Hyperfuzzy and SuperHyperfuzzy Integral: A method of studying the Fuzzy Integral under hierarchical uncertainty

Uncertainty modeling is fundamental to decision-making across diverse domains, and numerous frameworks—such as Fuzzy Sets [1, 2], Rough Sets [3, 4], Hesitant Fuzzy Sets [5, 6], Neutrosophic Sets [7, 8], and Plithogenic Sets [9, 10]—have been developed to capture different facets of imprecision. Among these, Hyperfuzzy Sets and their recursive generalization, SuperHyperfuzzy Sets, assign set-valued membership degrees at multiple hierarchical levels to represent uncertainty more richly [11]. The Fuzzy Integral provides a systematic method to aggregate single-valued fuzzy membership functions with respect to a fuzzy measure, thereby yielding a precise evaluation of overall set importance. In this paper, we extend this framework by introducing the Hyperfuzzy Integral and the SuperHyperfuzzy Integral, defined over Hyperfuzzy Sets and SuperHyperfuzzy Sets, respectively, and we investigate their fundamental properties.