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AIMS Mathematics

2022, Volume 7, Issue 7: 13278-13291. doi: 10.3934/math.2022733
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Research article

A study of fixed point sets based on Z-soft rough covering models

  • 1.
    Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad, Pakistan
  • 2.
    Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
  • 3.
    Department of Mathematics, Islamabad Model College for Girls, F-6/2, Islamabad, Pakistan
  • Received: 10 March 2022 Revised: 12 April 2022 Accepted: 22 April 2022 Published: 13 May 2022

  • MSC : 47H10, 54H25

  • Z-soft rough covering models are important generalizations of classical rough set theory to deal with uncertain, inexact and more complex real world problems. So far, the existing study describes various forms of approximation operators and their properties by means of soft neighborhoods. In this paper, we propose the notion of $ Z $-soft rough covering fixed point set (briefly, $\mathcal{Z}$-$\mathcal{SRCFP}$-set) induced by covering soft set. We study the conditions that the family of $ \mathcal{Z} $-$ \mathcal{SRCFP} $-sets become lattice structure. For any covering soft set, the $ \mathcal{Z} $-$ \mathcal{SRCFP} $-set is a complete and distributive lattice, and at the same time, it is also a double p-algebra. Furthermore, when soft neighborhood forms a partition of the universe, then $ \mathcal{Z} $-$ \mathcal{SRCFP} $-set is both a boolean lattice and a double stone algebra. Some main theoretical results are obtained and investigated with the help of examples.

    Citation: Imran Shahzad Khan, Choonkil Park, Abdullah Shoaib, Nasir Shah. A study of fixed point sets based on Z-soft rough covering models[J]. AIMS Mathematics, 2022, 7(7): 13278-13291. doi: 10.3934/math.2022733

    Related Papers:

  • Z-soft rough covering models are important generalizations of classical rough set theory to deal with uncertain, inexact and more complex real world problems. So far, the existing study describes various forms of approximation operators and their properties by means of soft neighborhoods. In this paper, we propose the notion of $ Z $-soft rough covering fixed point set (briefly, $\mathcal{Z}$-$\mathcal{SRCFP}$-set) induced by covering soft set. We study the conditions that the family of $ \mathcal{Z} $-$ \mathcal{SRCFP} $-sets become lattice structure. For any covering soft set, the $ \mathcal{Z} $-$ \mathcal{SRCFP} $-set is a complete and distributive lattice, and at the same time, it is also a double p-algebra. Furthermore, when soft neighborhood forms a partition of the universe, then $ \mathcal{Z} $-$ \mathcal{SRCFP} $-set is both a boolean lattice and a double stone algebra. Some main theoretical results are obtained and investigated with the help of examples.


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    [1] M. Akram, A. Adeel, J. C. R. Alcantud, Fuzzy $ N $-soft sets: A novel model with applications, J. Intell. Fuzzy Syst., 35 (2018), 4757–4771. http://doi.org/10.3233/JIFS-18244 doi: 10.3233/JIFS-18244
    [2] M. I. Ali, A note on soft sets, rough soft sets and fuzzy soft sets, Appl. Soft Comput., 11 (2011), 3329–3332. http://doi.org/10.1016/j.asoc.2011.01.003 doi: 10.1016/j.asoc.2011.01.003
    [3] M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, On some new operations in soft set theory, Comput. Math. Appl., 57 (2009), 1547–1553. http://doi.org/10.1016/j.camwa.2008.11.009 doi: 10.1016/j.camwa.2008.11.009
    [4] G. Birkhoff, Lattice Theory, 3rd (with corrections) edn, American Mathematical Society, Providence, 1995.
    [5] T. Beaubouef, F. E. Petry, G. Arora, Information-theoretic measures of uncertainty for rough sets and rough relational databases, Inform. Sci., 109 (1998), 185–195. https://doi.org/10.1016/S0020-0255(98)00019-X doi: 10.1016/S0020-0255(98)00019-X
    [6] Z. Bonikowski, E. Bryniariski, V. W. Skardowska, Extension and intensions in the rough set theory, Inform. Sci., 107 (1998), 149–167.
    [7] E. Bryniarski, A calculus of rough sets of the first order, B. Pol. Acad. Sci., 1989, 71–77.
    [8] D. Chen, W. Zhang, D. Yeung, E. Tsang, Rough approximations on a complete completely distributive lattice with applications to generalized rough sets, Inform. Sci., 176 (2006), 1829–1848.
    [9] J. Dai, W. Wang, Q. Xu, An uncertainty measure for incomplete decision tables and its applications, IEEE T. Syst. Man Cy., 2012, http://dx.doi.org/10.1109/TSMCB.2012.2228480
    [10] J. Dai, W. Wang, Q. Xu, H. Tian, Uncertainty measurement for interval-valued decision systems based on extended conditional entropy, Knowl.-Based Syst., 27 (2012), 443–450. https://doi.org/10.1016/j.knosys.2011.10.013 doi: 10.1016/j.knosys.2011.10.013
    [11] J. Dai, Q. Xu, W. Wang, H. Tian, Conditional entropy for incomplete decision systems and its application in data mining, Int. J. Gen. Syst., 41 (2012), 713–728. https://doi.org/10.1080/03081079.2012.685471 doi: 10.1080/03081079.2012.685471
    [12] I. Duentsch, G. Gediga, Uncertainty measures of rough set prediction, Artif. Intell., 106 (1998), 109–137. https://doi.org/10.1016/S0004-3702(98)00091-5 doi: 10.1016/S0004-3702(98)00091-5
    [13] A. A. Estaji, M. Vatandoost, R. Pourkhandani, Fixed points of covering upper and lower approximation operators, Soft Comput. 23 (2019), 11447–11460. https://doi.org/10.1007/s00500-019-04113-0 doi: 10.1007/s00500-019-04113-0
    [14] F. Fatimah, D. Rosadi, R. F. Hakim, J. C. R. Alcantud, N-soft sets and their decision making algorithms, Soft Comput., 22 (2018), 3829–3842. https://doi.org/10.1007/s00500-017-2838-6 doi: 10.1007/s00500-017-2838-6
    [15] F. Feng, X. Liu, V. L. Fotea, Y. B. Jun, Soft sets and soft rough sets, Inform. Sci., 181 (2011), 1125–1137. https://doi.org/10.1016/j.ins.2010.11.004 doi: 10.1016/j.ins.2010.11.004
    [16] F. Feng, Soft rough sets applied to multicriteria group decision making, Ann. Fuzzy Math. Inf., 2 (2011), 69–80.
    [17] S. Hirano, S. Tsumoto, Rough set theory and granular computing, Berlin: Springer, 2003.
    [18] Z. Huang, J. Li, Discernibility measures for fuzzy $ \beta $- covering and their application, IEEE T. Cybernetics, 2021.
    [19] Z. Huang, J. Li, Y. Qian, Noise-tolerant fuzzy covering based multigranulation rough sets and feature subset selection, IEEE T. Fuzzy Syst., 2021. https://doi.org/10.1109/TFUZZ.2021.3093202
    [20] Z. Huang, J. Li, A fitting model for attribute reduction with fuzzy $ \beta $-covering, Fuzzy Set. Syst., 413 (2021), 114–137.
    [21] H. Jiang, J. Zhan, D. Chen, Covering based variable precision (I, T)-fuzzy rough sets with applications to multi-attribute decision-making, IEEE T. Fuzzy Syst., 2018.
    [22] J. Y. Liang, Z. Shi, The information entropy, rough entropy and knowledge granulation in rough set theory, Int. J. Uncertain. Fuzz., 12 (2004), 37–46.
    [23] T. Y. Lin, Y. Y. Yao, L. A. Zadeh, Rough sets, granular computing and data mining, Studies in fuzziness and soft computing, Physica-Verlag, Heidelberg, 2001.
    [24] J. Y. Liang, K. S. Chin, C. Dang, R. C. M. Yam, A new method for measuring uncertainty and fuzziness in rough set theory, Int. J. Gen. Syst. 31 (2002), 331–342. https://doi.org/10.1080/0308107021000013635 doi: 10.1080/0308107021000013635
    [25] G. Liu, Y. Sai, A comparison of two types of rough sets induced by coverings, Int. J. Approx. Reason., 50 (2009), 521–528. https://doi.org/10.1016/j.ijar.2008.11.001 doi: 10.1016/j.ijar.2008.11.001
    [26] Q. Li, W. Zhu, Lattice structures of fixed points of the lower approximations of two types of covering-based rough sets, 2012.
    [27] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6 doi: 10.1016/S0898-1221(03)00016-6
    [28] P. K. Maji, A. R. Roy, R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl., 44 (2002), 1077–1083. https://doi.org/10.1016/S0898-1221(02)00216-X doi: 10.1016/S0898-1221(02)00216-X
    [29] D. Molodtsov, Soft set theory - first results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
    [30] D. Molodtsov, The theory of soft sets (in Russian), URSS Publishers, Moscow, 2004.
    [31] Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci., 11 (1982), 341–356. https://doi.org/10.4018/978-1-59140-560-3.ch095 doi: 10.4018/978-1-59140-560-3.ch095
    [32] Z. Pawlak, Rough sets: Theoretical aspects of reasoning about data, Kluwer Academic Publishers, Dordrecht, 1991.
    [33] Z. Pawlak, A. Skowron, Rough sets: Some extensions, Inform. Sci., 177 (2007), 28–40. https://doi.org/10.1016/j.ins.2006.06.006 doi: 10.1016/j.ins.2006.06.006
    [34] J. Pomykala, Approximation, similarity and rough constructions, ILLC Prepublication series, University of Amsterdam, 1993.
    [35] B. Praba, G. Gomathi, M. Aparajitha, A lattice structure on minimal soft rough sets and its applications, New Math. Nat. Comput., 16 (2020), 255–269. https://doi.org/10.1142/S1793005720500155 doi: 10.1142/S1793005720500155
    [36] N. Shah, N. Rehman, M. Shabir, M. I. Ali, Another approach to roughness of soft graphs with applications in decision making, Symmetry, 10 (2018). https://doi.org/10.3390/sym10050145
    [37] C. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., 27 (1948), 379–423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x doi: 10.1002/j.1538-7305.1948.tb01338.x
    [38] A. Skowron, J. Stepaniuk, Tolerance approximation spaces, Fundam. Inform., 27 (1996), 245–253. https://doi.org/10.3233/FI-1996-272311 doi: 10.3233/FI-1996-272311
    [39] N. Shah, M. I. Ali, M. Shabir, N. Rehman, Uncertainty measure of Z-soft covering rough models based on a knowledge granulation, J. Intell. Fuzzy Syst., 38 (2020), 1637–1647. https://doi.org/10.3233/JIFS-182708 doi: 10.3233/JIFS-182708
    [40] N. Rehman, N. Shah, M. I. Ali, C. Park, Uncertainty measurement for neighborhood based soft covering rough graphs with applications, RACSAM Rev. R. Acad. A, 113 (2019), 2515–2535. https://doi.org/10.1007/s13398-019-00632-5 doi: 10.1007/s13398-019-00632-5
    [41] R. Slowinski, D. Vanderpooten, Similarity relation as a basis for rough approximations, 1995.
    [42] M. Wierman, Measuring uncertainty in rough set theory, Int. J. Gen. Syst., 28 (1999), 283–297. https://doi.org/10.1080/03081079908935239 doi: 10.1080/03081079908935239
    [43] W. X. Xu, W. X. Zhang, Measuring roughness of generalized rough sets induced by a covering, Fuzzy Set. Syst., 158 (2007), 2443–2455. https://doi.org/10.1016/j.fss.2007.03.018 doi: 10.1016/j.fss.2007.03.018
    [44] Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sci., 101 (1998), 239–259. https://doi.org/10.1016/S0020-0255(98)10006-3 doi: 10.1016/S0020-0255(98)10006-3
    [45] S. Yüksel, Z. G. Ergül, N. Tozlu, Soft covering based rough sets and their application, The Scientific World J., 2014, 1–9. https://doi.org/10.1155/2014/970893
    [46] S. Yüksel, N. Tozlu, T. H. Dizman, An application of multicriteria group decision making by soft covering based rough sets, Filomat, 29 (2015), 209–219. https://doi.org/10.2298/FIL1501209Y doi: 10.2298/FIL1501209Y
    [47] L. A. Zadeh, Fuzzy sets, Inform. Sci., 8 (1965), 338–353. https://doi.org/10.21236/AD0608981 doi: 10.21236/AD0608981
    [48] W. Zhu, F. Y. Wang, On three types of covering-based rough sets, IEEE T. Knowl. Data En., 2007. https://doi.org/10.1109/TKDE.2007.1044
    [49] J. Zhan, K. Zhu, A novel soft rough fuzzy set: Z-soft rough fuzzy ideals of hemirings and corresponding decision making, Soft Comput., 21 (2017), 1923–1936. https://doi.org/10.1007/s00500-016-2119-9 doi: 10.1007/s00500-016-2119-9
    [50] J. Zhan, M. I. Ali, N. Mehmood, On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods, Appl. Soft Comput. 56 (2017), 446–457. https://doi.org/10.1016/j.asoc.2017.03.038 doi: 10.1016/j.asoc.2017.03.038
    [51] J. Zhan, J. C. R Alcantud, A novel type of soft rough covering and its application to multicriteria group decision making, Artif. Intell. Rev., 2018, 1–30.
    [52] J. Zhan, B. Sun, J. C. R. Alcantud, Covering based multigranulation (I, T)-fuzzy rough set models and applications in multi-attribute group decision-making, Inform. Sci., 476 (2019), 290–318.
    [53] H. Zhang, J. Zhan, Rough soft lattice implication algebras and corresponding decision making methods, Int. J. Mach Lear. Cyber., 8 (2017), 1301–1308. https://doi.org/10.1007/s13042-016-0502-6 doi: 10.1007/s13042-016-0502-6
    [54] L. Zhang, J. Zhan, J. C. R. Alcantud, Novel classes of fuzzy soft $ \beta $-coverings-based fuzzy rough sets with applications to multi-criteria fuzzy group decision making, Soft Comput., 2018, 1–25. https://doi.org/10.1007/s00500-018-3470-9
    [55] W. Zhu, F. Wang, On three types of covering-based rough sets, IEEE T. Knowl. Data En., 19 (2007), 1131–1144 https://doi.org/10.1109/TKDE.2007.1044 doi: 10.1109/TKDE.2007.1044
    [56] W. Zhu, F. Wang, Reduction and axiomization of covering generalized rough sets, Inform. Sci., 152 (2003), 217–230. https://doi.org/10.1016/S0020-0255(03)00056-2 doi: 10.1016/S0020-0255(03)00056-2
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