August 17, 2025

Foundations of (m, n; L)-SuperHyperFuzzy Sets

Key Points

The introduction of (m, n; L)–superhyperfuzzy sets enhances the modeling of uncertainty across various domains, particularly in decision-making.
Enumeration of relationships reveals that superhyperfuzzy sets relate closely to other structures, such as neutrosophic and plithogenic sets.
This research employs rigorous definitions and mathematical structure, focusing on how these memberships operate within residuated lattices.
The findings suggest a path forward in uncertainty modeling, potentially impacting fields ranging from artificial intelligence to decision support systems.

Abstract

Uncertainty modeling plays a crucial role in decision-making across diverse domains 1, and numerous mathematical frameworks have been proposed to capture various aspects of imprecision. These include Fuzzy Sets 2, Rough Sets 3,4, Vague Sets 5,6, Intuitionistic Fuzzy Sets 7,8, Hesitant Fuzzy Sets 9,10, Neutrosophic Sets 11,12, and Plithogenic Sets 13,14. Among these developments, Hyperfuzzy Sets 15,16 and their recursive generalizations, SuperHyperfuzzy Sets 17, extend the classical notion by assigning setvalued membership degrees at multiple hierarchical levels. In this paper, we formally define the concept of (𝑚, 𝑛; 𝐿)–SuperHyperFuzzy Sets and investigate their relationships with related structures, including SuperHyperFuzzy Sets, SuperHyperNeutrosophic Sets, and SuperHyperPlithogenic Sets. An (𝑚, 𝑛; 𝐿)–SuperHyperFuzzy Set maps nonempty 𝑚-level subsets of a base set to nonempty families of 𝑛-level degree-sets valued in a complete commutative residuated lattice 𝐿, supporting 𝑡-norm/𝑡-conorm aggregation.

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