HyFS is characterized by the hyperbolic constraint r_O(x) · r_P(x) ≤ 1, which geometrically extends the valid information space to the entire unit square in the amplitude plane, thereby permitting the independent assignment of optimistic and pessimistic degrees. Simultaneously, the integration of a complex phase term e^(iω(x)) enables the representation of periodic uncertainty and constructive and destructive interference patterns. We rigorously establish the fundamental set-theoretic operations—union, intersection, and complement—and prove that the Complex-HyFS structure is closed under these operations. We further demonstrate that the proposed complement operator satisfies the extended De Morgan laws and preserves the hyperbolic constraint without degeneracy. Additionally, a systematic methodology for constructing Complex-HyFSs from classical fuzzy sets via complement generators is presented. Finally, the applicability of the framework is discussed in the contexts of bio-signal analysis and seasonal forecasting, illustrating the potential of Complex-HyFS as a generalized framework for modeling high-entropy and periodic uncertainty.