Hypernetwork Theory: The Structural Kernel

Modelling across engineering, systems science, and formal methods remains limited by binary relations, implicit semantics, and diagram-centred notations that obscure multilevel structure and hinder mechanisation. Hypernetwork Theory (HT) addresses these gaps by treating the n-ary relation as the primary modelling construct. Each relation is realised as a typed hypersimplex - alpha (conjunctive, part-whole) or beta (disjunctive, taxonomic) - bound to a relation symbol R that fixes arity and ordered roles. Semantics are embedded directly in the construct, enabling hypernetworks to represent hierarchical and heterarchical systems without reconstruction or tool-specific interpretation. This paper presents the structural kernel of HT. It motivates typed n-ary relational modelling, formalises the notation and axioms (A1-A5) for vertices, simplices, hypersimplices, boundaries, and ordering, and develops a complete algebra of structural composition. Five operators - merge, meet, difference, prune, and split - are defined by deterministic conditions and decision tables that ensure semantics-preserving behaviour and reconcile the Open World Assumption with closure under rules. Their deterministic algorithms show that HT supports reproducible and mechanisable model construction, comparison, decomposition, and restructuring. The resulting framework elevates hypernetworks from symbolic collections to structured, executable system models, providing a rigorous and extensible foundation for mechanisable multilevel modelling.