Linear Diophantine HyperFuzzy Set and SuperHyperFuzzy Set

Abstract

Uncertainty modeling underpins decision-making across diverse domains, and numerous frameworks—such as Fuzzy Sets, Rough Sets, Hesitant Fuzzy Sets, and Plithogenic Sets—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. The Linear Diophantine Fuzzy Set further refines this approach by imposing weighted linear Diophantine constraints on membership and non-membership grades. 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. A Linear Diophantine HyperFuzzy Set assigns each element set-valued membership and nonmembership grades, constrained by a linear Diophantine relation. A (m,n)-Linear Diophantine SuperHyperFuzzy Set assigns each element set-valued membership and nonmembership grades, constrained by a linear Diophantine relation. We also examine the algorithms associated with these notions. These extensions offer a more structured, hierarchical means of applying Linear Diophantine Fuzzy Set methodology in practical uncertain environments.