Abstract
Three-way decisions, as a better way than two-way decisions, has played an important role in many fields. As an extension of semiconcept, preconcept constitutes a new approach for data analysis. In contrast to preconcept, formal concept or semiconcept is too conservative about dealing with data. Hence, we want to further apply three-way decisions to preconcept. In this work, we introduce three-way preconcept by an example. This new notion combines preconcept with the assistant of three-way decisions. After that, we attain a generalized double Boolean algebra consisting of three-way preconcept. Furthermore, we give two form operators, approximation operators from lattice and set equivalence relation approximation operators, respectively. Finally, we present a conclusion with some summary and future issues that need to be addressed.
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References
Burgmann C, Wille R (2006) The basic theorem on preconcept lattices. In: Formal concept analysis. Springer, Berlin, pp 80–88. https://doi.org/10.1007/11671404_5
Duntsch N, Gediga G (2002) Modal-style operators in qualitative data analysis. In: IEEE international conference on data mining. Proceedings IEEE Computer Society (online). https://doi.org/10.1109/icdm.2002.1183898
Ganter B, Wille R, Franzke C (1997) Formal concept analysis: mathematical foundations. Springer-Verlag, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59830-2
Grätzer G (1978) General lattice theory. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7633-9
Hu M, Deng X, Yao Y (2019) An application of bayesian confirmation theory for three-way decision. In: Rough sets. Springer, Berlin, pp 3–15 (online). https://doi.org/10.1007/978-3-030-22815-6_1
Jia X, Shang L, Zhou B, Yao Y (2016) Generalized attribute reduct in rough set theory. Knowl-Based Syst 91:204–218. https://doi.org/10.1016/j.knosys.2015.05.017
Jiao L, Yang H, Li S (2019) Three-way decision based on decision-theoretic rough sets with single-valued neutrosophic information. Int J Mach Learn Cybern. https://doi.org/10.1007/s13042-019-01023-3
Li J, Mei C, Xu W, Qian Y (2015) Concept learning via granular computing: a cognitive viewpoint. Inf Sci 298:447–467. https://doi.org/10.1016/j.ins.2014.12.010
Li W, Huang Z, Li Q (2016) Three-way decisions based software defect prediction. Knowl Based Syst 91:263–274. https://doi.org/10.1016/j.knosys.2015.09.035
Liang D, Xu Z, Liu D, Wu Y (2018) Method for three-way decisions using ideal TOPSIS solutions at pythagorean fuzzy information. Inf Sci 435:282–295. https://doi.org/10.1016/j.ins.2018.01.015
Luksch P, Wille R (1991) A mathematical model for conceptual knowledge systems. In: Studies in classification, data analysis, and knowledge organization. Springer, Berlin, pp 156–162 (online). https://doi.org/10.1007/978-3-642-76307-6_21
Mao H (2019) Approximation operators for semiconcepts. J Intell Fuzzy Syst 36(4):3333–3343. https://doi.org/10.3233/jifs-18104
Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11(5):341–356
Qian T, Wei L, Qi J (2019) A theoretical study on the object (property) oriented concept lattices based on three-way decisions. Soft Comput 23(19):9477–9489
Qi J, Wei L, Yao Y (2014) Three-way formal concept analysis. In: Rough sets and knowledge technology. Springer, Berlin, pp 732–741. (online). https://doi.org/10.1007/978-3-319-11740-967
Ren R, Wei L (2016) The attribute reductions of three-way concept lattices. Knowl-Based Syst 99:92–102. https://doi.org/10.1016/j.knosys.2016.01.045
Sharma HK, Kumari K, Kar S (2020) A rough set approach for forecasting models. Decis Mak Appl Manag Eng 3(1):1–21
Vormbrock B, Wille R (2005) Semiconcept and protoconcept algebras: the basic theorems. In: Formal concept analysis. Springer, Berlin, pp 34–48 (online). https://doi.org/10.1007/11528784_2
Wan Q, Wei L (2015) Approximate concepts acquisition based on formal contexts. Knowl Based Syst 75:78–86. https://doi.org/10.1016/j.knosys.2014.11.020
Wille R (1982) Restructuring lattice theory: an approach based on hierarchies of concepts. In: Ordered sets. Springer, Netherlands, pp 445–470 (online). https://doi.org/10.1007/978-94-009-7798-3_15
Wille R (2000) Boolean concept logic. In: Lecture notes in computer science. Springer, Berlin, pp 317–331. https://doi.org/10.1007/2F10722280_22
Wille R (2004) Preconcept algebras and generalized double boolean algebras. In: Concept lattices. Springer, Berlin, pp 1–13 (online). https://doi.org/10.1007/978-3-540-24651-0_1
Xu F, Xing Z, Yin H (2016) Attribute reductions and concept lattices in interval-valued intuitionistic fuzzy rough set theory: construction and properties. J Intell Fuzzy Syst 30(2):1231–1242. https://doi.org/10.3233/ifs-151848
Yang L, Xu L (2009) On rough concept lattices. Electron Notes Theor Comput Sci 257:117–133
Yao Y (2004) A comparative study of formal concept analysis and rough set theory in data analysis. In: Rough sets and current trends in computing. Springer, Berlin, pp 59–68 (online). https://doi.org/10.1007/978-3-540-25929-9_6
Yao Y (2004) Concept lattices in rough set theory. In: IEEE annual meeting of the fuzzy information, 2004. Processing NAFIPS 04. IEEE (online). https://doi.org/10.1109/nafips.2004.1337404
Yao Y (2009) Three-way decision: an interpretation of rules in rough set theory. In: Rough sets and knowledge technology. Springer, Berlin, pp 642–649. (online). https://doi.org/10.1007/978-3-642-02962-2_81
Yao Y (2010) Three-way decisions with probabilistic rough sets. Inf Sci 180(3):341–353. https://doi.org/10.1016/j.ins.2009.09.021
Yao Y (2018) Three-way decision and granular computing. Int J Approx Reason 103:107–123. https://doi.org/10.1016/j.ijar.2018.09.005
Zhan J, Jiang H, Yao Y (2020) Three-way multi-attribute decision-making based on outranking relations. IEEE Trans Fuzzy Syst 29:2844–2858
Zhan J, Ye J, Ding W, Liu P (2021) A novel three-way decision model based on utility theory in incomplete fuzzy decision systems. IEEE Trans Fuzzy Syst 30:2210–2226
Zhang H, Ma Q (2020) Three-way decisions with decision-theoretic rough sets based on pythagorean fuzzy covering. Soft Comput 24(24):18671–18688
Zhang K, Zhan J, Wu W-Z (2020) On multi-criteria decision-making method based on a fuzzy rough set model with fuzzy \(\alpha \)-neighborhoods. IEEE Trans Fuzzy Syst 29:2491–2505
Acknowledgements
This work is supported by the National Natural Science Foundation of China (61572011), the Natural Science Foundation of Hebei Province (A201820117), Hebei Graduate Innovation Project (CXZZSS2020004).
Funding
This work is supported by the National Natural Science Foundation of China (61572011), the Natural Science Foundation of Hebei Province (A201820117), Hebei Graduate Innovation Innovation Project (CXZZSS2020004).
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Mao, H., Cheng, Y. & Liu, X. Three-way preconcept and two forms of approximation operators. Soft Comput 27, 855–865 (2023). https://doi.org/10.1007/s00500-022-07532-8
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DOI: https://doi.org/10.1007/s00500-022-07532-8