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Rough sets and matroids from a lattice-theoretic viewpoint. (English) Zbl 1403.06018
Summary: This paper studies rough sets via matroidal approaches from a lattice-theoretic viewpoint. We firstly give a new interpretation of definable sets of Pawlak rough set model, i.e., the set of definable sets defines uniquely a matroid, in which it is the family of open and closed sets. Then we induce two equivalence relations on a given universe based on a matroid defined on this universe. One of the equivalence relations actually is defined on the set of all atoms of a geometric lattice corresponding to the matroid, another is based on the transitivity of circuits. Properties of these two equivalence relations are then studied. Besides, we also investigate the connections between relation-based rough sets and matroids. Finally, we point out that a geometric lattice can induce a series of coverings of a universe, on which the corresponding matroid is defined, and further relations of approximations based on the induced coverings are studied.

06D75 Other generalizations of distributive lattices
05B35 Combinatorial aspects of matroids and geometric lattices
68T37 Reasoning under uncertainty in the context of artificial intelligence
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[1] Davey, B. A.; Priestley, H. A., Introduction to lattices and order, (1990), Cambridge University Press Cambridge · Zbl 0701.06001
[2] Bonikowski, Z.; Bryniarski, E.; Wybraniec-Skardowska, U., Extensions and intentions in the rough set theory, Inf. Sci., 107, 149-167, (1998) · Zbl 0934.03069
[3] Cattaneo, G., Abstract approximation spaces for rough theories, (Polkowski, L.; Skowron, A., Rough Sets in Knowledge Discovery 1: Methodology and Applications (Applications, Cases Studies and Software Systems), Studies in Fuzziness and Soft Computing, vol. 18, (1998), Physica-Verlag Heidelberg), 59-98 · Zbl 0927.68087
[4] Cattaneo, G.; Ciucci, D., Algebraic structures for rough sets, (Peters, J. F.; etal., Transactions on Rough Sets II, LNCS, vol. 3135, (2009), Springer-Verlag Heidelberg), 67-116
[5] Cattaneo, G.; Ciucci, D., Lattices with interior and closure operators and abstract approximation spaces, (Peters, J. F.; etal., Transactions on Rough Sets X, LNCS, vol. 5656, (2009), Springer-Verlag Heidelberg), 67-116 · Zbl 1248.06005
[6] Deng, H. Y., The properties of rough sets, J. Hunan Norm. Univ., 30, 3, 16-18, (2007)
[7] Feng, F.; Li, C.; Davvaz, B.; Ali, M. I., Soft sets combined with fuzzy sets and soft rough sets: A tentative approach, Soft Comput., 14, 899-911, (2010) · Zbl 1201.03046
[8] Feng, F.; Liu, X.; Leoreanu-Fotea, V.; Jun, Y. B., Soft sets and soft rough sets, Inf. Sci., 181, 1125-1137, (2011) · Zbl 1211.68436
[9] Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst., 17, 191-209, (1990) · Zbl 0715.04006
[10] Goetschel, R.; Voxman, W., Fuzzy matroids, Fuzzy Sets Syst., 32, 291-302, (1988) · Zbl 0651.05024
[11] Grätzer, G., Lattice theory: foundation, (2011), Birkhäuser Basel · Zbl 1233.06001
[12] Jarvinen, J., Lattice theory for rough sets, (Peters, J. F.; etal., Transactions on Rough Sets VI, LNCS, vol. 4374, (2007), Springer-Verlag Heidelberg), 400-498 · Zbl 1186.03069
[13] Lawler, E., Combinatorial optimization: networks and matroids, (1976), Holt, Rinehart and Winston New York · Zbl 0413.90040
[14] Lashin, E. F.; Kozae, A. M.; Khadra, A. A.A.; Medhat, T., Rough set theory for topological spaces, Int. J. Approx. Reason., 40, 35-43, (2005) · Zbl 1099.68113
[15] Li, X. N.; Liu, S. Y.; Li, S. G., Connectedness of refined goetschel-voxman fuzzy matroids, Fuzzy Sets Syst., 161, 2709-2723, (2010) · Zbl 1205.05042
[16] Li, X. N.; Liu, S. Y., A new approach to the axiomatization of rough set, Proceedings of the Seventh International Conference on Fuzzy Systems and Knowledge Discovery (FSKD), 1936-1939, (2007)
[17] Li, X. N.; Liu, S. Y., Matroidal approaches to rough sets via closure operators, Int. J. Approx. Reason., 53, 513-527, (2012) · Zbl 1246.68233
[18] Mao, H., Characterization and reduct of concept lattices through matroid theory, Inf. Sci., 281, 338-354, (2014) · Zbl 1355.68248
[19] Miao, D. Q.; Han, S. Q.; Li, D. G.; Sun, L. J., Rough group, rough subgroup and their properties, (Slezak, D.; etal., RSFDGrC 2005, LNAI, vol. 3641, (2005), Springer-Verlag Berlin), 104-113 · Zbl 1134.20310
[20] Oxley, J. G., Matroid theory, (1992), Oxford University Press New York · Zbl 0784.05002
[21] Pawlak, Z., Rough sets, Int. J. Comput. Inf. Sci., 11, 341-356, (1982) · Zbl 0501.68053
[22] Pawlak, Z., Rough sets: theoretical aspects of reasoning about data, system theory, Knowledge Engineering and Problem Solving, vol. 9, (1991), Kluwer Academic Publishers Dordrecht, Netherlands · Zbl 0758.68054
[23] Pawlak, Z.; Skowron, A., Rudiments of rough sets, Inf. Sci., 177, 1, 3-27, (2007) · Zbl 1142.68549
[24] Pawlak, Z.; Skowron, A., Rough sets: some extensions, Inf. Sci., 177, 1, 28-40, (2007) · Zbl 1142.68550
[25] Pawlak, Z.; Skowron, A., Rough sets and Boolean reasoning, Inf. Sci., 177, 1, 41-73, (2007) · Zbl 1142.68551
[26] Pei, Z.; Pei, D. W.; Zheng, L., Topology vs generalized rough sets, Int. J. Approx. Reason., 52, 2, 231-239, (2010) · Zbl 1232.03044
[27] Pomykala, J. A., Approximation operations in approximation space, Bull. Pol. Acad. Sci. Math., 35, 653-662, (1987) · Zbl 0642.54002
[28] Shi, F. G., (l, m)- fuzzy matroids, Fuzzy Sets Syst., 160, 2387-2400, (2010) · Zbl 1229.05082
[29] Skowron, A., The rough set theory and evidence theory, Fundam. Inf., 13, 245-262, (1990) · Zbl 0752.94023
[30] Tsumoto, S.; Tanaka, H., AQ, rough sets and matroid theory, (Ziarko, W., Rough Sets, Fuzzy Sets and Knowledge Discovery, (1993), Spring-Verlag London), 290-297 · Zbl 0819.68048
[31] Tsumoto, S.; Tanaka, H., Algebraic specification of empirical inductive learning methods based on rough sets and matroid theory, LNCS, 985, 224-243, (1995)
[32] Wang, S. P.; Zhu, Q. X.; Zhu, W.; Min, F., Matroidal structure of rough sets and its characterization to attribute reduction, Knowl. Based Syst., 36, 155-161, (2012)
[33] Wang, S. P.; Zhu, Q. X.; Zhu, W.; Min, F., Rough set characterization for 2-circuit matroid, Fundam. Inf., 129, 377-393, (2014) · Zbl 1285.68186
[34] Welsh, D., Matroid theory, (1976), Academic press London · Zbl 0343.05002
[35] White, N., Matroid applications, (1992), Cambridge University Press Cambridge · Zbl 0742.00052
[36] Wu, W. Z.; Zhang, W. X., Constructive and axiomatic approaches of fuzzy approximation operators, Inf. Sci., 159, 233-254, (2004) · Zbl 1071.68095
[37] Yang, T.; Li, Q. G., Reduction about approximation spaces of covering generalized rough sets, Int. J. Approx. Reason., 51, 335-345, (2010) · Zbl 1205.68433
[38] Yao, Y. Y.; Lin, T. Y., Generalization of rough sets using modal logic, Intell. Autom. Soft Comput., 2, 103-120, (1996)
[39] Yao, Y. Y., Constructive and algebraic methods of the theory of rough sets, Inf. Sci., 109, 21-47, (1998) · Zbl 0934.03071
[40] Yao, Y. Y., Relational interpretational of neighborhood operators and rough set approximation operators, Inf. Sci., 111, 239-259, (1998) · Zbl 0949.68144
[41] Yao, Y. Y., A comparative study of fuzzy sets and rough sets, Infor. Sci., 109, 227-242, (1998) · Zbl 0932.03064
[42] Yao, Y. Y., On generalizing rough set theory, (Wang, G., Proceedings of the Ninth International Conference on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, (RSFDGrC 2003), LNAI, vol. 2639, (2003), Springer Heidelberg), 44-51 · Zbl 1026.68669
[43] Yao, Y. Y.; Yao, B. X., Covering based rough set approximations, Inf. Sci., 200, 91-107, (2012) · Zbl 1248.68496
[44] Zakowski, W., Approximations in the space (u, π), Demonstr. Math., 16, 761-769, (1983) · Zbl 0553.04002
[45] Zhu, W., Topological approaches to covering rough sets, Infor. Sci., 177, 1499-1508, (2007) · Zbl 1109.68121
[46] Zhu, W.; Wang, S. P., Rough matroids based on relations, Inf. Sci., 232, 241-252, (2013) · Zbl 1293.05036
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