Search Advanced

AIMS Mathematics

2022, Volume 7, Issue 6: 9911-9925. doi: 10.3934/math.2022553
Previous Article Next Article
Research article

Rough topological structure based on reflexivity with some applications

  • 1.
    Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt
  • 2.
    Mathematics Department, College of Science and Arts, Methnab, Qassim University, Buridah, Saudi Arabia
  • 3.
    Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt
  • Received: 21 November 2021 Revised: 05 March 2022 Accepted: 07 March 2022 Published: 18 March 2022

  • MSC : 54A05, 54A10, 03E20

  • Recently, topological structures have emerged as one of the most popular rough sets (RS) research topics. It can be stated that it is a fundamental and significant subject in the theory of RS. This study introduces a debate about the structure of rough topological space based on the reflexive relation. To create the rough topological space, we use the representation of RS. We also look at the relationships between approximation operators, closure operators, and interior operators. Also, the relationship between topological space in the universe that is not limited or restricted to be ended, and RS induced by reflexive relations is investigated. Furthermore, we define the relationships between the set of all topologies that satisfy the requirement of compactness $ C_{2} $ and the set of all reflexive relations. Finally, we present a medical application that addresses the issue of dengue fever. The proposed structures are used to determine the impact factors for identifying dengue fever.

    Citation: El-Sayed A. Abo-Tabl, Mostafa K. El-Bably. Rough topological structure based on reflexivity with some applications[J]. AIMS Mathematics, 2022, 7(6): 9911-9925. doi: 10.3934/math.2022553

    Related Papers:

  • Recently, topological structures have emerged as one of the most popular rough sets (RS) research topics. It can be stated that it is a fundamental and significant subject in the theory of RS. This study introduces a debate about the structure of rough topological space based on the reflexive relation. To create the rough topological space, we use the representation of RS. We also look at the relationships between approximation operators, closure operators, and interior operators. Also, the relationship between topological space in the universe that is not limited or restricted to be ended, and RS induced by reflexive relations is investigated. Furthermore, we define the relationships between the set of all topologies that satisfy the requirement of compactness $ C_{2} $ and the set of all reflexive relations. Finally, we present a medical application that addresses the issue of dengue fever. The proposed structures are used to determine the impact factors for identifying dengue fever.


    加载中


    [1] A. A. Abo Khadra, M. K. El-Bably, Topological approach to tolerance space, Alex. Eng. J., 47 (2008), 575–580.
    [2] E. A. Abo-Tabl, On topological properties of generalized rough multisets, Ann. Fuzzy Math. Inf., 19 (2020), 95-107. https://doi.org/10.30948/afmi.2020.19.1.95 doi: 10.30948/afmi.2020.19.1.95
    [3] E. A. Abo-Tabl, Topological approaches to generalized definitions of rough multiset approximations, Inter. J. Mach. Learn. Cyb., 6 (2015), 399-407. https://doi.org/10.1007/s13042-013-0196-y doi: 10.1007/s13042-013-0196-y
    [4] E. A. Abo-Tabl, Topological structure of generalized rough multisets, Life Sci. J., 11 (2014), 290-299.
    [5] E. A. Abo-Tabl, On links between rough sets and digital topology, Appl. Math., 5 (2014), 941–948. https://doi.org/10.4236/am.2014.56089 doi: 10.4236/am.2014.56089
    [6] E. A. Abo-Tabl, Rough sets and topological spaces based on similarity, Inter. J. Mach. Learn. Cyb., 4 (2013), 451-458. https://doi.org/10.1007/s13042-012-0107-7 doi: 10.1007/s13042-012-0107-7
    [7] E. A. Abo-Tabl, Topological approximations of multisets, Egypt. Math. Soc., 21 (2013), 123-132. https://doi.org/10.1016/j.joems.2012.12.001 doi: 10.1016/j.joems.2012.12.001
    [8] E. A. Abo-Tabl, A comparison of two kinds of definitions of rough approximations based on a similarity relation, Inform. Sciences, 181 (2011), 2587-2596. https://doi.org/10.1016/j.ins.2011.01.007 doi: 10.1016/j.ins.2011.01.007
    [9] R. Abu-Gdairi, M. A. El-Gayar, T. M. Al-shami, A. S. Nawar, M. K. El-Bably, Some topological approaches for generalized rough sets and their decision-making applications, Symmetry, 14 (2022). https://doi.org/10.3390/sym14010095 doi: 10.3390/sym14010095
    [10] R. Abu-Gdairi, M. A. El-Gayar, M. K. El-Bably, K. K. Fleifel, Two views for generalized rough sets with applications, Mathematics, 18 (2021), 2275. https://doi.org/10.3390/math9182275 doi: 10.3390/math9182275
    [11] M. I. Ali, M. K. El-Bably, E. A. Abo-Tabl, Topological approach to generalized soft rough sets via near concepts, Soft Comput., 26 (2022), 499-509. https://doi.org/10.1007/s00500-021-06456-z doi: 10.1007/s00500-021-06456-z
    [12] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, Some methods for generating topologies by relations, B. Malays. Math. Sci. Soc., 31 (2008), 35-46.
    [13] T. M. Al-shami, D. Ciucci, Subset neighborhood rough sets, Knowl.-Based Syst., 237 (2022), 107868. https://doi.org/10.1016/j.knosys.2021.107868 doi: 10.1016/j.knosys.2021.107868
    [14] T. M. Al-shami, W. Q. Fu, E. A. Abo-Tabl, New rough approximations based on $E$-neighborhoods, Complexity, 2021 (2021), 6666853. https://doi.org/10.1155/2021/6666853 doi: 10.1155/2021/6666853
    [15] T. M. Al-shami, An improvement of rough sets' accuracy measure using containment neighborhoods with a medical application, Inform. Sciences, 569 (2021), 110-124. https://doi.org/10.1016/j.ins.2021.04.016 doi: 10.1016/j.ins.2021.04.016
    [16] T. M. Al-shami, H. Isik, A. S. Nawar, R. A. Hosny, Some Topological Approaches for Generalized Rough Sets via Ideals, Math. Probl. Eng., 2021 (2021), Article ID 5642982. https://doi.org/10.1155/2021/5642982
    [17] A. A. Azzam, A. M. Khalil, S-G Li, Medical applications via minimal topological structure, J. Intell. Fuzzy Syst., 39 (2020), 4723-4730. https://doi.org/10.3233/JIFS-200651 doi: 10.3233/JIFS-200651
    [18] M. K. El-Bably, K. K. Fleifel, O. A. Embaby, Topological approaches to rough approximations based on closure operators, Granular Comput., 7 (2022), 1-14. https://doi.org/10.1007/s41066-020-00247-x doi: 10.1007/s41066-020-00247-x
    [19] M. K. El-Bably, T. A. Al-shami, Different kinds of generalized rough sets based on neighborhoods with a medical application, Int. J. Biomath., 4 (2021), 2150086. https://doi.org/10.1142/S1793524521500868 doi: 10.1142/S1793524521500868
    [20] M. K. El-Bably, E. A. Abo-Tabl, A topological reduction for predicting of a lung cancer disease based on generalized rough sets, J. Intell. Fuzzy Syst., 41 (2021), 335-346. https://doi.org/10.3233/JIFS-210167 doi: 10.3233/JIFS-210167
    [21] M. K. El-Bably, A. A. El Atik, Soft $\beta$-rough sets and its application to determine COVID-19, Turk. J. Math., 45 (2021), 1133-1148.
    [22] M. K. El-Bably, M. I. Ali, E. A. Abo-Tabl, New topological approaches to generalized soft rough approximations with medical applications, J. Math., 2021 (2021), 2559495. https://doi.org/10.1155/2021/2559495 https://doi.org/10.1155/2021/2559495 doi: 10.1155/2021/2559495
    [23] 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.
    [24] M. E. Sayed, A. G. A. Q. A. Qubati, M. K. El-Bably, Soft pre-rough sets and its applications in decision making, Math. Biosci. Eng., 17 (2020), 6045-6063.
    [25] K. P. Girish, S. J. John, Multiset topologies induced by multiset relations, Inform. Sciences, 188 (2012), 298-313. https://doi.org/10.1016/j.ins.2011.11.023 doi: 10.1016/j.ins.2011.11.023
    [26] K. P. Girish, S. J. John, Rough multisets and its multiset topology, Springer-Verlag Berlin Heidelberg.
    [27] A. Jimenez-Vargas, M. Isabel Ramirez, Algebraic reflexivity of non-canonical isometries on Lipschitz spaces, Mathematics, 9 (2021). https://doi.org/10.3390/math9141635 doi: 10.3390/math9141635
    [28] M. Kondo, On the structure of generalized rough sets, Inform. Sciences, 176 (2006), 589-600. https://doi.org/10.1016/j.ins.2005.01.001 doi: 10.1016/j.ins.2005.01.001
    [29] S. Li, T. Li, Z. Zhang, H. Chen, J. Zhang, Parallel computing of approximations in dominance-based rough sets approach, Knowl.-Based Syst., 87 (2015), 102-111. https://doi.org/10.1016/j.knosys.2015.05.003 doi: 10.1016/j.knosys.2015.05.003
    [30] Z. Li, T. Xie, Q. Li, Topological structure of generalized rough sets, Comput. Math. Appl., 63 (2012), 1066-1071.
    [31] T. Y. Lin, Granular computing on binary relations (I), in: Rough sets in knowledge discovery, Physica-Verlag, Heidelberg, 1 (1998), 107-121.
    [32] A. S. Nawar, M. A. El-Gayar, M. K. El-Bably, R. A. Hosny, $\theta \beta$-ideal approximation spaces and their applications, AIMS Math., 7 (2021), 2479-2497.
    [33] A. S. Nawar, M. K. El-Bably, A. A. El Atik, Certain types of coverings based rough sets with application, J. Intell. Fuzzy Syst., 39 (2020), 3085-3098. https://doi.org/10.3233/JIFS-191542 doi: 10.3233/JIFS-191542
    [34] Z. Pawlak, Rough sets: Theoretical aspects of reasoning about data, Kluwer Academic Publishers, Boston, 1991.
    [35] Z. Pawlak, Rough sets, IADIS-Int. J. Comput. S., 11 (1982), 341-356. https://doi.org/10.4018/978-1-59140-560-3 doi: 10.4018/978-1-59140-560-3
    [36] Z. Pei, D. Pei, L. Zheng, Topology vs generalized rough sets, Int. J. Approx. Reason., 1999,471-487. https://doi.org/10.1016/j.ijar.2010.07.010 doi: 10.1016/j.ijar.2010.07.010
    [37] L. Polkowski, Approximate mathematical morphology: Rough set approach, in: Rough fuzzy hybridization: A new trend in decision-making, Springer, Heidelberg, 52 (2011), 231-239.
    [38] L. Polkowski, Mathematical morphology of rough sets, B. Pol. Acad. Sci.-Math., 41 (1993), 241-273.
    [39] J. A. Pomykala, Approximation operations in approximation space, B. Pol. Acad. Sci., 35 (1987), 653-662.
    [40] P. Agrawal, A. Gautam, R. Jose, M. Farooqui, J. Doneria, Myriad manifestations of dengue fever: Analysis in retrospect, Int. J. Med. Sci. Public Health, 8 (2019), 6-9. https://doi.org/10.5455/ijmsph.2019.0514224092018 doi: 10.5455/ijmsph.2019.0514224092018
    [41] Q. Qiao, Topological structure of rough sets in reflexive and transitive relations, Proceedings of the fifth international conference on Bio medical engineering and informatics, 2012, 1585-1589.
    [42] K. Qin, J. Yang, Z. Pei, Generalized rough sets based on reflexive and transitive relations, Inform. Sciences, 178 (2008), 4138-4141. https://doi.org/10.1016/j.ins.2008.07.002 doi: 10.1016/j.ins.2008.07.002
    [43] M. A. Ragusa, F. Wu, Regularity criteria for the 3D magneto-hydrodynamics equations in anisotropic Lorentz spaces, Symmetry-Basel, 13 (2021). https://doi.org/10.3390/sym13040625
    [44] M. S. Raza, U. Qamar, A parallel approach to calculate lower and upper approximations in dominance based rough set theory, Appl. Soft Comput., 84 (2019), 105699.
    [45] W. Sierpenski, C. Krieger, General Topology, University of Toronto, Toronto, 1956.
    [46] A. Wiweger, On topological rough sets, B. Pol. Acad. Sci.-Math., 37 (1988), 51-62.
    [47] H. Zhang, Y. Ouyang, Z. Wang, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sciences, 179 (2009), 471-473.
    [48] Y. L. Zhang, C. Q. Li, Topological properties of a pair of relation-based approximation operators, Filomat, 31 (2017), 6175-6183. https://doi.org/10.2298/FIL1719175Z doi: 10.2298/FIL1719175Z
    [49] Z. Zhao, On some types of covering rough sets from topological points of view, Int. J. Approx. Reason., 68 (2016), 1-14. https://doi.org/10.1016/j.ijar.2015.09.003 doi: 10.1016/j.ijar.2015.09.003
    [50] W. Zhu, Relationship between generalized rough sets based on binary relation and covering, Inform. Sciences, 179 (2009), 210-225. https://doi.org/10.1016/j.ins.2008.09.015 doi: 10.1016/j.ins.2008.09.015
    [51] W. Zhu, Generalized rough sets based on relations, Inform. Sciences, 177 (2007), 4997-5011. https://doi.org/10.1016/j.ins.2007.05.037 doi: 10.1016/j.ins.2007.05.037
    [52] World Health Organization: Dengue and severe dengue fact sheet, World Health Organization Geneva, Switzerland. Available from: http://www.who.int/mediacentre/factsheets/fs117/en(2016).
  • Reader Comments
  • © 2022 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
AIMS Mathematics
1.8 3.4

Metrics

Article views(1516) PDF downloads(99) Cited by(7)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(1)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog