Modeling Uncertainty with Interval-Valued Intuitionistic Fuzzy Filters in Hoop Algebras

This paper systematically investigates interval-valued intuitionistic fuzzy (IVIF) sets and filters within the framework of hoop algebras, unifying and extending classical fuzzy set theory and intuitionistic fuzzy sets (IFS) in algebraic logic. We clarify the foundational relationships among fuzzy sets, IFS, and hoop algebras, and introduce novel characterizations of IVIF filters, including necessary and sufficient conditions for their existence and structure. Theoretical advancements include the demonstration that IVIF filters can be described via their endpoint functions, the establishment of a bounded distributive lattice of IVIF filters, and the identification of congruence relations induced by these filters. Algorithmic and numerical aspects are addressed through explicit pseudocode and detailed examples, illustrating how the verification and construction of IVIF filters can be performed in finite hoop algebras. Practical implications are highlighted in decision-making scenarios where modeling uncertainty and vagueness with interval-valued membership and non-membership degrees offers enhanced flexibility and robustness. Our results lay a rigorous foundation for further applications of IVIF filters in fuzzy logic, artificial intelligence, and multi-criteria decision analysis.