Extension of topological structures using lattices and rough sets

Mostafa A. El-Gayar; Radwan Abu-Gdairi; Mostafa A. El-Gayar; Radwan Abu-Gdairi

doi:10.3934/math.2024366

AIMS Mathematics

2024, Volume 9, Issue 3: 7552-7569. doi: 10.3934/math.2024366

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Research article

Extension of topological structures using lattices and rough sets

Mostafa A. El-Gayar ^{1
,
,},
Radwan Abu-Gdairi ²

1.
Department of Mathematics, Faculty of Science, Helwan University, Helwan, 11795, Egypt
2.
Mathematics Department, Faculty of Science, Zarqa University, Zarqa, 13110, Jordan

Received: 13 December 2023 Revised: 13 January 2024 Accepted: 22 January 2024 Published: 22 February 2024
MSC : 54A05, 54B10, 54D30, 54G99

This paper explores the application of rough set theory in analyzing ambiguous data within complete information systems. The study extends topological structures using equivalence relations, establishing an extension of topological lattice within lattices. Various relations on topological spaces generate different forms of exact and rough lattices. Building on Zhou's work, the research investigates rough sets within the extension topological lattice and explores the isomorphism between topology and its extension. Additionally, the paper investigates the integration of lattices and rough sets, essential mathematical tools widely used in problem-solving. Focusing on computer science's prominent lattices and Pawlak's rough sets, the study introduces extension lattices, emphasizing lower and upper extension approximations' adaptability for practical applications. These approximations enhance pattern recognition and model uncertain data with finer granularity. While acknowledging the benefits, the paper stresses the importance of empirical validations for domain-specific efficacy. It also highlights the isomorphism between topology and its extension, revealing implications for data representation, decision-making, and computational efficiency. This isomorphism facilitates accurate data representations and streamlines computations, contributing to improved efficiency. The study enhances the understanding of integrating lattices and rough sets, offering potential applications in data analysis, decision support systems, and computational modeling.
- topological structures,
- extension topology,
- rough sets,
- lattices,
- topological lattice
Citation: Mostafa A. El-Gayar, Radwan Abu-Gdairi. Extension of topological structures using lattices and rough sets[J]. AIMS Mathematics, 2024, 9(3): 7552-7569. doi: 10.3934/math.2024366

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Abstract

This paper explores the application of rough set theory in analyzing ambiguous data within complete information systems. The study extends topological structures using equivalence relations, establishing an extension of topological lattice within lattices. Various relations on topological spaces generate different forms of exact and rough lattices. Building on Zhou's work, the research investigates rough sets within the extension topological lattice and explores the isomorphism between topology and its extension. Additionally, the paper investigates the integration of lattices and rough sets, essential mathematical tools widely used in problem-solving. Focusing on computer science's prominent lattices and Pawlak's rough sets, the study introduces extension lattices, emphasizing lower and upper extension approximations' adaptability for practical applications. These approximations enhance pattern recognition and model uncertain data with finer granularity. While acknowledging the benefits, the paper stresses the importance of empirical validations for domain-specific efficacy. It also highlights the isomorphism between topology and its extension, revealing implications for data representation, decision-making, and computational efficiency. This isomorphism facilitates accurate data representations and streamlines computations, contributing to improved efficiency. The study enhances the understanding of integrating lattices and rough sets, offering potential applications in data analysis, decision support systems, and computational modeling.

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