A Unified Framework for U-Structures and Functorial Structure: Managing Super, Hyper, SuperHyper, Tree, and Forest Uncertain Over/Under/OffModels

Abstract

A Classical Structureis a mathematical framework on a nonempty set퐻equipped with one or more operations satisfying specified axioms. Here, “Structure” refers broadly to any formal system—from set theory and logic to probability, statistics, algebra, geometry, and beyond. In the domain of fuzzy sets and their extensions, a multitude of models has emerged—fuzzy sets, vague sets, intuitionistic fuzzysets, neutrosophic sets, picture fuzzy sets, hesitant fuzzy sets, plithogenic sets, and more. While this diversity reflects an active researchcommunity, it also introduces conceptual overlap. We therefore group all such variants under the termUncertain Sets.We propose a unified framework for their systematic management by defining the notions of theUncertain Model, the U-Structure, and the U-Off Structure. We then revisit and extend the HyperUncertain, SuperUncertain, SuperHyperUncertain, TreeUncertain, and ForestUncertainstructures, along with their Off counterparts. To encompass systems beyond Uncertain Sets, we further introduce Functorial Set and Functorial Structure. This comprehensive approach establishes a robust foundation for organizing and applying uncertainty models across diversemathematical and applied domains