Hyperfuzzy and Super-Hyperfuzzy Extensions of Fuzzy Metric Spaces, Queuing Models, and Ontologies

Uncertainty modeling underpins decision-making in domains ranging from engineering to artificial intelligence. A variety of frameworks—such as Fuzzy Sets 1, Rough Sets 2, 3, Picture Fuzzy Sets 4, 5, Soft Sets 6, 7, and Plithogenic Sets 8—have been proposed to capture different dimensions of imprecision. Hyperfuzzy Sets and their recursive extension, SuperHyperfuzzy Sets, further enrich this landscape by assigning set-valued membership degrees at multiple hierarchical levels. Building on the concept of a fuzzy metric space—a set 𝑋 equipped with a continuous 𝑡-norm ∗ and a function 𝑀: 𝑋²× (0, ∞) → 0, 1 satisfying positivity, symmetry, the triangle inequality, and continuity—we introduce Hyperfuzzy and SuperHyperfuzzy metrics that model proximity with multi-level, set-valued memberships. We also investigate how Hyperfuzzy and SuperHyperfuzzy constructs can be applied to extend fuzzy queues and fuzzy ontologies, illustrating their potential to enhance performance evaluation and knowledge representation in real-world systems.