HyperLattice-valued and SuperHyperLattice-valued Uncertain Sets with Decision Applications

Abstract

Fuzzy set theory enriches classical sets by assigning to each element a graded membership in [0,1], thereby capturing partial inclusion and uncertainty. The notion of an Uncertain Set further abstracts this idea by allowing membership to take values in a general degree-domain, providing a unified language that subsumes fuzzy, intuitionistic fuzzy, neutrosophic, plithogenic, and related models. On the algebraic side, a hyperlattice replaces one lattice operation by a multivalued hyperoperation, enabling the representation of ambiguous or non-deterministic combinations, while a superhyperlattice iterates this structure through powerset lifting to obtain higher-order layers of interaction. Motivated by these developments, we introduce HyperLattice-valued and SuperHyperLattice-valued Uncertain Sets as lattice-valued uncertainty frameworks whose degrees range over hyperlattices and their superextensions. We establish basic definitions, show that the proposed formalisms generalize existing lattice-valued models (including L-fuzzy, L-neutrosophic, and L-plithogenic sets), and discuss fundamental structural properties and canonical embeddings between the resulting classes.