HyperCloudy Number and (m,n)-SuperHyperCloudy Number

Abstract

Uncertainty modeling is fundamental to decision-making across diverse domains, and numerous frameworks, such as Fuzzy Sets, Rough Sets, Hesitant Fuzzy Sets, and Plithogenic Sets, have been developed to capture different facets of imprecision. Among these, HyperFuzzy Sets and their recursive generalization, SuperHyperFuzzy Sets, assign set-valued membership degrees at multiple hierarchical levels to represent uncertainty more richly. A Cloudy Number assigns to each real value a closed interval within [0,1], modeling uncertainty via lower and upper membership levels. Cloudy Numbers have deep connections with Fuzzy Numbers and real numbers, and substantial related research has been conducted. However, the counterparts of HyperFuzzy Numbers and SuperHyperFuzzy Numbers in the setting of Cloudy Numbers remain unexplored. In this paper, we introduce two new constructs: the HyperCloudy Number and the (m,n)-SuperHyperCloudy Number. Inspired by the HyperFuzzy Number and the (m,n)-SuperHyperFuzzy Number, these concepts hierarchically generalize the existing notion of a Cloudy Number. We establish their fundamental properties and demonstrate their applicability using concrete examples.

Received: 12 September 2025 | Revised: 23 October 2025 | Accepted: 29 October 2025

Conflicts of Interest

The author declares that he has no conflicts of interest to this work.

Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study. This research is purely theoretical, involving no data collection or analysis. We encourage future researchers to pursue empirical investigations to further develop and validate the concepts introduced here.

Author Contribution Statement

Takaaki Fujita: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing — original draft, Writing — review & editing, Visualization, Supervision, Project administration.