Topological Approximations Using Maximal Neighborhoods and Primal Structures with Advanced Decision-Making Strategies for Heart Failure Diagnosis

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DOI: https://doi.org/10.28924/2291-8639-24-2026-349
Published: Oct 15, 2026
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M. A. El-Gayar, O. A. Embaby, M. K. El-Bably, R. A. Hosny, Topological Approximations Using Maximal Neighborhoods and Primal Structures with Advanced Decision-Making Strategies for Heart Failure Diagnosis, Int. J. Anal. Appl., 24 (2026), 349.

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M. A. El-Gayar, O. A. Embaby, M. K. El-Bably, R. A. Hosny

Abstract

This paper develops a unified framework for modeling uncertainty in structured and evolving data environments by integrating primal structures with generalized approximation methods based on \(\kappa\)-neighborhood systems. Classical rough set theory is primarily founded on equivalence relations, which may limit its ability to effectively represent imprecise, dynamic, and sensitive information arising in complex real-world applications. To overcome these limitations, we introduce novel lower and upper approximation operators defined through \(\kappa\)-neighborhood spaces and primal sets, with particular emphasis on maximal neighborhood structures. The proposed approximation operators provide enhanced flexibility in the construction of rough approximations, particularly within topological and supra-topological settings. They enable approximation sets to capture more effectively the uncertainty associated with structural asymmetry, deformation, and changes in the underlying data. The fundamental properties of the proposed operators are systematically established and illustrated through suitable examples. Furthermore, a comprehensive comparative analysis with classical and recent rough set models demonstrates the effectiveness of the proposed framework in improving data representation and analysis. The practical applicability of the proposed approach is demonstrated through applications in digital image processing, including pattern recognition, noise removal, and feature extraction. In addition, a decision-making case study illustrates the effectiveness of the proposed classification methods and indicates improved performance compared with conventional rough set approaches. Overall, the proposed framework establishes a versatile algebraic–topological approach to uncertainty modeling, offering a rigorous foundation for further theoretical developments and a broad range of practical applications in data analysis, image processing, and decision support.

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