Approximation operators by using finite family of reflexive relations

  • Authors

    • Hassan Abu-Donia
    • Amgad Salama
    2015-05-17
    https://doi.org/10.14419/ijamr.v4i2.4517
  • Rough set, Lower approximations, Upper approximations, Right neighborhood, Reflexive relation, Accuracy measure

  • In this paper, we generalize the two types of Yao’s lower and upper approximations, using finite number of reflexive relations. Moreover, we give a comparison between these types and study some properties.

  • References

    1. [1] H. M. Abu-Donia, â€Multi Knowledge based rough approximations and applicationsâ€, Knowledge-Based systems, Vol.26, No.X, (2012), pp.20-29.

      [2] H.M. Abu-Donia and A.S. Salama, â€Generalization of Pawlak’s rough approximation spaces by using δβ-open setsâ€, International Journal of Approximate Reasoning, Vol.53, (2012), pp.1094-1105.

      [3] H. M. Abu-Donia, â€New Rough Set Approximation Spacesâ€, Abstract and Applied Analysis, Vol.2013, (2013), pp.1-7.

      [4] H.M. Abu-Donia, â€Comparison between different kinds of approximations by using a family of binary relationsâ€, Knowledge-Based Systems, Vol.21, (2008), pp.911-919.

      [5] M. Banerjee and M.K. Chakraborty, â€Rough consequence and rough algebra. In: Rough Sets, Fuzzy Sets and Knowledge Discovery â€, Proc. Int. Workshop on Rough Sets and Knowledge Discovery (RSKD ’93), Banff, Canada 1993, Ed. Ziarko, W.P. (London: Springer-Verlag), (1994), pp.196-207.

      [6] G. Cattaneo, D. Ciucci, â€Algebraic structures for rough setsâ€, LNCS, Vol.3135, (2004), pp.208-252.

      [7] G. Cattaneo, â€Abstract approximation spaces for rough theoriesâ€, in: L. Polkowski, A. Skowron (Eds.), Physica-Verlag, Heidelberg, , (Chapter 4) Vol.1, No.X, (1998), pp.59-98.

      [8] G. Cattaneo, â€Generalized rough sets (preclusivity fuzzy intuitionstic (BZ)lattice)â€, studia logica, Vol.53, (1997), pp.47-77.

      [9] M. Chuchro, â€On rough sets in topological Boolean algebras, in: W. Ziarko (Ed.)â€, Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer-Verlag, Berlin, (1994), pp.157-160.

      [10] S. Comer, â€An algebraic approach to the approximation of informationâ€, Fundamenta Informaticae, Vol.14, (1991), pp.492-502.

      [11] N.E. Tayar, R.S. Tsai, P.A. Carrupt, B. Testa, â€Octan-1-ol-water partition coefficients of zwitterionic _-amino acid, determination by centrifugal partition chromatography and factorization into steric/hydrophobic and polar componentsâ€, J. Chem. Soc., Perkin. Trans., Vol.2, (1992), pp.79-84.

      [12] P. Eklund, M.A. Galan, Werner Gahler, â€Partially Ordered Monads for Monadic Topologies, Rough Sets and Kleene Algebrasâ€, Electronic Notes in Theoretical Computer Science, Vol.225, No.2, (2009), pp.67-81.

      [13] T. P. Hong, Y. L. Liou and S. L. Wang, â€Fuzzy rough sets with hierarchical quantitative attributesâ€, Expert Systems with Applications, Vol.36, No.3, (2009), pp.6790-6799.

      [14] M. A. Khan, and M. Banerjee, â€Formal reasoning with rough sets in multiple-source approximation systemsâ€, Int. J. Approximate Reasoning, Vol.49, No.2, (2008), pp.466-477.

      [15] M. Kondo, â€On the Structure of Generalized Rough Setsâ€, Information Sciences, Vol.176, No.5, (2005), pp.589-600.

      [16] J. Kortelainen, â€On Relationship between modified sets, topological space and rough setsâ€, On Relationship between modified sets, Vol.61, (1994), pp.91-95.

      [17] T.Y. Lin, Q. Liu, â€Rough approximate operators: axiomatic rough set theoryâ€, in: W. Ziarko (Ed.), Rough Sets, Fuzzy Sets and Knowledge Discovery, Springer, Berlin, (1994), pp.256-260.

      [18] G. Liu and Y. Sai, â€A comparison of two types of rough sets induced by coveringsâ€, International Journal of Approximate Reasoning, Vol.50, (2009), pp.521-528.

      [19] J.-S. Mi, W.-X. Zhang, â€An axiomatic characterization of a fuzzy generalization of rough setsâ€, Information Sciences, Vol.160, No.(1-4), (2004), pp.235-249.

      [20] J.N. Mordeson, â€Rough set theory applied (fuzzy) ideal to theoryâ€, Fuzzy Sets and Systems, Vol.121, (2001), pp.315-324.

      [21] E. Orlowska, â€Semantics analysis of inductive reasoningâ€, Theoretical Computer Science, Vol.43, (1986), pp.81-89.

      [22] Z. Pawlak, A. Skowron, â€Rough sets: Some extensionsâ€, Information Sciences, Vol.177, (2007), pp.28-40.

      [23] Z. Pawlak, A. Skowron, â€Rudiments of rough setsâ€, Journal of Information Sciences, Vol.177, (2007), pp.3-27.

      [24] Z. Pawlak, A. Skowron, â€rough sets and Boolean reasoningâ€, Journal of Information Sciences, Vol.177, (2007), pp.41-73.

      [25] Z. Pawlak, â€Rough Sets: Theoretical Aspects of Reasoning about Data, System Theory, Knowledge Engineering and Problem Solvingâ€, Kluwer Academic Publishers, Dordrecht, The Netherlands, Vol.9, (1991).

      [26] Z. Pawlak, â€Rough Probabilityâ€, Bull Polish Acad Sci, Vol.32, (1984), pp.607-612.

      [27] Z. Pawlak, â€Rough setsâ€, International Journal of Computer and Information Sciences, Vol.11, (1982), pp.341-356.

      [28] Z. Pawlak, â€Information systems, Theoretical Foundationsâ€, Information Systems, Vol.6, (1981), pp.205-218.

      [29] J.A. Pomykala, â€Approximation operations in approximation spaceâ€, Bulletin of the Polish Academy of Sciences: Mathematics, Vol.35, (1987), pp.653-662.

      [30] K. Qin, Z. Pei, â€On the topological properties of fuzzy rough setsâ€, Fuzzy Sets and Systems, Vol.151, No.3, (2005), pp.601-613.

      [31] Y. Qian, J. Liang, Y. Yao, C. Dang â€MGRS: A multi-granulation rough setâ€, Information Sciences, Vol.180, (2010), pp.949-970.

      [32] E.A. Rady, A.M. Kozae, M.M.E. Abd El-Monsef, â€Generalized rough setsâ€, Chaos, Solitons and Fractals, Vol.21, (2004), pp.49-53.

      [33] C. Rauszer, â€Rough logic for multi-agent systemsâ€, In: Logic at Work 1992. LNCS (LNAI) Springer, Heidelberg, Vol.808, (1994), pp.161-181.

      [34] R. Slowinski, D. Vanderpooten, â€A generalized definition of rough approximations based on similarityâ€, EEE Transactions on Knowledge and Data Engineering, Vol.12, No.2, (2000), pp.331-336.

      [35] R. Slowinski, D. Vanderpooten, â€Similarity relation as a basis for rough approximationsâ€, in: P.P. Wang (Ed.), Advances in Machine Intelligence and Soft-Computing, Department of Electrical Engineering, Duke University, Durham, NC, USA,, (1997), pp.17-33.

      [36] A. Skowron, â€Rough sets and vague conceptsâ€, Fundamenta Informaticae, Vol.64, No.1-4, (2005), pp.417-431.

      [37] J. Stepaniuk and A. Skowron, â€Tolerance approximation spaces, Fundamenta Informaticaeâ€, journal, Vol.27, (1996), pp.245-253.

      [38] H. Thiele, â€On axiomatic characterisations of crisp approximation operatorsâ€, Information Sciences, Vol.129, (2000), pp.221-226.

      [39] Wei-Hua Xu and Wen-Xiu Zhang, â€Measuring roughness of generalized rough sets induced by a coveringâ€, Fuzzy Sets and Systems, Vol.158, (2007), pp.2443 - 2455.

      [40] W.-Z. Wu, Y. Leung, J.-S. Mi, â€On characterizations of (I,T)-fuzzy rough approximation operators â€, Fuzzy Sets and Systems, Vol.154, No.1, (2005), pp.76-102.

      [41] Y. Yao, â€On generalizing Pawlak approximation operatorsâ€, in: LNAI, Vol.1424, (1998), pp.298-307.

      [42] Y. Yao, â€Relational interpretations of neighborhood operators and rough set approximation operatorsâ€, Information Sciences, Vol.101, (1998), pp. 239-259.

      [43] Y.Y. Yao, â€Constructive and algebraic methods of the theory of rough setsâ€, Journal of Information Sciences, Vol.109, (1998), pp.21-47.

      [44] Y. Y. Yao, â€Generalized rough set modelsâ€, in: Rough Sets in Knowledge Discovery, Polkowski, L. and Skowron, A. (Eds.), Physica-Verlag, Heidelberg, V (1998), pp.286-318.

      [45] Y.Y. Yao, â€Two views of the theory of rough sets in finite universesâ€, International Journal of Approximate Reasoning, Vol.15, (1996), pp.291-317.

      [46] Y.Y. Yao, T.Y. Lin, â€Generalization of rough sets using modal logic, Intelligent Automation and Soft Computingâ€, an International Journal, Vol.2, (1996), pp.103-120.

      [47] W. Zakowski, â€On a concept of rough setsâ€, Demonstratio Mathematica, Vol.XV, (1982), pp.1129-1133.

      [48] H. P. Zhang, Y. Ouyang, Zhudeng Wang, â€Note on Generalized rough sets based on reflexive and transitive relationsâ€, Information Sciences, Vol.179, (2009), pp.471-473.

      [49] W. Zhu, â€Relationship among basic concepts in covering-based rough setsâ€, Information Sciences, Vol.179, (2009), pp.2478-2486.

      [50] W. Zhu, â€Relationship between generalized rough sets based on binary relation and coveringâ€, Information Sciences, Vol.179 No. (3) 16 , (2009), pp.210-225.

      [51] W. Zhu, â€Topological approaches to covering rough setsâ€, Information Sciences, Vol.177, No.6, (2007), pp.1499-1508.

      [52] W. Zhu, F.-Y. Wang, â€On three types of covering rough setsâ€, IEEE Transactions on Knowledge and Data Engineering, Vol.19, No.8, (2007), pp.1131-1144.

      [53] W. Zhu, F.-Y. Wang, â€Relationships among three types of covering rough setsâ€, in: IEEE GrC, (2006), pp.43-48.

      [54] W. Zhu, F.-Y. Wang, â€Axiomatic systems of generalized rough setsâ€, in: RSKT 2006, LNAI, Vol.4062, (2006), pp.216-

      221.

      [55] W. Zhu, F.-Y. Wang, â€A new type of covering rough setsâ€, in: IEEE IS 2006, Vol.4-6, (2006), pp.444-449.

      [56] W. Zhu, F.-Y. Wang, â€Binary relation based rough setsâ€, in: IEEE FSKD 2006, LNAI, Vol.4223, (2006), pp.276-285.

      [57] W. Ziarko, (ed.), â€Rough sets, fussy sets and knowledge discovery (RSKD’93)â€, Workshops in Computing, Springer- Verlag and British Coputer Society, London, Berlin,(1994).

      [58] W. Ziarko, â€Variable precision rough set modelâ€, Journal of Computer and System Sciences, Vol.46, (1993), pp.39-59.

  • Downloads

  • How to Cite

    Abu-Donia, H., & Salama, A. (2015). Approximation operators by using finite family of reflexive relations. International Journal of Applied Mathematical Research, 4(2), 376-392. https://doi.org/10.14419/ijamr.v4i2.4517