Iterated Hypernode Model, Iterated UberGraph, and Iterated Relational MetaHyperGraph

The Hypernode Model and UberGraph represent nested vertices and recursive hyperedge membership, respectively, while Meta-HyperGraphs describe relations among hypergraphs. We study three finite, typed extensions of these models. An Iterated Hypernode Model is equipped with explicit cross-level links between vertex occurrences; an Iterated UberGraph treats an entire UberGraph as a fundamental vertex at the next outer level; and an Iterated Relational MetaHyperGraph uses an edge-witnessed lifting of admissible relations. We establish well-definedness, canonical injective representations, and basic properties of nesting depth and relation lifting. Examples illustrate the constructions and clarify the distinction between an incidence-preserving embedding and a recoverable higher-level encoding.