Year 2020,
Volume: 49 Issue: 4, 1270 - 1294, 06.08.2020
Pankaj Kumar Singh
Surabhi Tiwari
References
- [1] F.G. Arenas, Alexandroff spaces, Acta Math. Univ. Comenian. 68, 17–25, 1999.
- [2] M. Banerjee and M.K. Chakraborty, Rough Consequence and Rough Algebra, in:
Rough Sets, Fuzzy Sets and Knowledge Discovery, Proceedings of the International
Workshop on Procedings of Rough Sets and Knowledge Discovery, (RSKD’93), Banff,
Alberta, Canada, 1993, W.P. Ziarko, (Ed.), Springer Verlag, London, 196–207, 1994.
- [3] M. Barr and C. Wells, Toposes, Triples and Theories, Springer-Verlag, 1985.
- [4] W. Bartol, J. Miro, K. Pioro, and F. Rossello, On the coverings by tolerance classes,
Inform. Sci. 166 (1-4), 193–211, 2004.
- [5] G. Birkhoff, Lattice theory, Amer. Math. Soc. Colloq. Publ. 25, 1973.
- [6] R. Biswas, Rough metric spaces, Bull. Pour. Les. Sous. Ens. Flous. Appl.(France) 68,
21–32, 1996.
- [7] J. Dai, S. Gao, and G. Zheng, Generalized rough set models determined by multiple
neighborhoods generated from a similarity relation, Soft Comput. 13, 2081–2094, 2013.
- [8] J.P. Doignon and J.C. Falmagne, Knowledge Spaces, Springer, Heidelberg, 1999.
- [9] A. Dvurecenskij and S. Pulmannova, New trends in quantum structures, Springer
Science and Business Media, 2013.
- [10] P. Eklund, M.A. Galán, and W. Gahler, Partially ordered monads for monadic topologies,
Rough Sets and Kleene Algebras, Elect. Notes Theoret. Comp. Sci. 225, 67–81,
2009.
- [11] B. Ganter and R. Wille, Formal Concept Analysis: Mathematical Foundations,
Springer Science and Business Media, 2012.
- [12] F. Geerts and B. Kuijpers, Topological formulation of termination properties of iterates
of functions, Inform. Process. Lett. 89 (1), 31–35, 2004.
- [13] M.A. Hajri, K. Belaid, and L.J. Belaid, Scattered spaces, compactification and an
application to image classification problem, Tatra Mt. Math. Publ. 66 (1), 1–12, 2016.
- [14] P.C. Hammer, Extended topology: Structure of isotonic functions, J. Reine Angew.
Math. 213, 174–186, 1964.
- [15] T. Herawan, Roughness of sets involving dependency of attributes in information
systems, Int J. Softw. Eng. Appl. 9 (7), 111–126, 2015.
- [16] J.R. Isbell, Uniform Space, in: Math. Surveys Monogr No. 12, Amer. Math. Soc.,
1964.
- [17] J. Järvinen, Approximations and rough sets based on tolerances, in: Lecture Notes in
Comput. Sci. 2005, 182–189, Springer-Verlag, Heidelberg, 2001.
- [18] J. Järvinen, On the structure of rough approximations, Fund. Inform. 53, 135-153,
2002.
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49–58, Springer-Verlag, Berlin, Heidelberg, 2004.
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4374, 400–498, 2007.,
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Galois connections and fixed points, Fund. Inform. 76, 129–145, 2007.
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Demonstr. Math. XLV (3), 2012.
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and Near, Demonstr. Math. XLVI (1), 2013.
- [25] M. Kondo, On the structure of generalized rough sets, Inform. Sci. 176, 589–600,
2006.
- [26] M. Kondo and W.A. Dudek, Topological structures of rough sets induced by equivalence
relations, J. Advance Comput. Intell. Intell. Inform. 10 (5), 621–624, 2006.
- [27] E.F. Lashin, A.M. Kozae, A.A. Abo Khadra, and T. Medhat, Rough set theory for
topological spaces, Internat. J. Approx. Reason. 40, 35–43, 2005.
- [28] T.J. Li and Y.L. Jing, Rough set model on granular structures and rule induction,
Intern. J. Database Theory Appl. 4 (1), 7–18, 2011.
- [29] Z. Li, T. Xie, and Q. Li, Topological structure of generalized rough sets, Comput.
Math. Appl. 63, 1066–1071, 2012.
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sets and topology, Methodologies for Intelligent Systems. Berlin, Germany: Springer-
Verlag, 1994.
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Conferences on Machine Learning and Cybernetics, 2353–2355, Shanghai,
2004.
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- [33] B.P. Mathew and S.J. John, On rough topological spaces, Intern. J. Math. Archive 3
(9), 3413–3421, 2012.
- [34] T. Medhat, Topological spaces and covering rough sets, Advances Inform. Sci. Service
Sci. 3 (8), 283–289, 2011.
- [35] K. Monks, A category of topological spaces encoding a cyclic set theoretic dynamics,
in: Proceedings of the International Conference on the Collatz Problem and Related
Topics, 2000.
- [36] S.A. Naimpally and B.D. Warrack, Proximity Spaces, Cambridge Tract in Math.
Math. Phys, Cambridge University Press 59, 1970.
- [37] P. Pagliani and M. Chakraborty, A Geometry of Approximation, Springer, Dordrecht,
2008.
- [38] Z. Pawlak, Information systems-theoretical foundations, Inf. Syst 6 (3), 205–218, 1981.
- [39] Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci. 11 (5), 341–356, 1982.
- [40] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic
Publishers Boston, 1991.
- [41] Z. Pawlak and W. Marek, Rough sets and information systems, ICS. PAS. Reports
441, 481–485, 1981.
- [42] Z. Pawlak and A. Skowron, Rudiment of rough sets, Inform. Sci. 177, 3–27, 2007.
- [43] J.F. Peters, Near sets. Special theory about nearness of objects, Fund. Inform. 75(3-4),
407–433, 2007.
- [44] J.F. Peters, A. Skowron, and J. Stepaniuk, Nearness of objects: Extension of approximation
space model, Fund. Inform. 79 (3-4), 497–512, 2007.
- [45] J.F. Peters, A. Skowron, and J. Stepaniuk , Nearness of visual objects. Application of
rough sets in proximity spaces, Fund. Inform. 128, 159–176, 2013.
- [46] L. Polkowski, Rough Sets: Mathematical Foundations, Heidelberg New York, Physica-
Verl., 2002.
- [47] L. Polkowski and A. Skowron, Rough sets in Knowledge Discovery 1: Methodology and
Applications, Stud. Fuzziness Soft Comput. 18, Physica-Verlag, Heidelberg, Germany,
1998.
- [48] L. Polkowski and A. Skowron, Rough Sets in Knowledge Discovery 2: Applications,
Case Studies and Software Systems, Stud. Fuzziness Soft Comput. 19, Physica-Verlag,
Heidelberg, Germany , 1998.
- [49] A.N. Prior, Tense-logic and the continuity of time, Studia Logica 13 (1), 133–148,
1962.
- [50] K. Qin and Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and
Systems 151, 601–613, 2005.
- [51] K. Qin, J. Yang, and Z. Pei, Generalized rough sets based on reflexive and transitive
relations, Inform. Sci. 178, 4138–4141, 2008.
- [52] S. Ramanna, A.H. Meghdadi, and J.F. Peters, Nature-inspired framework for measuring
visual image resemblance: A near rough set approach, Theoret. Comput. Sci.
412, 5926–5938, 2011.
- [53] H. Rasiowa, An Algebraic Approach to Non-Classical Logics, Stud. Logic Found. Math.
78, North-Holland Publishing Company, Amsterdam, London - PWN, Warsaw, 1974.
- [54] M. Restrepo, C. Cornelis, and J. Gomez, Characterization of neighborhood operators
for covering based rough sets, using duality and adjoining, in: Fourth International
Workshop Proceedings Eureka-2013, 98–104, 2013.
- [55] A.S. Salama, Some Topological properties of rough sets with tools for data mining,
Int. J comp. sci. Issues 8 (2), 588–595, 2011.
- [56] A. Skowron, On the topology in information systems, Bull. Pol. Acad. Sci. Math 36,
477–480, 1988.
- [57] M.H. Stone, Topological representation of distributive lattices and Brouwerian logics,
J. Symb. Log. 3 (2), 90–91, 1938.
- [58] A. Tarski, Grundzüge der Syntemenkalküls I, Fund. Math. 25 (1), 503–526, 1935.
- [59] M.L. Thivagar, C. Richard, and N.R. Paul, Mathematical innovations of a modern
topology in medical events, Int. J. inform. Sci. 2 (4), 33–36, 2012.
- [60] N.D. Thuan, Covering rough sets from a topological point of view, Int. J. Comp.
Theory Eng. 1 (5), 606–609, 2009.
- [61] S. Tiwari, Ultrafilter completeness in ε-approach nearness Spaces, Math. Comput. Sci.
7, 107–111, 2013.
- [62] S. Tiwari and P.K. Singh, An approach of proximity in rough set theory, Fund. Inform.
166 (3), 251–271, 2019.
- [63] S. Tiwari and P.K. Singh, Čech rough proximity spaces, Mat. Vesnik, accepted, 2019.
- [64] M. Vlach, Algebraic and topological aspects of rough set theory, in: IEEE Fourth
International Workshop on Computational Intelligence and Application 2008 (1),
23–30, 2008.
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Computer Sci. 6 (3), 147–150, 2010.
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Boston MA, 411–425, 1997.
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topological approximations, in: Transaction on Rough Sets III, LNCS 3400, 230–243,
2005.
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on rough Sets and Current Trend in Computing 2010, 237–246, 2010.
- [70] M.Wolski, Granular computing: Topological and categorical aspects of near and rough
set approaches to granulation of knowledge, in: Transaction on Rough Sets XVI, LNCS
7736, 34–52, 2013.
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fuzzy topological spaces, International Conference on Fuzzy Systems and Knowledge
Discovery, LNCS 3613, 167–174, 2005.
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infinite universes of discourse, in: Transactions on Rough Sets XIII, LNCS 6499,
175–206, 2011.
- [73] Q.E. Wu, T. Wang, Y.X. Huang, and J.S. Li, Topology theory on rough sets, IEEE
Transactions on Systems, man and Cybernetics, Part B (Cybernetics) 38 (1), 68–77,
2008.
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covering rough sets, IEEE International Conference on Granular Computing, 601–604,
2012.
- [75] Y.Y. Yao, Constructive and algebraic methods of the theory of rough sets, Inform. Sci.
109 (1-4), 21–47, 1998.
- [76] Y.Y. Yao, Relational interpretations of neighborhood operator and rough set approximation
operators, Inform. Sci. 111 (1-4), 239–259, 1998.
- [77] Y.Y. Yao and T.Y. Lin, Generalization of rough sets using modal logic, Intell. Autom.
Soft Co. 2 (2), 103–120, 1996.
- [78] Y.Y. Yao and B. Yao, Covering based rough sets approximations, Inform. Sci. 200,
91–107, 2012.
- [79] Y.Y. Yao and N. Zhong, Potential applications of granular computing in knowledge
discovery and data mining, in: Proceedings of World Multiconference on Systemics,
Cybernetics and Informatics 5, 573–580, 1999.
- [80] W. Zakowski, Approximations in the space (U, Π), Demonstr. Math. 16 (3), 761–769,
1983.
- [81] Y.L. Zhang, J. Li, and C. Li, Topological structure of relational-based generalized
rough sets, Fund. Inform. 147 (4), 477–491, 2016.
- [82] N. Zhong, J.Z. Dong, and S. Ohsuga, Using rough sets with heuristics to feature
selection, J. Intell. Inf. Syst. 16 (3), 199–214, 2001.
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T. Knowl. Data Eng. 15 (4), 952–960, 2003.
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Arizona, Tucson, AZ, USA, May 2002.
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1508, 2007.
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sets, Inform. Sci. 152, 217–230, 2003.
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Lecture Notes in Comput. Sci 3975, 566–571, 2006.
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International Conference on Fuzzy Systems and Knowledge Discovery 1, 289–293,
2007.
Topological structures in rough set theory: A survey
Year 2020,
Volume: 49 Issue: 4, 1270 - 1294, 06.08.2020
Pankaj Kumar Singh
Surabhi Tiwari
Abstract
Rough set theory provides a mathematical tool to study vague, imprecise, inconsistent and uncertain knowledge. The topological notions are closely related to the notions and results of rough set theory, so the conjoint study of rough set theory and topology becomes essential. Researchers have widely discussed the topological aspects and their applications in rough set theory. This study highlights the interdependencies of topology and classical rough set theory and the significant work done in this area during the last twenty years.
References
- [1] F.G. Arenas, Alexandroff spaces, Acta Math. Univ. Comenian. 68, 17–25, 1999.
- [2] M. Banerjee and M.K. Chakraborty, Rough Consequence and Rough Algebra, in:
Rough Sets, Fuzzy Sets and Knowledge Discovery, Proceedings of the International
Workshop on Procedings of Rough Sets and Knowledge Discovery, (RSKD’93), Banff,
Alberta, Canada, 1993, W.P. Ziarko, (Ed.), Springer Verlag, London, 196–207, 1994.
- [3] M. Barr and C. Wells, Toposes, Triples and Theories, Springer-Verlag, 1985.
- [4] W. Bartol, J. Miro, K. Pioro, and F. Rossello, On the coverings by tolerance classes,
Inform. Sci. 166 (1-4), 193–211, 2004.
- [5] G. Birkhoff, Lattice theory, Amer. Math. Soc. Colloq. Publ. 25, 1973.
- [6] R. Biswas, Rough metric spaces, Bull. Pour. Les. Sous. Ens. Flous. Appl.(France) 68,
21–32, 1996.
- [7] J. Dai, S. Gao, and G. Zheng, Generalized rough set models determined by multiple
neighborhoods generated from a similarity relation, Soft Comput. 13, 2081–2094, 2013.
- [8] J.P. Doignon and J.C. Falmagne, Knowledge Spaces, Springer, Heidelberg, 1999.
- [9] A. Dvurecenskij and S. Pulmannova, New trends in quantum structures, Springer
Science and Business Media, 2013.
- [10] P. Eklund, M.A. Galán, and W. Gahler, Partially ordered monads for monadic topologies,
Rough Sets and Kleene Algebras, Elect. Notes Theoret. Comp. Sci. 225, 67–81,
2009.
- [11] B. Ganter and R. Wille, Formal Concept Analysis: Mathematical Foundations,
Springer Science and Business Media, 2012.
- [12] F. Geerts and B. Kuijpers, Topological formulation of termination properties of iterates
of functions, Inform. Process. Lett. 89 (1), 31–35, 2004.
- [13] M.A. Hajri, K. Belaid, and L.J. Belaid, Scattered spaces, compactification and an
application to image classification problem, Tatra Mt. Math. Publ. 66 (1), 1–12, 2016.
- [14] P.C. Hammer, Extended topology: Structure of isotonic functions, J. Reine Angew.
Math. 213, 174–186, 1964.
- [15] T. Herawan, Roughness of sets involving dependency of attributes in information
systems, Int J. Softw. Eng. Appl. 9 (7), 111–126, 2015.
- [16] J.R. Isbell, Uniform Space, in: Math. Surveys Monogr No. 12, Amer. Math. Soc.,
1964.
- [17] J. Järvinen, Approximations and rough sets based on tolerances, in: Lecture Notes in
Comput. Sci. 2005, 182–189, Springer-Verlag, Heidelberg, 2001.
- [18] J. Järvinen, On the structure of rough approximations, Fund. Inform. 53, 135-153,
2002.
- [19] J. Järvinen, The ordered set of rough sets, in: Lecture Notes in Comput. Sci. 3066,
49–58, Springer-Verlag, Berlin, Heidelberg, 2004.
- [20] J. Järvinen, Lattice Theory for Rough Sets, in: Transactions on Rough Sets VI, LNCS
4374, 400–498, 2007.,
- [21] J. Järvinen, M. Kono, and J. Kortelainen, Modal-like operators in boolean algebras,
Galois connections and fixed points, Fund. Inform. 76, 129–145, 2007.
- [22] J.L. Kelley, General Topology, Van Nostrand, 1955.
- [23] M. Khare and S. Tiwari, L-approach merotopies and their categorical perspective,
Demonstr. Math. XLV (3), 2012.
- [24] M. Khare and S. Tiwari, Completion in a common supercategory of Met, UAP, wsAP
and Near, Demonstr. Math. XLVI (1), 2013.
- [25] M. Kondo, On the structure of generalized rough sets, Inform. Sci. 176, 589–600,
2006.
- [26] M. Kondo and W.A. Dudek, Topological structures of rough sets induced by equivalence
relations, J. Advance Comput. Intell. Intell. Inform. 10 (5), 621–624, 2006.
- [27] E.F. Lashin, A.M. Kozae, A.A. Abo Khadra, and T. Medhat, Rough set theory for
topological spaces, Internat. J. Approx. Reason. 40, 35–43, 2005.
- [28] T.J. Li and Y.L. Jing, Rough set model on granular structures and rule induction,
Intern. J. Database Theory Appl. 4 (1), 7–18, 2011.
- [29] Z. Li, T. Xie, and Q. Li, Topological structure of generalized rough sets, Comput.
Math. Appl. 63, 1066–1071, 2012.
- [30] T.Y. Lin, Q. Liu, and Y.Y. Yao, Logic systems for approximate reasoning: via rough
sets and topology, Methodologies for Intelligent Systems. Berlin, Germany: Springer-
Verlag, 1994.
- [31] W.J. Liu, Topological space properties of rough sets, in: Proceedings of the Third International
Conferences on Machine Learning and Cybernetics, 2353–2355, Shanghai,
2004.
- [32] E.G. Manes, Algebraic Theories, Springer, 1976.
- [33] B.P. Mathew and S.J. John, On rough topological spaces, Intern. J. Math. Archive 3
(9), 3413–3421, 2012.
- [34] T. Medhat, Topological spaces and covering rough sets, Advances Inform. Sci. Service
Sci. 3 (8), 283–289, 2011.
- [35] K. Monks, A category of topological spaces encoding a cyclic set theoretic dynamics,
in: Proceedings of the International Conference on the Collatz Problem and Related
Topics, 2000.
- [36] S.A. Naimpally and B.D. Warrack, Proximity Spaces, Cambridge Tract in Math.
Math. Phys, Cambridge University Press 59, 1970.
- [37] P. Pagliani and M. Chakraborty, A Geometry of Approximation, Springer, Dordrecht,
2008.
- [38] Z. Pawlak, Information systems-theoretical foundations, Inf. Syst 6 (3), 205–218, 1981.
- [39] Z. Pawlak, Rough sets, Int. J. Comput. Inf. Sci. 11 (5), 341–356, 1982.
- [40] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic
Publishers Boston, 1991.
- [41] Z. Pawlak and W. Marek, Rough sets and information systems, ICS. PAS. Reports
441, 481–485, 1981.
- [42] Z. Pawlak and A. Skowron, Rudiment of rough sets, Inform. Sci. 177, 3–27, 2007.
- [43] J.F. Peters, Near sets. Special theory about nearness of objects, Fund. Inform. 75(3-4),
407–433, 2007.
- [44] J.F. Peters, A. Skowron, and J. Stepaniuk, Nearness of objects: Extension of approximation
space model, Fund. Inform. 79 (3-4), 497–512, 2007.
- [45] J.F. Peters, A. Skowron, and J. Stepaniuk , Nearness of visual objects. Application of
rough sets in proximity spaces, Fund. Inform. 128, 159–176, 2013.
- [46] L. Polkowski, Rough Sets: Mathematical Foundations, Heidelberg New York, Physica-
Verl., 2002.
- [47] L. Polkowski and A. Skowron, Rough sets in Knowledge Discovery 1: Methodology and
Applications, Stud. Fuzziness Soft Comput. 18, Physica-Verlag, Heidelberg, Germany,
1998.
- [48] L. Polkowski and A. Skowron, Rough Sets in Knowledge Discovery 2: Applications,
Case Studies and Software Systems, Stud. Fuzziness Soft Comput. 19, Physica-Verlag,
Heidelberg, Germany , 1998.
- [49] A.N. Prior, Tense-logic and the continuity of time, Studia Logica 13 (1), 133–148,
1962.
- [50] K. Qin and Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and
Systems 151, 601–613, 2005.
- [51] K. Qin, J. Yang, and Z. Pei, Generalized rough sets based on reflexive and transitive
relations, Inform. Sci. 178, 4138–4141, 2008.
- [52] S. Ramanna, A.H. Meghdadi, and J.F. Peters, Nature-inspired framework for measuring
visual image resemblance: A near rough set approach, Theoret. Comput. Sci.
412, 5926–5938, 2011.
- [53] H. Rasiowa, An Algebraic Approach to Non-Classical Logics, Stud. Logic Found. Math.
78, North-Holland Publishing Company, Amsterdam, London - PWN, Warsaw, 1974.
- [54] M. Restrepo, C. Cornelis, and J. Gomez, Characterization of neighborhood operators
for covering based rough sets, using duality and adjoining, in: Fourth International
Workshop Proceedings Eureka-2013, 98–104, 2013.
- [55] A.S. Salama, Some Topological properties of rough sets with tools for data mining,
Int. J comp. sci. Issues 8 (2), 588–595, 2011.
- [56] A. Skowron, On the topology in information systems, Bull. Pol. Acad. Sci. Math 36,
477–480, 1988.
- [57] M.H. Stone, Topological representation of distributive lattices and Brouwerian logics,
J. Symb. Log. 3 (2), 90–91, 1938.
- [58] A. Tarski, Grundzüge der Syntemenkalküls I, Fund. Math. 25 (1), 503–526, 1935.
- [59] M.L. Thivagar, C. Richard, and N.R. Paul, Mathematical innovations of a modern
topology in medical events, Int. J. inform. Sci. 2 (4), 33–36, 2012.
- [60] N.D. Thuan, Covering rough sets from a topological point of view, Int. J. Comp.
Theory Eng. 1 (5), 606–609, 2009.
- [61] S. Tiwari, Ultrafilter completeness in ε-approach nearness Spaces, Math. Comput. Sci.
7, 107–111, 2013.
- [62] S. Tiwari and P.K. Singh, An approach of proximity in rough set theory, Fund. Inform.
166 (3), 251–271, 2019.
- [63] S. Tiwari and P.K. Singh, Čech rough proximity spaces, Mat. Vesnik, accepted, 2019.
- [64] M. Vlach, Algebraic and topological aspects of rough set theory, in: IEEE Fourth
International Workshop on Computational Intelligence and Application 2008 (1),
23–30, 2008.
- [65] S. Wang, P. Zhu, and W. Zhu, Structures of covering-based rough sets, Int. J. Math.
Computer Sci. 6 (3), 147–150, 2010.
- [66] A. Wasilewska, Topological Rough Algebras, Rough sets and data mining. Springer,
Boston MA, 411–425, 1997.
- [67] A. Wiweger, On topological rough sets, Bull. Pol. Acad. Sci. Math. 37, 51–62, 1988.
- [68] M. Wolski, Formal concept analysis and rough set theory from the perspective of finite
topological approximations, in: Transaction on Rough Sets III, LNCS 3400, 230–243,
2005.
- [69] M. Wolski, Rough sets in terms of Discrete Dynamical system, International Conference
on rough Sets and Current Trend in Computing 2010, 237–246, 2010.
- [70] M.Wolski, Granular computing: Topological and categorical aspects of near and rough
set approaches to granulation of knowledge, in: Transaction on Rough Sets XVI, LNCS
7736, 34–52, 2013.
- [71] W.Z. Wu, Study on relationship between fuzzy rough approximation operators and
fuzzy topological spaces, International Conference on Fuzzy Systems and Knowledge
Discovery, LNCS 3613, 167–174, 2005.
- [72] W.Z. Wu and J.S. Mi, Some mathematical structures of generalized rough sets in
infinite universes of discourse, in: Transactions on Rough Sets XIII, LNCS 6499,
175–206, 2011.
- [73] Q.E. Wu, T. Wang, Y.X. Huang, and J.S. Li, Topology theory on rough sets, IEEE
Transactions on Systems, man and Cybernetics, Part B (Cybernetics) 38 (1), 68–77,
2008.
- [74] T. Yang, Q. Li, and B. Zhou, On the topological structures of granular reducts with
covering rough sets, IEEE International Conference on Granular Computing, 601–604,
2012.
- [75] Y.Y. Yao, Constructive and algebraic methods of the theory of rough sets, Inform. Sci.
109 (1-4), 21–47, 1998.
- [76] Y.Y. Yao, Relational interpretations of neighborhood operator and rough set approximation
operators, Inform. Sci. 111 (1-4), 239–259, 1998.
- [77] Y.Y. Yao and T.Y. Lin, Generalization of rough sets using modal logic, Intell. Autom.
Soft Co. 2 (2), 103–120, 1996.
- [78] Y.Y. Yao and B. Yao, Covering based rough sets approximations, Inform. Sci. 200,
91–107, 2012.
- [79] Y.Y. Yao and N. Zhong, Potential applications of granular computing in knowledge
discovery and data mining, in: Proceedings of World Multiconference on Systemics,
Cybernetics and Informatics 5, 573–580, 1999.
- [80] W. Zakowski, Approximations in the space (U, Π), Demonstr. Math. 16 (3), 761–769,
1983.
- [81] Y.L. Zhang, J. Li, and C. Li, Topological structure of relational-based generalized
rough sets, Fund. Inform. 147 (4), 477–491, 2016.
- [82] N. Zhong, J.Z. Dong, and S. Ohsuga, Using rough sets with heuristics to feature
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