HyperInterval-valued and SuperHyperInterval-valued Fuzzy/Neutrosophic Set

We study uncertainty models built from interval families over a finite universe. An interval set collects allsubsets bounded between a designated lower and upper set. A HyperInterval set assigns to each base intervala nonempty family of admissible refinements, while a SuperHyperInterval set of order ? maps elements of the?-fold iterated powerset to (?−1)-nested families, enabling hierarchical evidence organization. On the numericside, an interval-valued fuzzy set attaches to each element an interval of admissible memberships, and aninterval-valued neutrosophic set assigns independent intervals for truth, indeterminacy, and falsity. Building onthese primitives, we introduce HyperInterval- and SuperHyperInterval-valued fuzzy/neutrosophic sets, defineconjunctive “core” (intersection) and disjunctive “hull” semantics, and prove embedding theorems showingthat classical interval, fuzzy, and neutrosophic models appear as singleton or degenerate cases. Realisticexamples from commute planning, delivery scheduling, and clinical assessment illustrate the methodology.