Recursive Intuitionistic Fuzzy SuperHyperGraphs

Abstract

Finite hypergraphs generalize ordinary graphs by allowing each hyperedge to connect an arbitrary nonempty subset of vertices, thereby providing a natural framework for genuinely multiway interactions. To represent hierarchical and multi-layer structures, SuperHyperGraphs further extend this framework via iterated powerset constructions, so that set-valued objects formed at one level may serve as vertices at higher levels. Independently, recursive hypergraphs enrich the edge structure by allowing a hyperedge to contain not only vertices but also lower-level hyperedges, yielding nested incidence relations under a prescribed recursion depth. In this paper, we unify these two directions and introduce Recursive Intuitionistic Fuzzy SuperHyper Graphs. The proposed model combines hierarchical supervertices, recursively defined superhyperedges, and intuitionistic fuzzy membership/non-membership grades in the sense of Atanassov, enabling the representation of higher-order systems that are simultaneously hierarchical, recursive, and uncertain. We formulate the structure on a well-founded recursive universe, establish the fundamental axioms (including vertex–edge consistency and covering conditions), and study basic structural properties. In particular, we discuss induced substructures, isomorphisms, and level-induced (depth-truncated) structures, and clarify how the model re duces to standard intuitionistic fuzzy hypergraph-type objects in special cases. The proposed framework provides a mathematically consistent foundation for modeling complex relational systems with nested inter actions and uncertainty across multiple levels of organization.