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AIMS Mathematics

2022, Volume 7, Issue 6: 9603-9626. doi: 10.3934/math.2022534
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Research article

A new approach for operations on neutrosophic soft sets based on the novel norms for constructing topological structures

  • 1.
    Finance and Accounting Department, Ho Chi Minh City University of Law, Ho Chi Minh City, Vietnam
  • 2.
    Department of Mathematics, Tien Giang High School for The Gifted, Tien Giang Province, Vietnam
  • 3.
    Faculty of Education and Basic Sciences, Tien Giang University, Tien Giang Province, Vietnam
  • 4.
    Faculty of Electrical Engineering and Computer Science, VŠB-Technical University of Ostrava, Ostrava-Poruba, Czech Republic
  • 5.
    Faculty of Information Technology, HUTECH University, Ho Chi Minh City, Vietnam
  • Received: 24 December 2021 Revised: 29 January 2022 Accepted: 08 February 2022 Published: 15 March 2022

  • MSC : 54A40, 54E55, 54D10

  • Neutrosophic sets have recently emerged as a tool for dealing with imprecise, indeterminate, inconsistent data, while soft sets may have the potential to deal with uncertainties that classical methods cannot control. Combining these two types of sets results in a unique hybrid structure, a neutrosophic soft set (NS-set), for working effectively in uncertain environments. This paper focuses on determining operations on NS-sets through two novel norms. Accordingly, the $ {\rm{min}}-{\rm{n}}{\rm{o}}{\rm{r}}{\rm{m}} $ and $ {\rm{max}}-{\rm{n}}{\rm{o}}{\rm{r}}{\rm{m}} $ are well-defined here for the first time to construct the intersection, union, difference, AND, OR operations. Then, the topology, open set, closed set, interior, closure, regularity concepts on NS-sets are introduced based on these just constructed operations. All the properties in the paper are stated in theorem form, which is proved convincingly and logically. In addition, we also elucidate the relationship between the topology on NS-sets and the fuzzy soft topologies generated by the truth, indeterminacy, falsity degrees by theorems and counterexamples.

    Citation: Tram B.T. Tran, My-Phuong Ngo, Quang-Thinh Bui, Vaclav Snasel, Bay Vo. A new approach for operations on neutrosophic soft sets based on the novel norms for constructing topological structures[J]. AIMS Mathematics, 2022, 7(6): 9603-9626. doi: 10.3934/math.2022534

    Related Papers:

  • Neutrosophic sets have recently emerged as a tool for dealing with imprecise, indeterminate, inconsistent data, while soft sets may have the potential to deal with uncertainties that classical methods cannot control. Combining these two types of sets results in a unique hybrid structure, a neutrosophic soft set (NS-set), for working effectively in uncertain environments. This paper focuses on determining operations on NS-sets through two novel norms. Accordingly, the $ {\rm{min}}-{\rm{n}}{\rm{o}}{\rm{r}}{\rm{m}} $ and $ {\rm{max}}-{\rm{n}}{\rm{o}}{\rm{r}}{\rm{m}} $ are well-defined here for the first time to construct the intersection, union, difference, AND, OR operations. Then, the topology, open set, closed set, interior, closure, regularity concepts on NS-sets are introduced based on these just constructed operations. All the properties in the paper are stated in theorem form, which is proved convincingly and logically. In addition, we also elucidate the relationship between the topology on NS-sets and the fuzzy soft topologies generated by the truth, indeterminacy, falsity degrees by theorems and counterexamples.


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    [1] J. Han, J. Pei, M. Kamber, Data mining: Concepts and techniques, New York: Elsevier, 2011.
    [2] R. H. Hariri, E. M. Fredericks, K. M. Bowers, Uncertainty in big data analytics: Survey, opportunities, and challenges, J. Big Data, 6 (2019), 44. https://doi.org/10.1186/s40537-019-0206-3 doi: 10.1186/s40537-019-0206-3
    [3] L. A. Zadeh, Fuzzy Sets, Inform. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
    [4] W. L. Gau, D. J. Buehrer, Vague sets, IEEE T. Syst. Man Cyber., 23 (1993), 610–614. https://doi.org/10.1109/21.229476 doi: 10.1109/21.229476
    [5] 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
    [6] F. Smarandache, Neutrosophic set–A generalization of the intuitionistic fuzzy set, 2006 IEEE International Conference on Granular Computing, 2006, 38–42. https://doi.org/10.1109/GRC.2006.1635754 doi: 10.1109/GRC.2006.1635754
    [7] H. Sun, W. Lv, A. O. Khadidos, R. Kharabsheh, Research on the influence of fuzzy mathematics simulation model in the development of Wushu market, Appl. Math. Nonlinear Sci., 2021. https://doi.org/10.2478/amns.2021.2.00062 doi: 10.2478/amns.2021.2.00062
    [8] L. Zhang, X. Tian, Application of fuzzy mathematics calculation in quantitative evaluation of students' performance of basketball jump shot, Appl. Math. Nonlinear Sci., 2021. https://doi.org/10.2478/amns.2021.1.00074 doi: 10.2478/amns.2021.1.00074
    [9] Y. Wang, A. O. Khadidos, The Influence of X fuzzy mathematical method on basketball tactics scoring, Appl. Math. Nonlinear Sci., 2021. https://doi.org/10.2478/amns.2021.2.00057 doi: 10.2478/amns.2021.2.00057
    [10] Y. Zhang, M. Cui, L. Shen, Z. Zeng, Memristive fuzzy deep learning systems, IEEE T. Fuzzy Syst., 29 (2020), 2224–2238. https://doi.org/10.1109/TFUZZ.2020.2995966 doi: 10.1109/TFUZZ.2020.2995966
    [11] Y. Zheng, Z. Xu, X. Wang, The fusion of deep learning and fuzzy systems: A state-of-the-art survey, IEEE T. Fuzzy Syst., 2021. https://doi.org/10.1109/TFUZZ.2021.3062899 doi: 10.1109/TFUZZ.2021.3062899
    [12] Q. T. Bui, B. Vo, H. A. N. Do, N. Q. V. Hung, V. Snasel, F-Mapper: A Fuzzy Mapper clustering algorithm, Knowl.-Based Syst., 189 (2020), 105107. https://doi.org/10.1016/j.knosys.2019.105107 doi: 10.1016/j.knosys.2019.105107
    [13] Q. T. Bui, B. Vo, V. Snasel, W. Pedrycz, T. P. Hong, SFCM: A fuzzy clustering algorithm of extracting the shape information of data, IEEE T. Fuzzy Syst., 29 (2021), 75–89. https://doi.org/10.1109/TFUZZ.2020.3014662 doi: 10.1109/TFUZZ.2020.3014662
    [14] F. Smarandache, A unifying field in logics: Neutrosophic logic, neutrosophic set, neutrosophic probability and statistics, American Research Press, 1999.
    [15] M. Das, D. Mohanty, K. C. Parida, On the neutrosophic soft set with rough set theory, Soft Comput., 25 (2021), 13365–13376. https://doi.org/10.1007/s00500-021-06089-2 doi: 10.1007/s00500-021-06089-2
    [16] B. Vo, T. Tran, T. P. Hong, N. L. Minh, Using soft set theory for mining maximal association rules in text data, J. Univ. Comput. Sci., 22 (2016), 802–821.
    [17] P. K. Maji, A neutrosophic soft set approach to a decision making problem, Ann. Fuzzy Math. Inform., 3 (2012), 313–319.
    [18] P. K. Maji, Neutrosophic soft set, Ann. Fuzzy Math. Inform., 5 (2013), 157–168.
    [19] Deli, S. Broumi, Neutrosophic soft matrices and NSM-decision making, J. Intell. Fuzzy Syst., 28 (2015), 2233–2241. https://doi.org/10.3233/IFS-141505 doi: 10.3233/IFS-141505
    [20] S. Jha, R. Kumar, L. H. Son, J. M. Chatterjee, M. Khari, N. Yadav, et al., Neutrosophic soft set decision making for stock trending analysis, Evolving Syst., 10 (2019), 621–627. https://doi.org/10.1007/s12530-018-9247-7 doi: 10.1007/s12530-018-9247-7
    [21] Arockiarani, A fuzzy neutrosophic soft set model in medical diagnosis, 2014 IEEE Conference on Norbert Wiener in the 21st Century, 2014, 1–8. https://doi.org/10.1109/NORBERT.2014.6893943 doi: 10.1109/NORBERT.2014.6893943
    [22] J. S. Chai, G. Selvachandran, F. Smarandache, V. C. Gerogiannis, L. H. Son, Q. T. Bui, et al., New similarity measures for single-valued neutrosophic sets with applications in pattern recognition and medical diagnosis problems, Complex Intell. Syst., 7 (2021), 703–723. https://doi.org/10.1007/s40747-020-00220-w doi: 10.1007/s40747-020-00220-w
    [23] F. G. Lupiáñez, On neutrosophic sets and topology, Procedia Comput. Sci., 120 (2017), 975–982. https://doi.org/10.1016/j.procs.2018.01.090 doi: 10.1016/j.procs.2018.01.090
    [24] T. Y. Ozturk, Some structures on neutrosophic topological spaces, Appl. Math. Nonlinear Sci., 6 (2021), 467–478. https://doi.org/10.2478/amns.2020.2.00069 doi: 10.2478/amns.2020.2.00069
    [25] J. C. R. Alcantud, Soft open bases and a novel construction of soft topologies from bases for topologies, Mathematics, 8 (2020), 672. https://doi.org/10.3390/math8050672 doi: 10.3390/math8050672
    [26] G. Cantin, C. J. Silva, Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models, AIMS Math., 4 (2019), 1145–1169. https://doi.org/10.3934/math.2019.4.1145 doi: 10.3934/math.2019.4.1145
    [27] M. E. Sayed, M. A. E. 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
    [28] T. Bera, N. K. Mahapatra, Introduction to neutrosophic soft topological space, OPSEARCH, 54 (2017), 841–867. https://doi.org/10.1007/s12597-017-0308-7 doi: 10.1007/s12597-017-0308-7
    [29] T. Bera, N. K. Mahapatra, On neutrosophic soft topological space, Neutrosophic Sets Syst., 9 (2018), 299–324. https://doi.org/10.1016/j.fiae.2017.09.004 doi: 10.1016/j.fiae.2017.09.004
    [30] T. Y. Ozturk, Ç. G. Aras, S. Bayramov, A new approach to operations on neutrosophic soft sets and to neutrosophic soft topological spaces, Commun. Math. Appl., 10 (2019), 481–493. https://doi.org/10.26713/cma.v10i3.1068 doi: 10.26713/cma.v10i3.1068
    [31] Ç. G. Aras, T. Y. Ozturk, S. Bayramov, Separation axioms on neutrosophic soft topological spaces, Turk. J. Math., 43 (2019), 498–510. doi: 10.3906/mat-1805-110
    [32] G. A. Çiğdem, B. Sadi, Neutrosophic soft continuity in neutrosophic soft topological spaces, Filomat, 34 (2020), 3495–3506. https://doi.org/10.2298/FIL2010495G doi: 10.2298/FIL2010495G
    [33] T. Y. Ozturk, A. Benek, A. Ozkan, Neutrosophic soft compact spaces, Afr. Mat., 32 (2021), 301–316. https://doi.org/10.1007/s13370-020-00827-9 doi: 10.1007/s13370-020-00827-9
    [34] P. Revathi, K. Chitirakala, A. vadivel, Soft e-separation axioms in neutrosophic soft topological spaces, J. Phys.: Conf. Ser., 2070 (2021), 012028. doi: 10.1088/1742-6596/2070/1/012028
    [35] J. C. R. Alcantud, An operational characterization of soft topologies by crisp topologies, Mathematics, 9 (2021), 1656. https://doi.org/10.3390/math9141656 doi: 10.3390/math9141656
    [36] J. C. R. Alcantud, T. M. Al-shami, A. A. Azzam, Caliber and chain conditions in soft topologies, Mathematics, 9 (2021), 2349. https://doi.org/10.3390/math9192349 doi: 10.3390/math9192349
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