Fractional Powersets and SuperHyperStructures: Toward a Framework for Fractional Set Theory and Discrete Hierarchical Systems

Abstract

Hyperstructures build on powersets to model multivalued relations on a base set; SuperHyperstructures iterate the powerset to capture layered hierarchies and richer composition. Prior work typically fixes the iteration height to a nonnegative integer. This paper asks whether fractional, inverse, and complex (including imaginary) “heights" can be incorporated coherently. We introduce the notions of an m-root powerset (peeling a specified number of subset layers), a negative powerset (a partial inverse of iterated powersets under a given presentation), and a complex-height powerset defined at the level of observables via operator-theoretic interpolation. We characterize when these operators are well defined—by exponential-tower size conditions in the finite case and by the beth hierarchy in the infinite case—and establish exact inverse laws on their natural domains.
Lifting from carriers to operations, we obtain root and negative SuperHyperStructures that preserve incidence, compose naturally, and recover the original structures after the appropriate number of lifts. Conceptually, the framework provides a principled, continuous interpolation across hierarchical levels and a reversible mechanism for descending them, suggesting applications to discrete modeling, policy design, and multi-resolution analysis.