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Ma, Liwen

On some types of neighborhood-related covering rough sets. (English) Zbl 1246.03068

Int. J. Approx. Reasoning 53, No. 6, 901-911 (2012).
Summary: Covering rough sets are natural extensions of the classical rough sets by relaxing the partitions to coverings. Recently, the concept of neighborhood has been applied to define different types of covering rough sets. In this paper, by introducing a new notion of complementary neighborhood, we consider some types of neighborhood-related covering rough sets, two of which are firstly defined. We first show some basic properties of the complementary neighborhood. We then explore the relationships between the considered covering rough sets and investigate the properties of them. It is interesting that the set of all the lower and upper approximations belonging to the considered types of covering rough sets, equipped with the binary relation of inclusion \(\subseteq \), constructs a lattice. Finally, we also discuss the topological importance of the complementary neighborhood and investigate the topological properties of the lower and upper approximation operators.

Cited in 54 Documents

MSC:

03E72 Theory of fuzzy sets, etc.

Keywords:

covering rough set; neighborhood; complementary neighborhood; topology
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\textit{L. Ma}, Int. J. Approx. Reasoning 53, No. 6, 901--911 (2012; Zbl 1246.03068)
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References:

[1] Pawlak, Z., Rough sets, International Journal of Computer and Information Sciences, 11, 341-356 (1982) · Zbl 0501.68053
[2] Slowinski, R.; Vanderpooten, D., A generalized definition of rough approximations based on similarity, IEEE Transactions on Knowledge and Data Engineering, 12, 331-336 (2000)
[3] Meng, Z.; Shi, Z., A fast approach to attribute reduction in incomplete decision systems with tolerance relation-based rough sets, Information Sciences, 179, 2774-2793 (2009) · Zbl 1191.68667
[4] Skowron, A.; Stepaniuk, J., Tolerance approximation spaces, Fundamenta Informaticae, 27, 245-253 (1996) · Zbl 0868.68103
[5] Yao, Y. Y., Constructive and algebraic methods of theory of rough sets, Information Sciences, 109, 21C47 (1998) · Zbl 0934.03071
[6] Kondo, M., On the structure of generalized rough sets, Information Sciences, 176, 589-600 (2005) · Zbl 1096.03065
[7] Zhu, W., Generalized rough sets based on relations, Information Sciences, 177, 4997-5011 (2007) · Zbl 1129.68088
[9] Li, T.; Ruan, D.; Geert, W.; Song, J.; Xu, Y., A rough sets based characteristic relation approach for dynamic attribute generalization in data mining, Knowledge-Based System, 20, 485-494 (2007)
[10] Bonikowski, Z.; Bryniarski, E.; Wybraniec-Skardowska, U., Extensions and intentions in the rough set theory, Information Sciences, 107, 149-167 (1998) · Zbl 0934.03069
[11] Pomykala, J. A., Approximation operations in approximation space, Bulletin of the Polish Academy of Science, 35, 653-662 (1987) · Zbl 0642.54002
[12] Zakowski, W., Approximations in the space (U,Π), Demonstratio Mathematica, 16, 761-769 (1983) · Zbl 0553.04002
[13] Zhu, W.; Wang, F. Y., Reduction and axiomization of covering generalized rough sets, Information Sciences, 152, 217-230 (2003) · Zbl 1069.68613
[14] Zhu, W., Relationship among basic concepts in covering-based rough sets, Information Sciences, 179, 2478-2486 (2009) · Zbl 1178.68579
[15] Zhu, W.; Wang, F. Y., On three types of covering rough sets, IEEE Transactions on Knowledge and Data Engineering, 19, 1131-1144 (2007)
[16] Pomykala, J. A., On definability in the nondeterministic information system, Bulletin of the Polish Academy of Science, Mathematics, 36, 193-210 (1988) · Zbl 0677.68110
[17] Zhu, W., Topological approaches to covering rough sets, Information Sciences, 177, 1499-1508 (2007) · Zbl 1109.68121
[19] Liu, G.; Sai, Y., A comparison of two types of rough sets induced by coverings, International Journal of Approximate Reasoning, 50, 521-528 (2009) · Zbl 1191.68689
[20] Zhu, P., Covering rough sets based on neighborhoods: An approach without using neighborhoods, International Journal of Approximate Reasoning, 52, 461-472 (2011) · Zbl 1229.03047
[21] Yang, T.; Li, Q., Reduction about approximation spaces of covering generalized rough sets, International Journal of Approximate Reasoning, 51, 335-345 (2010) · Zbl 1205.68433
[22] Feng, T.; Zhang, S.; Mi, J., The reduction and fusion of fuzzy covering systems based on the evidence theory, International Journal of Approximate Reasoning, 53, 87-103 (2012) · Zbl 1242.68326
[23] Wang, C.; Chen, D.; Wu, C.; Hu, Q., Data compression with homomorphism in covering information systems, International Journal of Approximate Reasoning, 52, 519-525 (2011) · Zbl 1217.68240
[24] Zhu, P.; Wen, Q., Entropy and co-entropy of a covering approximation space, International Journal of Approximate Reasoning, 53, 528-540 (2012) · Zbl 1246.68237
[25] Pei, Z.; Pei, D. W.; Zheng, Li, Topology vs generalized rough sets, International Journal of Approximate Reasoning, 52, 231-239 (2011) · Zbl 1232.03044
[26] Li, X.; Liu, S., Matroidal approaches to rough sets via closure operators, International Journal of Approximate Reasoning, 53, 513-527 (2012) · Zbl 1246.68233
[27] Salama, A. S., Some topological properties of rough sets with tools for data mining, International Journal of Computer Science, 8, 588-595 (2011)
[28] Yang, X. B.; Zhang, M.; Dou, H. L.; Yang, J. Y., Neighborhood systems-based rough sets in incomplete information system, Knowledge-Based Systems, 24, 858-867 (2011)
[29] Samanta, P.; Chakraborty, M. K., Covering based approaches to rough sets and implication lattices, Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, 5908, 127-134 (2009)
[30] Liu, B. X., Analytical models and decision models in rough set pairs (2011), Science Press: Science Press Beijing, (in Chinese)
[31] Yao, Y. Y., Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences, 111, 239-259 (1998) · Zbl 0949.68144
[32] Xu, W. H.; Zhang, W. X., Measuring roughness of generalized rough sets induced by a covering, Fuzzy Sets and Systems, 158, 2443-2455 (2007) · Zbl 1127.68106
[33] Pawlak, Z., ROUGH SETS - Theoritical Aspects of Reasoning About Data (1991), Kluwer Academic Publisher: Kluwer Academic Publisher Dordrecht
[34] Engelking, R., General Topology (1977), Poblish Scientific Publishers warszaw
[35] Kelly, J., General Topology (1995), Van Nostrand Company: Van Nostrand Company New York
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