• Jun, Young-Bae (Department of Mathematics Education (and RINS) Gyeongsang National University) ;
  • Song, Seok-Zun (Department of Mathematics Jeju National University)
  • Received : 2011.06.02
  • Published : 2012.10.31


As a generalization of the concept of a fuzzy implicative filter which is introduced by Liu and Li [3], the notion of (${\in}$, ${\in}{\vee}q_k$)-fuzzy implicative filters is introduced, and related properties are investigated. The relationship between (${\in}$, ${\in}{\vee}q_k$)-fuzzy filters and (${\in}$, ${\in}{\vee}q_k$)-fuzzy implicative filters is established. Conditions for an (${\in}$, ${\in}{\vee}q_k$)-fuzzy filter to be an (${\in}$, ${\in}{\vee}q_k$)-fuzzy implicative filter are considered. Characterizations of an (${\in}$, ${\in}{\vee}q_k$)-fuzzy implicative filter are provided, and the implication-based fuzzy implicative filters of an $R_0$-algebra is discussed.


Supported by : National Research Foundation of Korea (NRF)


  1. Y. B. Jun and L. Liu, Filters of $R_{0}$-algebras, Int. J. Math. Math. Sci. 2006 (2006), Article ID 93249, 9 pages.
  2. Y. B. Jun, S. Z. Song, and J. Zhan, Generalizations of $({\in},{\in}Vq)$-fuzzy filters in $R_{0}$- algebras, Int. J. Math. Math. Sci. (submitted).
  3. L. Liu and K. Li, Fuzzy implicative and Boolean filters of $R_{0}$-algebras, Inform. Sci. 171 (2005), no. 1-3, 61-71.
  4. X. Ma, J. Zhan, and Y. B. Jun, On $({\in},{\in}Vq)$-fuzzy filters of $R_{0}$-algebras, MLQ Math. Log. Q. 55 (2009), no. 5, 493-508.
  5. D. W. Pei and G. J. Wang, The completeness and applications of the formal system $\mathcal{L}^{\ast}$, Sci. China Ser. F 45 (2002), no. 1, 40-50.
  6. P. M. Pu and Y. M. Liu, Fuzzy topology I, Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980), no. 2, 571-599.
  7. G. J. Wang, Non-Classical Mathematical Logic and Approximate Reasoning, Science Press, Beijing, 2000.
  8. G. J. Wang, On the logic foundation of fuzzy reasoning, Inform. Sci. 117 (1999), no. 1-2, 47-88.
  9. M. S. Ying, A new approach for fuzzy topology. I, Fuzzy Sets and Systems 39 (1991), no. 3, 303-321.
  10. M. S. Ying, On standard models of fuzzy modal logics, Fuzzy Sets and Systems 26 (1988), no. 3, 357-363.
  11. L. A. Zadeh, Toward a generalized theory of uncertainty (GTU)-an outline, Inform. Sci. 172 (2005), no. 1-2, 1-40.