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

Abstract

A Hyperstructure is built upon the powerset, providing a framework to model multivalued relations among elements of a set. Extending this idea, a SuperHyperstructure leverages the ?th iterated powerset to represent systems with multi-layered hierarchical relationships, enabling deeper abstraction and complexity. In most treatments, the exponent ? of the iterated powerset is a nonnegative integer. Motivated by fractional analysis in discrete settings, this paper investigates whether fractional and inverse layers can be meaningfully incorporated into set theory and superhyperstructural models. We further extend these notions to SuperHyperStructures by coupling carrier roots with lift/unlift of hyperoperations, yielding root and negative constructions that preserve incidence and recover the original structures after the appropriate number of lifts.