The essential aim of the current work is to enhance the application aspects of Pawlak rough sets. Using the notion of a j-neighborhood space and the related concept of θβ-open sets, different methods for generalizing Pawlak rough sets are proposed and their characteristics will be examined. Moreover, in the context of ideal notion, novel generalizations of Pawlak's models and some of their generalizations are presented. Comparisons between the suggested methods and the previous approximations are calculated. Finally, an application from real-life problems is proposed to explain the importance of our decision-making methods.
Citation: Ashraf S. Nawar, Mostafa A. El-Gayar, Mostafa K. El-Bably, Rodyna A. Hosny. θβ-ideal approximation spaces and their applications[J]. AIMS Mathematics, 2022, 7(2): 2479-2497. doi: 10.3934/math.2022139
The essential aim of the current work is to enhance the application aspects of Pawlak rough sets. Using the notion of a j-neighborhood space and the related concept of θβ-open sets, different methods for generalizing Pawlak rough sets are proposed and their characteristics will be examined. Moreover, in the context of ideal notion, novel generalizations of Pawlak's models and some of their generalizations are presented. Comparisons between the suggested methods and the previous approximations are calculated. Finally, an application from real-life problems is proposed to explain the importance of our decision-making methods.
[1] | Z. Pawlak, Rough sets, Int. J. Inform. Comput. Sci., 11 (1982), 341-356. doi: 10.1007/BF01001956. doi: 10.1007/BF01001956 |
[2] | R. Slowinski, D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE T. Data En., 12 (2000), 331-336. doi: 10.1109/69.842271. doi: 10.1109/69.842271 |
[3] | 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. doi: 10.1016/j.ins.2011.01.007. doi: 10.1016/j.ins.2011.01.007 |
[4] | K. Y. Qin, J. L. Yang, Z. Pei, Generalized rough sets based on reflexive and transitive relations, Inform. Sci., 178 (2008), 4138-4141. doi: 10.1016/j.ins.2008.07.002. doi: 10.1016/j.ins.2008.07.002 |
[5] | M. Kondo, On the structure of generalized rough sets, Inform. Sci., 176 (2006), 589-600. doi: 10.1016/j.ins.2005.01.001. doi: 10.1016/j.ins.2005.01.001 |
[6] | Y. Y. Yao, Two views of the theory of rough sets in finite universes, Int. J. Approx. Reason., 15 (1996), 291-317. doi: 10.1016/S0888-613X(96)00071-0. doi: 10.1016/S0888-613X(96)00071-0 |
[7] | A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for basic rough set concepts, In: International workshop on rough sets, fuzzy sets, data mining, and granular computing, Lecture Notes in Artificial Intelligence, Berlin, Heidelberg: Springer, 2005. doi: 10.1007/11548669_7. |
[8] | M. K. El-Bably, T. M. Al-shami, Different kinds of generalized rough sets based on neighborhoods with a medical application, Int. J. Biomath., 14 (2021), 2150086. doi: 10.1142/S1793524521500868. doi: 10.1142/S1793524521500868 |
[9] | R. Abu-Gdairi, M. A. El-Gayar, M. K. El-Bably, K. K. Fleifel, Two different views for generalized rough sets with applications, Mathematics, 9 (2021), 2275. doi: 10.3390/math9182275. doi: 10.3390/math9182275 |
[10] | Z. M. Yu, X. L. Bai, Z. Q. Yun, A study of rough sets based on 1-neighborhood systems, Inform. Sci., 248 (2013), 103-113. doi: 10.1016/j.ins.2013.06.031. doi: 10.1016/j.ins.2013.06.031 |
[11] | 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), 3045-3060. doi: 10.3233/JIFS-210167. doi: 10.3233/JIFS-210167 |
[12] | Y. Y. Yao, Three-way decision and granular computing, Int. J. Approx. Reason., 103 (2018), 107-123. doi: 10.1016/j.ijar.2018.09.005. doi: 10.1016/j.ijar.2018.09.005 |
[13] | M. El Sayed, M. A. El Safety, M. K. El-Bably, Topological approach for decision-making of COVID-19 infection via a nano-topology model, AIMS Mathematics, 6 (2021), 7872-7894. doi: 10.3934/math.2021457. doi: 10.3934/math.2021457 |
[14] | M. E. Abd El-Monsef, M. A. EL-Gayar, R. M. Aqeel, On relationships between revised rough fuzzy approximation operators and fuzzy topological spaces, Int. J. Granul. Comput. Rough Sets Intell. Syst., 3 (2014), 257-271. |
[15] | M. K. El-Bably, T. M. Al-shami, A. S. Nawar, A. Mhemdi, Corrigendum to "Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space", J. Intell. Fuzzy Syst., 2021, 1-9. doi: 10.3233/JIFS-211198. doi: 10.3233/JIFS-211198 |
[16] | M. K. El-Bably, K. K. Fleifel, O. A. Embaby, Topological approaches to rough approximations based on closure operators, Granul. Comput., 2021. doi: 10.1007/s41066-020-00247-x. doi: 10.1007/s41066-020-00247-x |
[17] | B. K. Tripathy, A. Mitra, Some topological properties of rough sets and their applications, Int. J. Granul. Comput. Rough Sets Intell. Syst., 1 (2010), 355-369. |
[18] | A. S. Nawar, Approximations of some near open sets in ideal topological spaces, J. Egypt. Math. Soc., 28 (2020), 5. doi: 10.1186/s42787-019-0067-0. doi: 10.1186/s42787-019-0067-0 |
[19] | M. E. Abd El-Monsef, O. A. Embaby, M. K. El-Bably, Comparison between rough set approximations based on different topologies, Int. J. Granul. Comput. Rough Sets Intell. Syst., 3 (2014), 292-305. |
[20] | W. S. Amer, M. I. Abbas, M. K. El-Bably, On j-near concepts in rough sets with some applications, J. Intell. Fuzzy Syst., 32 (2017), 1089-1099. doi: 10.3233/JIFS-16169. doi: 10.3233/JIFS-16169 |
[21] | M. Hosny, On generalization of rough sets by using two different methods, J. Intell. Fuzzy Syst., 35 (2018), 979-993. doi: 10.3233/JIFS-172078. doi: 10.3233/JIFS-172078 |
[22] | M. Hosny, Idealization of j-approximation spaces, Filomat, 34 (2020), 287-301. doi: 10.2298/FIL2002287H. doi: 10.2298/FIL2002287H |
[23] | M. E. Abd El-Monsef, M. A. EL-Gayar, R. M. Aqeel, A comparison of three types of rough fuzzy sets based on two universal sets, Int. J. Mach. Learn. Cyber., 8 (2017), 343-353. doi: 10.1007/s13042-015-0327-8. doi: 10.1007/s13042-015-0327-8 |
[24] | W. H. Xu, W. X. Zhang, Measuring roughness of generalized rough sets induced by a covering, Fuzzy Set. Syst., 158 (2007), 2443-2455. doi: 10.1016/j.fss.2007.03.018. doi: 10.1016/j.fss.2007.03.018 |
[25] | M. E. Abd El-Monsef, A. M. Kozae, M. K. El-Bably, On generalizing covering approximation space, J. Egypt. Math. Soc., 23 (2015), 535-545. doi: 10.1016/j.joems.2014.12.007. doi: 10.1016/j.joems.2014.12.007 |
[26] | 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. doi: 10.3233/JIFS-191542. doi: 10.3233/JIFS-191542 |
[27] | Y. R. Syau, E. B. Lin, Neighborhood systems and covering approximation spaces, Knowl.-Based Syst., 66 (2014), 61-67. doi: 10.1016/j.knosys.2014.04.017. doi: 10.1016/j.knosys.2014.04.017 |
[28] | F. F. Zhao, L. Q. Li, Axiomatization on generalized neighborhood system-based rough sets, Soft Comput., 22 (2018), 6099-6110. doi: 10.1007/s00500-017-2957-0. doi: 10.1007/s00500-017-2957-0 |
[29] | W. Yao, Y. H. She, L. X. Lu, Metric-based L-fuzzy rough sets: Approximation operators and definable sets, Knowl.-Based Syst., 163 (2019), 91-102. doi: 10.1016/j.knosys.2018.08.023. doi: 10.1016/j.knosys.2018.08.023 |
[30] | W. Yao, X. Q. Chen, Fuzzy partition and fuzzy rough approximation operators, J. Liaocheng Univ., 33 (2020), 1-4. |
[31] | H. C. Lu, A. M. Khalil, W. Alharbi, M. A. El-Gayar, A new type of generalized picture fuzzy soft set and its application in decision making, J. Intell. Fuzzy Syst., 40 (2021), 12459-12475. doi: 10.3233/JIFS-201706. doi: 10.3233/JIFS-201706 |
[32] | H. M. Abu-Donia, A. S. Salama, Generalization of Pawlaks rough approximation spaces by using δβ-open sets, Int. J. Approx. Reason., 53 (2012), 1094-1105. doi: 10.1016/j.ijar.2012.05.001. doi: 10.1016/j.ijar.2012.05.001 |
[33] | T. M. Al-Shami, B. A. Asaad, M.A. El-Gayar, Various types of supra pre-compact and supra pre-Lindelöf spaces, Missouri J. Math. Sci., 32 (2020), 1-20. doi: 10.35834/2020/3201001. doi: 10.35834/2020/3201001 |
[34] | D. Jankovic, T. R. Hamlet, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295-310. doi: 10.1080/00029890.1990.11995593. doi: 10.1080/00029890.1990.11995593 |
[35] | N. E. Tayar, R. S. Tsai, P. A. Carrupt, B. Testa, Octan-1-ol-water partition coefficients of zwitterionic α-amino acids. Determination by centrifugal partition chromatography and factorization into steric/hydrophobic and polar components, J. Chem. Soc. Perkin Trans. 2, 1992, 79-84. doi: 10.1039/P29920000079. doi: 10.1039/P29920000079 |
[36] | B. Walczak, D. L. Massart, Rough sets theory, Chemometr. Intell. Lab. Syst., 47 (1999) 1-16. doi: 10.1016/S0169-7439(98)00200-7. doi: 10.1016/S0169-7439(98)00200-7 |