Abstract
Fuzzy sets, introduced by Zadeh, and their extensions—including intuitionistic fuzzy sets and neutro-sophic sets—provide flexible mathematical frameworks for representing graded, hesitant, incomplete, and even mutually inconsistent information. These uncertainty models have been extended to network structures through fuzzy graphs, intuitionistic fuzzy graphs, and neutrosophic graphs. A common feature of such models is that uncertainty is typically assigned to vertices and/or edges, whereas the orientation of an edge is usually treated as fixed. This raises a natural question: can uncertainty be associated not only with vertices or edges, but also with the orientation itself ? More specifically, can one represent an interaction whose direction is not completely determined—for example, an edge that may be oriented from u to v to one degree, from v to u to another degree, or may instead be regarded as undirected or bidirected under uncertainty? To address this type of orientation un-certainty, the concepts of the Fuzzy-Oriented Mixed Graph and the Neutrosophic-Oriented Mixed Graph have recently been introduced. In this paper, we extend these orientation-uncertainty frameworks from ordinary graphs to higher-order network structures. Specifically, we introduce the FuzzDi-rected HyperGraph, the Single-Valued NeutroDirected HyperGraph, the FuzzDirected n-SuperHyperGraph, and the Single-Valued NeutroDirected n-SuperHyperGraph. These constructions provide mathematical frameworks for representing fuzzy and neutrosophic uncertainty in the orientations of hyperedges and superhyperedges, thereby extending orientation-uncertain graph modeling to hypergraph and SuperHyperGraph settings.