Ordered $j$-Approximation Spaces with an Application

Abstract

In 1982, Pawlak introduced the concept of rough sets as a mathematical framework for addressing real-world problems. However, the dependence on equivalence relations within this framework has often proven inadequate for handling more complex cases. The aim of this study is to develop a more effective approach for such situations. In particular, partial order relations are combined with $\mathit{j}$-neighborhoods to capture not only vague similarities among objects but also hierarchical dependencies and priority structures that frequently arise in real data. For this purpose, ordered $\mathit{j}$-neighborhood classes are defined by employing increasing and decreasing sets based on partial order relations together with $\mathit{j}$-neighborhoods. The fundamental properties of these classes, as well as their relationships with $\mathit{j}$-neighborhoods and ordered equivalence classes under various binary relations, are then examined. Building on these classes, ordered $\mathit{j}$-approximation spaces are introduced as a natural generalization of both ordered approximation spaces and Pawlak approximation spaces. The findings show that the proposed approximations yield more capable results in reducing boundary regions compared to some existing approaches. Moreover, generalized nano topologies are constructed from these new structures, and their applicability is demonstrated through a real-world example.

Keywords

• generalized nano topological spaces
• generalized rough set
• ordered $\mathit{j}$-approximation spaces
• ordered $\mathit{j}$-neighborhood classes

Supporting Institution

No grants were received from any public, private or non-profit organizations for this research.

Ethical Statement

It is declared that during the preparation process of this study, scientific and ethical principles were followed and all the studies benefited from are stated in the bibliography.

Thanks

The authors would like to express their sincere thanks to the editor and the reviewers for their helpful comments and suggestions.

References

1. [1] M. E. Abd El-Monsef, O. A. Embaby and M. K. El-Bably, Comparison between rough set approximations based on different topologies, International Journal of Granular Computing, Rough Sets and Intelligent Systems, Vol:3, No.4 (2014), 292–305.
2. [2] H. M. Abu-Donia, Comparison between different kinds of approximations by using a family of binary relations, Knowledge-Based Systems, Vol:21, No.8 (2008), 911–919.
3. [3] A. A. Allam, M. Y. Bakeir and E. A. Abo-Tabl, New approach for basic rough set concepts, International Workshop on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Lecture Notes in Artificial Intelligence, Vol:3641 (2005), 64–73.
4. [4] A. A. Allam, M. Y. Bakeir and E. A. Abo-Tabl, New approach for closure spaces by relations, Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis, Vol:22, No.3 (2006), 285–304.
5. [5] T. M. Al-Shami, An improvement of rough sets’ accuracy measure using containment neighborhoods with a medical application, Information Sciences, Vol:569 (2021), 110–124.
6. [6] T. M. Al-Shami and D. Ciucci, Subset neighborhood rough sets, Knowledge-Based Systems, Vol:237 (2022), Article 107868.
7. [7] T. M. Al-Shami and A. Mhemdi, Overlapping containment rough neighborhoods and their generalized approximation spaces with applications, Journal of Applied Mathematics and Computing, Vol:71 (2025), 869–900.
8. [8] T. M. Al-Shami, M. A. Nuwairan, Overlapping subset neighborhoods and applications via generalized approximation spaces, Filomat, Vol: 39, No.27 (2025), 9415–9447.

9. [9] M. Atef et al., Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space, Journal of Intelligent and Fuzzy Systems, Vol:39, No.3 (2020), 4515–4531.
10. [10] M. K. El-Bably and E. A. Abo-Tabl, A topological reduction for predicting of a lung cancer disease based on generalized rough sets, Journal of Intelligent and Fuzzy Systems, Vol:41, No.2 (2021), 3045–3060.
11. [11] M. K. El-Bably and T. M. Al-Shami, Different kinds of generalized rough sets based on neighborhoods with a medical application, International Journal of Biomathematics, Vol:14, No.8 (2021), Article 2150086.
12. [12] M. K. El-Bably et al., Corrigendum to “Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space”, Journal of Intelligent and Fuzzy Systems, Vol:41, No.6 (2021), 7353–7361.
13. [13] M. El-Sayed, M. A. El-Safty and M. K. El-Bably, Topological approach for decision making of COVID-19 infection via a nano-topology model, AIMS Mathematics, Vol:6, No.7 (2021), 7872–7894.
14. [14] R. Mareay, Generalized rough sets based on neighborhood systems and topological spaces, Journal of the Egyptian Mathematical Society, Vol:24, No.4 (2016), 603–608.
15. [15] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences, Vol:11, No.5 (1982), 341–356.
16. [16] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic Publishers, 1991.
17. [17] S. H. Shalil, S. A. El-Sheikh and S. A. Kandil, An application on an information system via nano ordered topology, Malaysian Journal of Mathematical Sciences, Vol:17, No.4 (2023), 509–529.
18. [18] M. L. Thivagar and C. Richard, On nano forms of weakly open sets, International Journal of Mathematics and Statistics Invention, Vol:1, No.1 (2013), 31–37.
19. [19] Y. Y. Yao, Two views of the theory of rough sets in finite universes, International Journal of Approximate Reasoning, Vol:15 (1996), 291–317.
20. [20] Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences, Vol:111 (1998), 239–259.