Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

MORE GENERAL FORMS OF (∈, ∈ VQk) FUZZY FILTERS OF ORDERED SEMIGROUPS

  • Khan, Asghar (Department of Mathematics, Abdul Wali Khan University) ;
  • Muhammad, Shakoor (Department of Mathematics, Abdul Wali Khan University) ;
  • Khalaf, Mohammed M. (Department of Mathematics, Faculty of Science in Zulfi, Majmaah University)
  • Received : 2016.12.14
  • Accepted : 2017.05.24
  • Published : 2017.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

In the paper [Y. B. Jun, B. Davvaz and A. Khan, Filters of ordered semigroups based on the fuzzy points, JIFS 24 (2013) 619-630]. Jun et al. discussed the notion of (${\in},{\in}{\vee}q_k$)-fuzzy left (resp., right) filters as a generalization of the notion of (${\in},{\in}{\vee}q$)-fuzzy left (resp., right) filters of ordered semigroups. In this article, we try to obtain a more general form that (${\in},{\in}{\vee}q_k$)-fuzzy left (resp., right) filters in ordered semigroups. The notion of (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp., right) filters is discussed, and several properties are investigated. Characterizations of an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp., right) filter are established. A condition for an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp., right) filter to be a fuzzy left (resp., right) filter is provided. The important achievement of the study with an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (right) filter is that the notion of an (${\in},{\in}{\vee}q_k$)-fuzzy left ( right) filter and hence an (${\in},{\in}{\vee}q$)-fuzzy left (resp. right) filter are special cases of an (${\in},{\in}{\vee}q_k^{\delta}$)-fuzzy left (resp. right) filter, and thus several results in published papers are becoming corollaries of our results obtained in this paper.

Keywords

References

  1. Y. B. Jun, B. Davvaz and A. Khan, Filters of ordered semigroups based on the fuzzy points, Journal of Intelligent & Fuzzy Systems, 24 (2013) 619-630.
  2. B. Davvaz and A. Khan, Generalized fuzzy filters in ordered semigroups, IJST 36 (2012), A1:77-86.
  3. N. Kehayopulu and M. Tsingelis, Fuzzy sets in ordered groupoids, Semigroup Forum 65 (2002), 128-132.
  4. S.K. Bhakat and P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems 51 (1992), 235-241. https://doi.org/10.1016/0165-0114(92)90196-B
  5. S.K. Bhakat and P. Das, (${\in}$, ${\in}$ ${\vee}q$)-fuzzy subgroup, Fuzzy Sets and Systems 80 (1996), 359-368. https://doi.org/10.1016/0165-0114(95)00157-3
  6. Y.B. Jun and S.Z. Song, Generalized fuzzy interior ideals in semigroups, Inform Sci 176 (2006), 3079-3093. https://doi.org/10.1016/j.ins.2005.09.002
  7. Y.B. Jun, K.J. Lee and C.H. Park, New types of fuzzy ideals in BCK/BCI-algebras, Comput. Math. Appl. 60 (2010), 771-785. https://doi.org/10.1016/j.camwa.2010.05.024
  8. O. Kazanci and S. Yamak, Generalized fuzzy bi-ideals of semigroup, Soft Computing 12 (2008), 1119-1124. https://doi.org/10.1007/s00500-008-0280-5
  9. N. Kuroki, On fuzzy semigroups, Inform Sci bf 53(3) (1991), 203-236. https://doi.org/10.1016/0020-0255(91)90037-U
  10. N. Mordeson, D.S. Malik and N. Kuroki, Fuzzy Semigroups, Springer, Berlin, 2003.
  11. J.N. Mordeson and D.S. Malik, Fuzzy Automata and Languages, Chapman & Hall/CRC, Boca Raton, FL, 2002.
  12. V. Murali, Fuzzy points of equivalent fuzzy subsets, Inform. Sci. 158 (2004), 277-288. https://doi.org/10.1016/j.ins.2003.07.008
  13. 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), 571-599. https://doi.org/10.1016/0022-247X(80)90048-7
  14. M. Shabir, Y.B. Jun and Y. Nawaz, Characterizations of regular semigroups by (${\alpha}$, ${\beta}$ )-fuzzy ideals, Comput. Math. Appl. 59 (2010), 161-175. https://doi.org/10.1016/j.camwa.2009.07.062
  15. M.S. Ying, A new approach for fuzzy topology (I), Fuzzy Sets and Systems 39 (1991), 303-321. https://doi.org/10.1016/0165-0114(91)90100-5
  16. M.S. Ying, On standard models of fuzzy modal logics, Fuzzy Sets and Systems 26 (1988), 357-363.17 https://doi.org/10.1016/0165-0114(88)90128-5