Research article

Rough topological structures by various types of maximal neighborhoods

  • Received: 19 August 2024 Revised: 25 September 2024 Accepted: 29 September 2024 Published: 18 October 2024
  • MSC : 03E99, 54A05, 91B06, 54E99

  • This manuscript centers on creating various topologies utilizing different sorts of maximal neighborhoods. The comparison of these topologies with the previous ones reveal that the earlier topology is weaker than the current ones. The core properties of the proposed topologies are examined, and the necessary conditions for achieving certain equivalences among them are outlined. Additionally, this study provides a distinctive characterization of these topologies by pinpointing the coarsest and largest one among all types, whereas previous methods were limited to characterizing only disjoint pairs of sets. Thereafter, these topologies are utilized to evolve new approximations. One of the major benefits of the current extension is that it adheres to all the properties of the original approximations without the constraints or limitations imposed by earlier versions. The significance of this paper lies not only in introducing new types of approximations based primarily on different kinds of topologies, but also in the fact that these approximations maintain the monotonic property for any given relation, enabling effective evaluation of uncertainty in the data. The monotonic property is crucial for various applications, as it guarantees that the approximation process is logically coherent and robust in the face of evolving information. The proposed models distinguish from their predecessors by their ability to compare all types of the suggested approximations. Moreover, comparisons reveal that the optimal approximations and accuracy are achieved with a specific type of generating topologies. The results demonstrate that topological notions can be a potent technique for studying rough set models. Furthermore, advanced topological features of approximate sets aid in finding rough measures, which assists in identifying missing feature values. Afterward, a numerical example is presented to highlight and emphasize the importance of the present results. Ultimately, the benefits of the followed manner are scrutinized and also some of their limitations are pointed out.

    Citation: Mona Hosny. Rough topological structures by various types of maximal neighborhoods[J]. AIMS Mathematics, 2024, 9(11): 29662-29688. doi: 10.3934/math.20241437

    Related Papers:

  • This manuscript centers on creating various topologies utilizing different sorts of maximal neighborhoods. The comparison of these topologies with the previous ones reveal that the earlier topology is weaker than the current ones. The core properties of the proposed topologies are examined, and the necessary conditions for achieving certain equivalences among them are outlined. Additionally, this study provides a distinctive characterization of these topologies by pinpointing the coarsest and largest one among all types, whereas previous methods were limited to characterizing only disjoint pairs of sets. Thereafter, these topologies are utilized to evolve new approximations. One of the major benefits of the current extension is that it adheres to all the properties of the original approximations without the constraints or limitations imposed by earlier versions. The significance of this paper lies not only in introducing new types of approximations based primarily on different kinds of topologies, but also in the fact that these approximations maintain the monotonic property for any given relation, enabling effective evaluation of uncertainty in the data. The monotonic property is crucial for various applications, as it guarantees that the approximation process is logically coherent and robust in the face of evolving information. The proposed models distinguish from their predecessors by their ability to compare all types of the suggested approximations. Moreover, comparisons reveal that the optimal approximations and accuracy are achieved with a specific type of generating topologies. The results demonstrate that topological notions can be a potent technique for studying rough set models. Furthermore, advanced topological features of approximate sets aid in finding rough measures, which assists in identifying missing feature values. Afterward, a numerical example is presented to highlight and emphasize the importance of the present results. Ultimately, the benefits of the followed manner are scrutinized and also some of their limitations are pointed out.



    加载中


    [1] E. A. Abo-Tabl, A comparison of two kinds of definitions of rough approximations based on a similarity relation, Inform. Sci., 181 (2011), 2587–2596. https://doi.org/10.1016/j.ins.2011.01.007 doi: 10.1016/j.ins.2011.01.007
    [2] M. E. Abd El-Monsef, O. A. Embaby, M. K. El-Bably, Comparison between rough set approximations based on different topologies, Int. J. Granular Comput., Rough Sets Intell. Syst, 3 (2014), 292–305. https://doi.org/10.1504/IJGCRSIS.2014.068032 doi: 10.1504/IJGCRSIS.2014.068032
    [3] A. A. Abo Khadra, B. M. Taher, M. K. El-Bably, Generalization of Pawlak approximatio space, Egypt. Math. Soc., Cairo, 3 Top., Geom., 2007,335–346.
    [4] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for basic rough set concepts, Int. Workshop Rough Sets Fuzzy Sets Data Mining Granul. Comput., 3641 (2005), 64–73. https://doi.org/10.1007/11548669_7 doi: 10.1007/11548669_7
    [5] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for closure spaces by relations, Acta Math. Acad. Paedagog. Nyiregyháziensis, 22 (2006), 285–304.
    [6] T. M. Al-shami, An improvement of rough sets' accuracy measure using containment neighborhoods with a medical application, Inform. Sci., 569 (2021), 110–124. https://doi.org/10.1016/j.ins.2021.04.016 doi: 10.1016/j.ins.2021.04.016
    [7] T. M. Al-shami, Maximal rough neighborhoods with a medical application, J Ambient. Intell. Humaniz. Comput., 2022. https://doi.org/10.1007/s12652-022-03858-1 doi: 10.1007/s12652-022-03858-1
    [8] T. M. Al-shami, D. Ciucci, Subset neighborhood rough sets, Knowl.-Based Syst., 237 (2022). https://doi.org/10.1016/j.knosys.2021.107868 doi: 10.1016/j.knosys.2021.107868
    [9] M. Atef, New categories of coverings in terms of rough fuzzy sets, Comp. Appl. Math., 43 (2024). https://doi.org/10.1007/s40314-024-02882-5 doi: 10.1007/s40314-024-02882-5
    [10] A. A. Azzam, T. M. Al-shami, Five generalized rough Approximation spaces produced by maximal rough neighborhoods, Symmetry, 15 (2023), https://doi.org/10.3390/sym15030751 doi: 10.3390/sym15030751
    [11] B. De Baets, E. Kerre, A revision of Bandler-Kohout compositions of relations, Math. Pannon., 4 (1993), 59–78.
    [12] J. Dai, S. Gao, G. Zheng, Generalized rough set models determined by multiple neighborhoods generated from a similarity relation, Soft Comput., 22 (2018), 2081–2094. https://doi.org/10.1007/s00500-017-2672-x doi: 10.1007/s00500-017-2672-x
    [13] M. El Sayed, M. A. El Safty, M. K. El-Bably, Topological approach for decision-making of COVID-19 infection via a nano-topology model, AIMS Math., 6 (2021), 7872–7894. https://doi.org/10.3934/math.2021457 doi: 10.3934/math.2021457
    [14] A. Ç. G$\ddot{u}$ler, E. D. Yildirim, O. B. $\ddot{O}$zbakir, Rough approximations based on different topologies via ideals, Turk. J. Math., 46 (2022), 1177–1192. https://doi.org/10.55730/1300-0098.3150 doi: 10.55730/1300-0098.3150
    [15] M. Hosny, On generalization of rough sets by using two different methods, J. Intell. Fuzzy Syst., 35 (2018), 979–993. https://doi.org/10.3233/JIFS-172078 doi: 10.3233/JIFS-172078
    [16] M. Hosny, Topological approach for rough sets by using $J$-nearly concepts via ideals, Filomat, 34 (2020), 273–286. https://doi.org/10.2298/FIL2002273H doi: 10.2298/FIL2002273H
    [17] M. Hosny, Idealization of $j$-approximation spaces, Filomat, 34 (2020), 287–301. https://doi.org/10.2298/FIL2002287H doi: 10.2298/FIL2002287H
    [18] M. Hosny, Topologies generated by two ideals and the corresponding $j$-approximations spaces with applications, J. Math., 2021. https://doi.org/10.1155/2021/6391266 doi: 10.1155/2021/6391266
    [19] M. Hosny, Generalization of rough sets using maximal right neighborhood systems and ideals with medical applications, AIMS Math., 7 (2022), 13104–13138. https://doi.org/10.3934/math.2022724 doi: 10.3934/math.2022724
    [20] M. Hosny, Idealization of rough sets from the perspective of topology generated by several types of maximal neighbourhoods, In Press.
    [21] M. Hosny, Near open sets by several types of maximal neighbourhoods and rough set, Submitted.
    [22] M. Hosny, M. Raafat, On generalization of rough multiset via multiset ideals, J. Intell. Fuzzy Syst., 33 (2017), 1249–1261. https://doi.org/10.3233/JIFS-17102 doi: 10.3233/JIFS-17102
    [23] Z. Huang, J. Li, Discernibility measures for fuzzy $\alpha$-covering and their application, IEEE Trans. Cybern., 52 (2022), 9722–9735. https://doi.org/10.1109/TCYB.2021.3054742 doi: 10.1109/TCYB.2021.3054742
    [24] Z. Huang, J. Li, Covering based multi-granulation rough fuzzy sets with applications to feature selection, Expert Syst. Appl., 238 (2023). https://doi.org/10.1016/j.eswa.2023.121908 doi: 10.1016/j.eswa.2023.121908
    [25] Z. Huang, J. Li, Noise-tolerant discrimination indexes for fuzzy $\beta$-covering and feature subset selection, IEEE Trans Neural Netw. Learn Syst., 35 (2024), 609–623. https://doi.org/10.1109/TNNLS.2022.3175922 doi: 10.1109/TNNLS.2022.3175922
    [26] J. Jarvinen, J. Kortelainen, A unifying study between model-like operators, topologies, and fuzzy sets, Fuzzy Sets Syst., 158 (2007), 1217–1225. https://doi.org/10.1016/j.fss.2007.01.011 doi: 10.1016/j.fss.2007.01.011
    [27] A. Kandil, S. A. El-Sheikh, M. Hosny, M. Raafat, Bi-ideal approximation spaces and their applications, Soft Comput., 24 (2020), 12989–13001. https://doi.org/10.1007/s00500-020-04720-2 doi: 10.1007/s00500-020-04720-2
    [28] A. M. Kozae, S. A. El-Sheikh, M. Hosny, On generalized rough sets and closure spaces. Int. J. Appl. Math., 23 (2010), 997–1023.
    [29] A. M. Kozae, S. A. El-Sheikh, E. H. Aly, M. Hosny, Rough sets and its applications in a computer network, Ann. Fuzzy Math. Inform., 6 (2013), 605–624.
    [30] Z. Li, T. Xie, Q. Li, Topological structure of generalized rough sets, Comput. Math. Appl., 63 (2012), 1066–1071. https://doi.org/10.1016/j.camwa.2011.12.011 doi: 10.1016/j.camwa.2011.12.011
    [31] X. Ma, Qi Liu, J. Zhan, A survey of decision making methods based on certain hybrid soft set models, Artif. Intell. Rev., 47 (2017), 507–530. https://doi.org/10.1007/s10462-016-9490-x doi: 10.1007/s10462-016-9490-x
    [32] E. A. Marei, Neighborhood system and decision making, Zagazig Univ., 2007.
    [33] S. Pal, P. Mitra, Case generation using rough sets with fuzzy representation, IEEE Trans. Knowl. Data Eng., 16 (2004), 293–300. https://doi.org/10.1109/TKDE.2003.1262181 doi: 10.1109/TKDE.2003.1262181
    [34] H. Mustafa, T. M. Al-shami, R. Wassef, Rough set paradigms via containment neighborhoods and ideals, Filomat, 37 (2023), 4683–4702. https://doi.org/10.2298/fil2314683m doi: 10.2298/fil2314683m
    [35] Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci., 11 (1982), 341–356. https://doi.org/10.1007/BF01001956 doi: 10.1007/BF01001956
    [36] Z. Pawlak, Rough concept analysis, Bull. Pol. Acad. Sci. Math., 33 (1985), 495–498. https://doi.org/10.1007/978-1-4471-3238-7_30 doi: 10.1007/978-1-4471-3238-7_30
    [37] Z. Pei, D. Pei, Li Zheng, Topology vs generalized rough sets, Int. J. Approx. Reason. 52 (2011), 231–239. https://doi.org/10.1016/j.ijar.2010.07.010 doi: 10.1016/j.ijar.2010.07.010
    [38] L. Polkowski, Rough sets: Mathematical foundations, Phys.-Verlag, 2002.
    [39] E. A. Rady, A. M. Kozae, M. M.E. Abd El-Monsef, Generalized rough sets, Chaos Solit. Fract., 21 (2004), 49–53. https://doi.org/10.1016/j.chaos.2003.09.044 doi: 10.1016/j.chaos.2003.09.044
    [40] A. Skowron, On topology in information system, Bull. Pol. Acad. Sci. Math., 36 (1988), 477–480.
    [41] D. I. Taher, R. Abu-Gdairi, M. K. El-Bably, M. A. El-Gayar, Decision-making in diagnosing heart failure problems using basic rough sets, AIMS Math., 9 (2024), 21816–21847. https://doi.org/10.3934/math.20241061 doi: 10.3934/math.20241061
    [42] A. Wiweger, On topological rough sets, Bull. Pol. Acad. Sci. Math., 37 (1989), 89–93.
    [43] Y. Y. Yao, Two views of the theory of rough sets in finite universes, Int. J. Approx. Reason., 15 (1996), 291–317. https://doi.org/10.1016/S0888-613X(96)00071-0 doi: 10.1016/S0888-613X(96)00071-0
    [44] Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci., 111 (1998), 239–259. https://doi.org/10.1016/S0020-0255(98)10006-3 doi: 10.1016/S0020-0255(98)10006-3
    [45] E. D. Yildirim, New topological approaches to rough sets via subset neighborhoods, J. Math., 2022, https://doi.org/10.1155/2022/3942708 doi: 10.1155/2022/3942708
    [46] W. Zhu, Topological approaches to covering rough sets, Inform. Sci., 177 (2007), 1499–1508. https://doi.org/10.1016/j.ins.2006.06.009 doi: 10.1016/j.ins.2006.06.009
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(291) PDF downloads(88) Cited by(0)

Article outline

Figures and Tables

Tables(4)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog