Linear Diophantine HyperFuzzy Set and SuperHyperFuzzy Set

Abstract

Uncertainty modeling underpins decision-making across diverse domains, and numerous frameworks—such as Fuzzy Sets [1, 2], Rough Sets [3, 4], Hesitant Fuzzy Sets [5, 6], and Plithogenic Sets [7, 8]—have been developed to capture different facets of imprecision. Hyperfuzzy Sets and their recursive generalization, SuperHyperfuzzy Sets, assign set-valued membership degrees at multiple hierarchical levels to represent uncertainty more richly [9]. The Linear Diophantine Fuzzy Set further refines this approach by imposing weighted linear Diophantine constraints on membership and non-membership grades [10–13]. In this paper, we define two new constructs—the Linear Diophantine Hyperfuzzy Set and the Linear Diophantine SuperHyperfuzzy Set—by integrating Diophantine constraints with hyperfuzzy and superhyperfuzzy frameworks, and we present a concise application example. These extensions offer a more structured, hierarchical means of applying Linear Diophantine Fuzzy Set methodology in practical uncertain environments.