Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data

Amr Zakaria; Sobhy El-Sheikh; Safaa Mohamed; Eman Said

doi:10.47000/tjmcs.1813065

Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data

Abstract

This paper presents a theoretical and computational advancement in rough set theory by developing a unified framework that combines neighborhood systems with ideals to better handle uncertainty in decision-making problems. We introduce new definitions of lower and upper approximations based on reflexive neighborhood systems via ideals, rigorously analyzing their mathematical properties through theorems and counterexamples. The proposed framework offers three key advantages: it significantly reduces uncertainty through more precise approximations that shrink boundary regions while improving accuracy measures; it generalizes classical rough set models when using minimal ideals; and it provides practical algorithms for computing these approximations efficiently. We present complete algorithmic implementations of these methods, including detailed pseudocode and Python implementations that demonstrate their computational feasibility. A comprehensive case study on school selection with incomplete data validates the framework's effectiveness in real-world applications. By bridging topological neighborhood concepts with ideal-based algebraic structures, this work advances rough set theory both theoretically and practically, offering ready-to-use computational tools for artificial intelligence and data mining applications. The accompanying algorithms and implementations ensure reproducible results while making these theoretical advances immediately applicable to practical problems in uncertainty management and knowledge discovery.

Keywords

References

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Details

Primary Language

English

Subjects

Topology

Journal Section

Research Article

Authors

Amr Zakaria ^*
0000-0003-4071-0042
Egypt

Sobhy El-Sheikh This is me
0000-0001-8092-8439
Egypt

Safaa Mohamed This is me
0000-0001-8092-8435
Egypt

Eman Said This is me
0000-0001-8092-8438
Egypt

Publication Date

October 13, 2026

Submission Date

October 29, 2025

Acceptance Date

February 23, 2026

Published in Issue

Year 2026 Volume: 18 Number: 3

DOI

https://doi.org/10.47000/tjmcs.1813065

IZ

https://izlik.org/JA63WJ64HF

Cite

RIS / Bibtex

APA

Zakaria, A., El-Sheikh, S., Mohamed, S., & Said, E. (2026). Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data. Turkish Journal of Mathematics and Computer Science, 18(3), 639-655. https://doi.org/10.47000/tjmcs.1813065

AMA

1.Zakaria A, El-Sheikh S, Mohamed S, Said E. Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data. TJMCS. 2026;18(3):639-655. doi:10.47000/tjmcs.1813065

Chicago

Zakaria, Amr, Sobhy El-Sheikh, Safaa Mohamed, and Eman Said. 2026. “Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data”. Turkish Journal of Mathematics and Computer Science 18 (3): 639-55. https://doi.org/10.47000/tjmcs.1813065.

EndNote

Zakaria A, El-Sheikh S, Mohamed S, Said E (October 1, 2026) Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data. Turkish Journal of Mathematics and Computer Science 18 3 639–655.

IEEE

[1]A. Zakaria, S. El-Sheikh, S. Mohamed, and E. Said, “Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data”, TJMCS, vol. 18, no. 3, pp. 639–655, Oct. 2026, doi: 10.47000/tjmcs.1813065.

ISNAD

Zakaria, Amr - El-Sheikh, Sobhy - Mohamed, Safaa - Said, Eman. “Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data”. Turkish Journal of Mathematics and Computer Science 18/3 (October 1, 2026): 639-655. https://doi.org/10.47000/tjmcs.1813065.

JAMA

1.Zakaria A, El-Sheikh S, Mohamed S, Said E. Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data. TJMCS. 2026;18:639–655.

MLA

Zakaria, Amr, et al. “Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 3, Oct. 2026, pp. 639-55, doi:10.47000/tjmcs.1813065.

Vancouver

1.Amr Zakaria, Sobhy El-Sheikh, Safaa Mohamed, Eman Said. Ideal-Based Rough Approximations: Algorithms and Applications for Incomplete Data. TJMCS. 2026 Oct. 1;18(3):639-55. doi:10.47000/tjmcs.1813065