Hierarchical Uncertainty Modeling via (m, n)-Superhyperuncertainand (h, k)-ary (m, n)-Superhyperuncertain Sets: Unified Extensions of Fuzzy, Neutrosophic, Soft, Rough, and Plithogenic Set Theories

Abstract

Various set-theoretic frameworks have been widely recognized for their effectiveness in handling uncertainty, including Fuzzy Sets, Neutrosophic Sets, Plithogenic Sets, Rough Sets, and Soft Sets. These foundational models have been further extended through the use of hyperstructures—based on the powerset construction—and superhyperstructures—based on the n-th-order powerset, obtained by iteratively applying the powerset operation [1, 2]. These extended constructs are collectively referred to as HyperUncertain Sets and SuperHyperUncertain Sets. Research on SuperHyperUncertain Sets is still in its early stages, and investigations into their properties, extended forms, and potential applications are expected to become increasingly sig-
nificant in the future. In this paper, we propose two new, more general frameworks: the (m, n)-SuperHyperUncertain Set and the (h, k)-ary (m, n)-SuperHyperUncertain Set. These new structures represent a concrete and refined reconsideration of the foundational concepts introduced in [1, 2]. It is anticipated that the concepts developed in this work can be effectively applied to
the modeling of more hierarchical forms of uncertainty, as well as to scenarios requiring complex membership functions. Since this paper conducts only theoretical analysis, we also hope that quantitative analysis using computational methods will be carried out in the future.