Properties of Basic Fuzzy Implication Algebra

  • Zhi-wei Li
  • Gui-hua Li
Part of the Advances in Soft Computing book series (AINSC, volume 54)


Fuzzy Implication algebra is a kind of algebraic abstraction of implicative connection of logic system which values in [0,1]. In this paper, the logic properties of implication operator were given on the frame of Basic Fuzzy implication algebra. Some lattice properties of it were obtained when the basic implication algebra was regular.


Fuzzy Logic Fuzzy Implication Algebra Basic Fuzzy Implication Algebra Regularity Lattice 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hajec, P.: Mathematics of fuzzy logic. Kluwer, Dordrecht (1998)Google Scholar
  2. 2.
    Li, Z.W., Zheng, C.Y.: Relations between Fuzzy Implication Algebra and MV Algebra. The Journal of Fuzzy Mathematics 9(1), 201–205 (2001)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Li, Z.W., Zheng, C.Y.: Implication Algebra and Heyting Algebra. International Congress of Mathematics: Abstracts of Short Communications and Poster Sessions, 4–5 (2002)Google Scholar
  4. 4.
    Ma, J., Chen, S., Xu, Y.: Fuzzy logic from the viewpoint of machine intelligence. Fuzzy Sets and Systems 157, 628–634 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Pei, D.W.: The Characterizations of MTL Algebras. Acta Mathmatica Sinica, Chinese Series 50, 1201–1206 (2007)Google Scholar
  6. 6.
    Rachunek, J., Salounova, D.: Truth values on generalizations of some commutative fuzzy structures. Fuzzy Sets and Systems 157, 3159–3168 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Wang, G.J.: On the logic foundation of fuzzy reasoning. Inform. Sci. 117, 47–88 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Wang, G.J.: A Formal Deductive System of Fuzzy Proposition Calculus. Chinese Science Bulletin 42(10), 1041–1045 (1997)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Wang, G.J.: Implication Lattices and Their Fuzzy Implication Space Representation Theorem. Acta Mathmatica Sinica, Chinese Series 41(1), 133–140 (1999)Google Scholar
  10. 10.
    Wu, H.B., Wang, G.J.: Total Complication Triple I Method Based on Complete BR0-Algebra. T. Mathematical research and exposition 26, 341–353 (2006) (in Chinese)zbMATHGoogle Scholar
  11. 11.
    Wu, W.M.: Fuzzy Implication Algebra. Fuzzy Systems and Mathematics 4(1), 56–63 (1990) (in Chinese)Google Scholar
  12. 12.
    Xu, Y.: Lattices Implication Algebra. J. Southwest Jiaotong Univ. 28, 20–27 (1993)Google Scholar
  13. 13.
    Yi, L.Z., Pei, Z., Song, W.: Results of Associated Implication Algebra on a Partial Ordered Set. Journal of Donghua University 24(2), 293–296 (2007) (in Chinese)Google Scholar
  14. 14.
    Zadeh, L.A.: Fuzzy sets. Inform. And Control 8, 338–353 (1965)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Zhi-wei Li
    • 1
  • Gui-hua Li
    • 2
  1. 1.School of Math. ScienceCapital Normal UniversityBeijingChina
  2. 2.Dept. Basic CourseBeijing Vocational Agricultural CollegeBeijingChina

Personalised recommendations