Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 908623, 6 pages
http://dx.doi.org/10.1155/2013/908623
Research Article

A Bijection between Lattice-Valued Filters and Lattice-Valued Congruences in Residuated Lattices

1School of Computer Science and Engineering, Xi’an University of Technology, Xi’an, Shaanxi 710048, China
2College of Computer Science and Technology, Taiyuan University of Technology, Taiyuan 030024, China

Received 22 January 2013; Accepted 1 July 2013

Academic Editor: Bin Liu

Copyright © 2013 Wei Wei et al. 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. F. Esteva and L. Godo, “Monoidal t-norm based logic: towards a logic for left-continuous t-norms,” Fuzzy Sets and Systems, vol. 124, no. 3, pp. 271–288, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  2. U. Höhle, “Commutative residuated monoid,” in Non-Classical Logics and Their Applications to Fuzzy Subsets, U. Höhle and E. P. Klement, Eds., pp. 53–106, Kluwer Academic, Dordrecht, The Netherlands, 1995. View at Google Scholar
  3. G. J. Wang, Non-Classical Mathematical Logic and Approximation Reasoning, Chinese Science Press, Beijing, China, 2000.
  4. N. Galatos, P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, vol. 151 of Studies in Logic and the Foundations of Mathematics, Elsevier B. V., Amsterdam, The Netherlands, 2007. View at MathSciNet
  5. M. Ward and R. P. Dilworth, “Residuated lattices,” Transactions of the American Mathematical Society, vol. 45, no. 3, pp. 335–354, 1939. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. P. T. Johnstone, Stone Spaces, vol. 3 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, Mass, USA, 1982. View at MathSciNet
  7. C. C. Chang, “Algebraic analysis of many valued logics,” Transactions of the American Mathematical Society, vol. 88, pp. 467–490, 1958. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. P. Hájek, Mathematics of Fuzzy Logic, Kluwer Academic, Dordrecht, The Netherlands, 1998.
  9. Y. Xu and K. Y. Qin, “On filters of lattice implication algebras,” Journal of Fuzzy Mathematics, vol. 1, no. 2, pp. 251–260, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. Bělohlávek, “Some properties of residuated lattices,” Czechoslovak Mathematical Journal, vol. 53, no. 1, pp. 161–171, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. K. Blount and C. Tsinakis, “The structure of residuated lattices,” International Journal of Algebra and Computation, vol. 13, no. 4, pp. 437–461, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. Li and N. Chen, Simplication of congruence relations in residuated lattices, Computer Engineering and Applications, accpeted (in Chinese).
  13. L. Liu and K. Li, “Fuzzy filters of BL-algebras,” Information Sciences, vol. 173, no. 1–3, pp. 141–154, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  14. J. M. Yang, S. Feng, and W. Yao, “Simplication of congruence relation in residuated lattice,” Journal of Hebei University of Science and Technology, vol. 33, no. 6, pp. 479–481, 2012. View at Google Scholar
  15. R. Belohlavek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic; Plenum Press, New York, NY, USA, 2002.
  16. J. Pavelka, “On fuzzy logic. II. Enriched residuated lattices and semantics of propositional calculi,” Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 25, no. 2, pp. 119–134, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. Papert, “Which distributive lattices are lattices of closed sets?” Proceedings of the Cambridge Philosophical Society, vol. 55, pp. 172–176, 1959. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet