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Mathematical Problems in Engineering
Volume 2013, Article ID 486321, 11 pages
http://dx.doi.org/10.1155/2013/486321
Research Article

Soft Rough Approximation Operators on a Complete Atomic Boolean Lattice

Mathematics Department, Faculty of Science, Zagazig University, Egypt

Received 23 May 2013; Revised 3 August 2013; Accepted 4 August 2013

Academic Editor: Sotiris Ntouyas

Copyright © 2013 Heba I. Mustafa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning About Data, Kluwer Academic Publishers, Boston, Mass, USA, 1991.
  2. D. Molodtsov, “Soft set theory-first results,” Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. P. K. Maji, R. Biswas, and A. R. Roy, “Fuzzy soft sets,” Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589–602, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets in a decision making problem,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1077–1083, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. P. K. Maji, R. Biswas, and A. R. Roy, “Soft set theory,” Computers & Mathematics with Applications, vol. 45, no. 4-5, pp. 555–562, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. R. Roy and P. K. Maji, “A fuzzy soft set theoretic approach to decision making problems,” Journal of Computational and Applied Mathematics, vol. 203, no. 2, pp. 412–418, 2007. View at Google Scholar · View at Zentralblatt MATH
  7. Y. Jiang, Y. Tang, Q. Chen, J. Wang, and S. Tang, “Extending soft sets with description logics,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 2087–2096, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H. Aktas and N. Cagman, “Soft sets and soft groups,” Information Sciences, vol. 177, no. 13, pp. 2726–2735, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Shabir and M. Naz, “On soft topological spaces,” Computers & Mathematics with Applications, vol. 61, no. 7, pp. 1786–1799, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. X. Ge, Z. Li, and Y. Ge, “Topological spaces and soft sets,” Journal of Computational Analysis and Applications, vol. 13, no. 5, pp. 881–885, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Z. Pawlak and A. Skowron, “Rough sets: some extensions,” Information Sciences, vol. 177, no. 1, pp. 28–40, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  12. A. Bargiela and W. Pedrycz, Granular Computing: An Introduction, Kluwer Academic Publishers, Hingham, Mass, USA, 2003. View at MathSciNet
  13. J. Järvinen, “Lattice theory for rough sets,” in Transactions on Rough Sets. VI, vol. 4374 of Lecture Notes in Computer Science, pp. 400–498, Springer, Berlin, Germany, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. G. Birkhoff, Lattice Theory, American Mathematical Society, Providence, NJ, USA, 3rd edition, 1993.
  15. G. Boole, An Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories of Logic and Probabilities, Walton and Maberley, London, UK, 1854. View at Publisher · View at Google Scholar · View at MathSciNet
  16. J.-H. Dai, “Rough 3-valued algebras,” Information Sciences, vol. 178, no. 8, pp. 1986–1996, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. H. Heijmans, Morphological Image Operators, Academic Press, New York, NY, USA, 1994.
  18. J. A. Goguen, “L-fuzzy sets,” Journal of Mathematical Analysis and Applications, vol. 18, no. 1, pp. 145–174, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. Järvinen, “Set operations for L-fuzzy sets,” in Rough Sets Amd Intelligent System Radigms, vol. 4585 of Lecture Notes in Computer Science, pp. 221–229, Springer, Berlin, Germany, 2007. View at Google Scholar
  20. P. Sussner and E. L. Esmi, “Morphological perceptrons with competitive learning: lattice-theoretical framework and constructive learning algorithm,” Information Sciences, vol. 181, no. 10, pp. 1929–1950, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  21. Y. Xu, D. Ruan, K. Qin, and J. Liu, Lattice-Valued Logic, vol. 132, Springer, Berlin, Germany, 2003. View at MathSciNet
  22. B. Ganter and P. Wille, Formal Concept Analysis, Springer, Berlin, Germany, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  23. J. Järvinen, “On the structure of rough approximations,” Fundamenta Informaticae, vol. 53, no. 2, pp. 135–153, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. J. Järvinen, M. Kondo, and J. Kortelainen, “Modal-like operators in Boolean lattices, Galois connections and fixed points,” Fundamenta Informaticae, vol. 76, no. 1-2, pp. 129–145, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. F. Feng, C. Li, B. Davvaz, and M. I. Ali, “Soft sets combined with fuzzy sets and rough sets: a tentative approach,” Soft Computing, vol. 14, no. 9, pp. 899–911, 2010. View at Google Scholar
  26. F. Feng, X. Liu, V. Leoreanu-Fotea, and Y. B. Jun, “Soft sets and soft rough sets,” Information Sciences, vol. 181, no. 6, pp. 1125–1137, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, Mass, USA, 1990. View at MathSciNet
  28. G. Gratzer, General Lattice Theory, Academic Press, New York, NY, USA, 1978. View at MathSciNet
  29. Z. Pawlak and A. Skowron, “Rudiments of rough sets,” Information Sciences, vol. 177, no. 1, pp. 3–27, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet