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The characterizations of upper approximation operators based on special coverings. (English) Zbl 1377.54003

The paper under review is about some characterizations of upper approximation operators based on special coverings. Let us recall some basic definitions of this subject:
\(U\) is the universe of discourse and \(P(U)\) denotes the family of all subsets of \(U\). If \(C\) is a family of subsets of \(U\), none of the sets in \(C\) is empty, and \(\bigcup C=U\), then \(C\) is called a covering of \(U\).
Definition:
(i) A mapping \(n:U\rightarrow P(U)\) is called a neighborhood operator.
(ii) A neighborhood system of an object \(x\in U\), denoted by \(NS(x)\), is a non-empty family of neighborhoods of \(x\).
The set \(\{NS(x):x\in U\}\) is called a neighborhood system of \(U\), and it is denoted by \(NS(U)\).
\(NS(U)\) is said to be serial, if for any \(x\in U\) and \(n(x)\in NS(x),n(x)\) is a non-empty space.
\(NS(U)\) is said to be reflexive, if for any \(x\in U\) and \(n(x)\in NS(x)\), \(x\in n(x)\).
\(NS(U)\) is said to be symmetric, if for any \(x,y\in U\) and \(n(x)\in NS(x),\) \( n(y)\in NS(y),x\in n(y)\) then \(y\in n(x).\)
\(NS(U)\) is said to be transitive, if for any \(x,y,z\in U\) and \(n(y)\in NS(y) \) and \(n(z)\in NS(z),x\in n(y)\) and \(y\in n(z)\) then \(x\in n(z).\)
Definition: (Covering approximation space) If \(U\) is a universe and \(C\) is a covering of \(U\), then we call \(U\) together with the covering \(C\) a covering approximation space, denoted by \((U,C)\).
Definition: Let \(NS(U)\) be a neighborhood system of \(U.\) The lower and upper operators of \(X\) are defined as follows: \[ \underline{apr}_{NS}(X):=\left\{ x\in U:\exists n(x)\in NS(x)\text{, } n(x)\subseteq X\right\} ; \]
\[ \overline{apr}_{NS}(X):=\left\{ x\in U:\forall n(x)\in NS(x)\text{, } n(x)\cap X\neq \varnothing \right\} . \]
This study contains 6 sections and a good deal of information is given in the introduction about the subject. The main ideas of generalized rough sets and covering approximations are recalled in the second section. In Section 3, the properties of \(NS(U)\) are given with some examples. The authors study in Section 4 the characterization of \(NS(U)\) for \(\underline{apr}_{NS}\) being a closure operator, while they consider in Section 5 the properties of \(S\) and \(\overline{apr}_{S}\) and they obtain general topological characterizations of a special covering \(S\) for the covering-based upper approximation operator \(\overline{apr}_{S}\) to be a closure operator. Finally, Section 6 concludes the paper.

MSC:

54A05 Topological spaces and generalizations (closure spaces, etc.)
54A40 Fuzzy topology
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[1] Cattaneo G., Abstract approximation spaces for rough theory in: Rough Sets in Knowledge Discovery 1: Methodology and Applications. 1998,59-98.; Cattaneo, G., Abstract approximation spaces for rough theory in: Rough Sets in Knowledge Discovery 1, Methodology and Applications, 59-98 (1998) · Zbl 0927.68087
[2] Cattaneo G., D.Ciucci, AIgebraic structures for rough sets, in: LNCS. 2004,3135,208-252.; Cattaneo, G.; Ciucci, D., AIgebraic structures for rough sets, LNCS, 3135, 208-252 (2004) · Zbl 1109.68115
[3] Kondo M., On the structure of generalized rough sets, Information Sciences. 2005,176,589-600.; Kondo, M., On the structure of generalized rough sets, Information Sciences, 176, 589-600 (2005) · Zbl 1096.03065
[4] Oin K., Pei Z., On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems. 2005,151,601-613.; Oin, K.; Pei, Z., On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems, 151, 601-613 (2005) · Zbl 1070.54006
[5] Skowron A., Stepaniuk J., Tolerance approximation spaces, Fundamenta Informaticae.1996,27,245-253.; Skowron, A.; Stepaniuk, J., Tolerance approximation spaces, Fundamenta Informaticae, 27, 245-253 (1996) · Zbl 0868.68103
[6] Slowinski R., Vanderpooten D., A generalized definition of rough approximations based on similarity IEEE Transactions on Knowledge and DataEngineering. 2000,12,331-336; Slowinski, R.; Vanderpooten, D., A generalized definition of rough approximations based on similarity, IEEE Transactions on Knowledge and DataEngineering, 12, 331-336 (2000)
[7] Y.Yao, A comparative study of fuzzy sets and rough sets, Information Sciences. 1998,109,227-242.; Yao, Y., A comparative study of fuzzy sets and rough sets, Information Sciences, 109, 227-242 (1998) · Zbl 0932.03064
[8] Y.Yao, Constructive and algebraic methods of theory of rough sets, Information Sciences. 1998,109,21-47.; Yao, Y., Constructive and algebraic methods of theory of rough sets, Information Sciences, 109, 21-47 (1998) · Zbl 0934.03071
[9] Zadeh L.A., Fuzzy sets, Information and Control. 1965,8,338-353.; Zadeh, L. A., Fuzzy sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606
[10] Oin K., Gao y., Pei Z., On covering rough sets, Lecture Notes inAl. 2007,4481,34-41.; Oin, K.; Gao, Y.; Pei, Z., On covering rough sets, Lecture Notes inAI, 4481, 34-41 (2007)
[11] Z.Pawlak, Rough sets, International Journal of Computer and Informa-tion Sciences. 1982,11,341-356.; Pawlak, Z., Rough sets, International Journal of Computer and Information Sciences, 11, 341-356 (1982) · Zbl 0501.68053
[12] Pawlak Z., Rough Sets: Theoretical Aspects of Reasoning about Data, KIuwer Academic Publishers, Boston. 1991.; Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning about Data (1991) · Zbl 0758.68054
[13] Pawlak Z., A. Skowron, Rudiments of rough sets, Information Sciences. 2007,177,3-27.; Pawlak, Z.; Skowron, A., Rudiments of rough sets, Information Sciences, 177, 3-27 (2007) · Zbl 1142.68549
[14] Pawlak Z., A. Skowron, Rough sets: some extensions, Information Sciences. 2007,177,28-40.; Pawlak, Z.; Skowron, A., Rough sets: some extensions, Information Sciences, 177, 28-40 (2007) · Zbl 1142.68550
[15] Pawlak Z., A. Skowron, Rough sets and boolean reasoning, Information Sciences. 2007,177,41-73.; Pawlak, Z.; Skowron, A., Rough sets and boolean reasoning, Information Sciences, 177, 41-73 (2007) · Zbl 1142.68551
[16] Zadeh L., Fuzzy logic = computing with words, IEEE Transactions on Fuzzy Systems. 1996,4,103-111.; Zadeh, L., Fuzzy logic = computing with words, IEEE Transactions on Fuzzy Systems, 4, 103-111 (1996) · Zbl 0947.03038
[17] W.Zhu, F.Wang, Properties of the third type of covering-based rough sets, in: ICMLC07.2007,3746-3751.; Zhu, W.; Wang, F., Properties of the third type of covering-based rough sets, ICMLC07. 3751, 3746 (2007)
[18] W.Zhu, Topological approaches to covering rough sets, Information Sciences.2007, 177,1499-1508.; Zhu, W., Topological approaches to covering rough sets, Information Sciences, 177, 1499-1508 (2007) · Zbl 1109.68121
[19] W.Zhu, F.Y. Wang, On three types of covering rough sets, IEEE Trans-actions on Knowledge and Data Engineering. 2007,19,1131-1144.; Zhu, W.; Wang, F. Y., On three types of covering rough sets, IEEE Trans-actions on Knowledge and Data Engineering, 19, 1131-1144 (2007)
[20] W.Zhu, Generalized rough sets based on relations, Information Sciences. 2007,177,4997-5011.; Zhu, W., Generalized rough sets based on relations, Information Sciences, 177, 4997-5011 (2007) · Zbl 1129.68088
[21] W.Zhu, Relationship between generalized rough sets based on binary relation and covering, Information Sciences. 2009,179,210-225.; Zhu, W., Relationship between generalized rough sets based on binary relation and covering, Information Sciences, 179, 210-225 (2009) · Zbl 1163.68339
[22] W.Zhu, Relationship among basic concepts in covering-based rough sets, Information Sciences. 2009,179,2478-2486.; Zhu, W., Relationship among basic concepts in covering-based rough sets, Information Sciences, 179, 2478-2486 (2009) · Zbl 1178.68579
[23] W.Zhu, F.Y.Wang, The fourth type of covering-based rough sets, Information Sciences. 2012,1016,1-13.; Zhu, W.; Wang, F. Y., The fourth type of covering-based rough sets, Information Sciences, 1016, 1-13 (2012)
[24] N. Fan, G. Hu, H. Liu, Study of definable subsets in covering approximation space of rough sets. Proceedings of the 2011IEEE International Conference on Information Reuse and Integration. 2011,1,21-24.; Fan, N.; Hu, G.; Liu, H., Study of definable subsets in covering approximation space of rough sets, Proceedings of the 2011IEEE International Conference on Information Reuse and Integration, 1, 21-24 (2011)
[25] X.Ge, Z. Li, Definable subset in covering approximation spaces. International Journal of Computational and Mathematical Sciences. 2011,5,31-34.; Ge, X.; Li, Z., Definable subset in covering approximation spaces, International Journal of Computational and Mathematical Sciences, 5, 31-34 (2011)
[26] N.Fan, G.Hu, W.Zhang, Study on conditions of neighborhoods forming a partition, Fuzzy Systems and Knowledge Discovery (FSKD). 2012,256-259.; Fan, N.; Hu, G.; Zhang, W., Study on conditions of neighborhoods forming a partition, Fuzzy Systems and Knowledge Discovery (FSKD), 256-259 (2012)
[27] Zakowski W., Approximations in the Space(U, Π), Demonstratio Mathematica 1983,16,761-769.; Zakowski, W., Approximations in the Space(U, Π), Demonstratio Mathematica, 16, 761-769 (1983) · Zbl 0553.04002
[28] D.Chen, C.Wang, A new aooproach to arrtibute reduction of consistent and inconsistent covering decision systems with covering rough sets, Information Sciences.2007,176,3500-3518; Chen, D.; Wang, C., A new aooproach to arrtibute reduction of consistent and inconsistent covering decision systems with covering rough sets, Information Sciences, 176, 3500-3518 (2007) · Zbl 1122.68131
[29] X.Ge, An application of covering approximation spaces on network security Computer and Mathematics with Applications.2010,60,1191-1199.; Ge, X., An application of covering approximation spaces on network security, Computer and Mathematics with Applications, 60, 1191-1199 (2010) · Zbl 1201.68025
[30] X.Ge, Connectivity of covering approximation spaces and its applications onepidemiological issue, Applied Soft computing.2014,25,445-451.; Ge, X., Connectivity of covering approximation spaces and its applications onepidemiological issue, Applied Soft computing, 25, 445-451 (2014)
[31] X. Bian, P.Wang, Z.Yu, X. Bai, B.Chen, Characterizations of coverings for upper approximation operators being closure operators, Information Sciences 2015,314,41-54.; Bian, X.; Wang, P.; Yu, Z.; Bai, X.; Chen, B., Characterizations of coverings for upper approximation operators being closure operators, Information Sciences, 314, 41-54 (2015) · Zbl 1387.68216
[32] Y.Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences.1998,111,239-259.; Yao, Y., Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences, 111, 239-259 (1998) · Zbl 0949.68144
[33] W.Zhu, W.X.Zhang, Neighborhood operators systems and approximations, Information Sciences. 2002,144,201-217.; Zhu, W.; Zhang, W. X., Neighborhood operators systems and approximations, Information Sciences, 144, 201-217 (2002) · Zbl 1019.68109
[34] Y.L. Zhang, M.K. Luo, Relationships beween covering-based rough sets and relation-based rough sets, Information Sciences. 2013,225,55-71.; Zhang, Y. L.; Luo, M. K., Relationships beween covering-based rough sets and relation-based rough sets, Information Sciences, 225, 55-71 (2013) · Zbl 1293.03028
[35] Pomykala J.A., Approximation operations in approximation space, Bulletin of the Polish Academy of Science Mathematics. 1987,35,653-662.; Pomykala, J. A., Approximation operations in approximation space, 35, 653-662 (1987) · Zbl 0642.54002
[36] Bryniarski E., A calculus of rough sets of the first order, Bulletin of the Polish Academy of Science Mathematics. 1989,16,71-78.; Bryniarski, E., A calculus of rough sets of the first order, 16, 71-78 (1989) · Zbl 0756.04002
[37] Bonikowski Z., Bryniarski E., U.W.Skardowska, Extensions and intentions in the rough set theory, Information Sciences. 1998,107,149-167.; Bonikowski, Z.; Bryniarski, E.; Skardowska, U. W., Extensions and intentions in the rough set theory, Information Sciences, 107, 149-167 (1998) · Zbl 0934.03069
[38] Thomas G.B., Thomas’Calculus(10th Edition), Addison Wesley Publishing Cmpany.2003; Thomas, G. B., Thomas’Calculus (2003)
[39] T.Yang, O.Li, B.Zhou, Reduction about approximation spaces of covering generalized rough sets, International Journal of Approximate Reasoning.2010,51,335-345.; Yang, T.; Li, O.; Zhou, B., Reduction about approximation spaces of covering generalized rough sets, International Journal of Approximate Reasoning, 51, 335-345 (2010) · Zbl 1205.68433
[40] G. Liu, Y.Sai, A comparison of two types of rough sets induced by coverings, International Journal of Approximate Reasoning.2009,50,521-528.; 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
[41] G. Liu, Using one axiom to characterize rough set and fuzzy rough set approximations, Information Sciences. 2013,223,285-296.; Liu, G., Using one axiom to characterize rough set and fuzzy rough set approximations, Information Sciences, 223, 285-296 (2013) · Zbl 1293.03024
[42] G. Liu, The axiomatization of the rough set upper approximation operations, Fundamenta Informaticae. 2006,69,331-342.; Liu, G., The axiomatization of the rough set upper approximation operations, Fundamenta Informaticae, 69, 331-342 (2006) · Zbl 1096.68150
[43] G. Liu, Axiomatic systems for rough sets and fuzzy rough sets, International Journal of Approximate Reasoning. 2008,48,857-867.; Liu, G., Axiomatic systems for rough sets and fuzzy rough sets, International Journal of Approximate Reasoning, 48, 857-867 (2008) · Zbl 1189.03056
[44] G. Liu, W.Zhu, The algebraic structures of generalized rough set theory, Information Sciences. 2008,178,4105-4113.; Liu, G.; Zhu, W., The algebraic structures of generalized rough set theory, Information Sciences, 178, 4105-4113 (2008) · Zbl 1162.68667
[45] X.Ge, X. Bai, Z.Yun, Topological characterizaitons of covering for special covering-based upper spproximation operators, Information Sciences. 2012,204,70-81.; Ge, X.; Bai, X.; Yun, Z., Topological characterizaitons of covering for special covering-based upper spproximation operators, Information Sciences, 204, 70-81 (2012) · Zbl 1250.68258
[46] Y.Yao, B.Yao, Covering based rough set approximation, Information Sciences. 2012,200,91-107.; Yao, Y.; Yao, B., Covering based rough set approximation, Information Sciences, 200, 91-107 (2012) · Zbl 1248.68496
[47] Engelking R., General topology, Heldermann Verlag, Berlin,1989.; Engelking, R., General topology (1989) · Zbl 0684.54001
[48] Samanta P., Chakraborty M, K., Covering based approaches to rough sets and inplication lattices, in: RSFDGRC 2009, LANI, 2009,5908, 127-134.; Samanta, P.; Chakraborty, M, K., Covering based approaches to rough sets and inplication lattices, RSFDGRC 2009, LANI, 5908, 127-134 (2009)
[49] Y.Zhang, C. Li, M. Lin, Y.Lin, Relationships between generalized rough sets based on covering and reflexive neighborhood system, Information Sciences. 2015,319,56-67.; Zhang, Y.; Li, C.; Lin, M.; Lin, Y., Relationships between generalized rough sets based on covering and reflexive neighborhood system, Information Sciences, 319, 56-67 (2015) · Zbl 1390.68688
[50] T. Lin, Neighborhood systems-application to qualitative fuzzy and rough sets, in:P.P. Wang(ed.), Advanced in machine intelligence and soft computing IV, Department of EIectrical Engineering Durham North Carolina,1997,130-141.; Lin, T.; Wang, P. P., Advanced in machine intelligence and soft computing IV, 130-141 (1997)
[51] Sierpiński W., General topology, University of Toronto, Toronto,1956.; Sierpiński, W., General topology (1956) · JFM 60.0502.01
[52] T. Lin, Y.Yao, Mining soft rules using rough sets and neighborhoods, in:Proceeding of the symposium on dodeling analysis and simulation, comptatinal engineering in systems application, INASCS Multi Conference, Lille, France, 1996,9-12.; Lin, T.; Yao, Y., Proceeding of the symposium on dodeling analysis and simulation, comptatinal engineering in systems application, 9-12 (1996)
[53] G. Liu, Approximations in rough sets vs granular computing for coverings, International Journal of Cognitive Infromatics and Natural Intelligence. 2010,4,61-74.; Liu, G., Approximations in rough sets vs granular computing for coverings, International Journal of Cognitive Infromatics and Natural Intelligence, 4, 61-74 (2010)
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