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September 5, 2025Open Access

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

Key Points

Introducing fractional analysis into set theory allows deeper insights into discrete hierarchical systems, enabling novel representations.
Superhyperstructures utilize the iterated powerset to model multi-layered relationships, presenting a unique framework for advanced structures.
Coupling carrier roots with hyperoperations preserves structures during fractional manipulation, leading to new theoretical constructs.
Extending traditional set theory concepts to include fractional aspects may enhance understanding of complex relational 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 𝑛th iterated powerset to represent systems with multi-layered hierarchical relationships, enabling deeper abstraction and complexity. In most treatments, the exponent 𝑛 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.

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

Key Points

Abstract

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