HyperInterval-Valued and SuoerHyperInterval-Valued Fuzzy/Neutrosophic Set

We study uncertainty models built from interval families over a finite universe. An interval set collects all subsets bounded between a designated lower and upper set. A HyperInterval set assigns to each base interval a nonempty family of admissible refinements, while a SuperHyperInterval set of order n maps elements of the n-fold iterated powerset to (n−1)-nested families, enabling hierarchical evidence organization. On the numeric side, an interval-valued fuzzy set attaches to each element an interval of admissible memberships, and an interval-valued neutrosophic set assigns independent intervals for truth, indeterminacy, and falsity. Building on these primitives, we introduce HyperInterval- and SuperHyperInterval-valued fuzzy/neutrosophic sets, define conjunctive “core” (intersection) and disjunctive “hull” semantics, and prove embedding theorems showing that classical interval, fuzzy, and neutrosophic models appear as singleton or degenerate cases. Realistic examples from commute planning, delivery scheduling, and clinical assessment illustrate the methodology. The framework unifies multi-source and hierarchical evidence, offering transparent bounds for conservative and exploratory decision policies.