Afrika Matematika

, Volume 25, Issue 3, pp 501–518 | Cite as

Characterization of regular LA-semigroups by interval-valued \((\overline{\alpha },\overline{\beta })\)-fuzzy ideals



The concept of interval-valued \((\overline{\alpha },\overline{\beta })\)-fuzzy ideals, interval-valued \((\overline{\alpha },\overline{\beta })\)-fuzzy generalized bi-ideals are introduced in LA-semigroups, using the ideas of belonging and quasi-coincidence of an interval-valued fuzzy point with an interval-valued fuzzy set and some related properties are investigated. We define the lower and upper parts of interval-valued fuzzy subsets of an LA-semigroup. Also regular LA-semigroups are characterized by the properties of the upper part of interval-valued \((\overline{\in }, \overline{\in }\vee \overline{q})\)-fuzzy left ideals, interval-valued \(( \overline{\in },\overline{\in }\vee \overline{q})\)-fuzzy quasi-ideals and interval-valued \((\overline{\in },\overline{\in }\vee \overline{q})\)-fuzzy generalized bi-ideals.


Interval-valued \((\overline{\alpha }, \overline{\beta })\)-fuzzy sub LA-semigroups Interval-valued \((\overline{\alpha }, \overline{\beta })\)-fuzzy ideals Interval-valued \((\overline{\alpha }, \overline{\beta })\)-fuzzy Interval-valued \((\overline{\alpha }, \overline{\beta })\)-fuzzy quasi-ideals 

Mathematics Subject Classification (2000)

16D25 20N99 03E72 


  1. 1.
    Aslam, M., Abdullah, S., Masood, M.: Bipolar Fuzzy Ideals in LA-semigroups. World Appl. Sci. J. 17(12), 1769–1782 (2012)Google Scholar
  2. 2.
    Abdullah, S., Aslam, M., Amin, N., Khan, T.: Direct product of finite fuzzy subsets of LA-semigroups. Ann. Fuzzy Math. Info. 3(2), 281–292 (2012)Google Scholar
  3. 3.
    Abdullah, S., Aslam, M., Khan, T.A., Naeem, M.: A new type of fuzzy normal subgroups and fuzzy cosets. J. Intell. Fuzzy Syst. doi:10.3233/IFS-2012-0612
  4. 4.
    Abdullah, S., Davvaz, B., Aslam, M.: (\(\alpha,\beta \))-intuitionistic fuzzy ideals of hemirings. Comput. Math. Appl. 62(8), 3077–3090 (2011)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bhakat, S.K., Das, P.: (\(\in,\in \vee q\))-fuzzy subgroup. Fuzzy Sets Syst. 80, 359–368 (1996)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bhakat, S.K., Das, P.: On the definition of a fuzzy subgroup. Fuzzy Sets Syst. 51, 235–241 (1992)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Biswas, R.: Rosenfeld’s fuzzy subgroups with interval-valued membership functions. Fuzzy Sets Syst. 63, 87–90 (1994)MATHCrossRefGoogle Scholar
  8. 8.
    Davvaz, B.: (\(\in,\in \vee q\))-fuzzy subnearrings and ideals. Soft Comput. 10, 206–211 (2006)MATHCrossRefGoogle Scholar
  9. 9.
    Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)MATHGoogle Scholar
  10. 10.
    Jun, Y.B., Song, S.Z.: Generalized fuzzy interior ideals in semigroups. Inf. Sci. 176, 3079–3093 (2006)Google Scholar
  11. 11.
    Kazanci, O., Yamak, S.: Generalized fuzzy bi-ideals of semigroups. Soft Comput. 12, 1119–1124 (2008)Google Scholar
  12. 12.
    Kazim, M.A., Naseerudin, M.: On almost semigroups. Alig. Bull. Math. 2, 1–7 (1972)MATHGoogle Scholar
  13. 13.
    Khan, A., Sarmin, N.H., Khan, F.M., Davvaz, B.: Regular AG-groupoids characterized by-fuzzy ideals. Iran. J. Sci. Technol. 2, 97–113 (2012)MathSciNetGoogle Scholar
  14. 14.
    Khan, A., Jun, Y.B., Mahmood, T.: Generalized fuzzy interior ideals in Abel Grassmann’s groupoids. IJMMS 2010, Article ID: 838392 (2010). doi:10.1155/2010/838392
  15. 15.
    Khan, A., Shabir, M., Jun, Y.B.: Generalized fuzzy Abel Grassmann’s groupoids. Int. J. Fuzzy Syst. 12(4), 340–349 (2010)MathSciNetGoogle Scholar
  16. 16.
    Kuroki, N.: On fuzzy semigroups. Inf. Sci. 53, 203–236 (1991)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Mordeson, J.N., Malik, D.S., Kuroki, N.: Fuzzy semigroups. In: Studies in Fuzziness and Soft Computing, vol. 131. Springer, Berlin (2003)Google Scholar
  18. 18.
    Mushtaq, Q., Yousaf, S.M.: On LA-semigroups. Alig. Bull. Math 8, 65–70 (1987)Google Scholar
  19. 19.
    Narayanan, Al, Manikantan, T.: Interval-valued fuzzy ideals generated by an interval valued fuzzy subset in semigroups. J. Appl. Math. Comput. 20, 455–464 (2006)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Pu, P.M., Liu, Y.M.: Fuzzy topology 1, neighbourhood structure of a fuzzy point and Moore-Smith convergence. J. Math. Anal. Appl. 76, 571–599 (1980)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Rosenfeld, A.: Fuzzy groups. J. Math. Anal. Appl. 35, 512–517 (1971)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Shabir, M., Khan, I.A.: Interval-valued fuzzy ideals generated by an interval-valued fuzzy subset in ordered semigroups. Mathw. Soft Comput. 15, 263–272 (2008)MATHMathSciNetGoogle Scholar
  23. 23.
    Shabir, M., Jun, Y.B., Nawaz, Y.: Characterization of regular semigroups by (\(\alpha,\beta \))-fuzzy ideals. Comput. Math. Appl. 59, 161–175 (2010)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Shabir, M., Nawaz, Y., Ali, M.: Characterization of semigroups by (\(\in ,\in \vee q\))-fuzzy ideals. World Appl. Sci. J. 14(12), 805–819 (2011)Google Scholar
  25. 25.
    Yaqoob, N., Abdullah, S., Rehman, N., Naeem, M.: Roughness and fuzziness in ordered ternary semigroups. World Appl. Sci. J. 17(12), 1683–1693 (2012)Google Scholar
  26. 26.
    Yaqoob, N., Chinram, R., Ghareeb, A., Aslam, M.: Left almost semigroups characterized by their interval valued fuzzy ideals. Afr. Mat. 1–15 (2011). doi:10.1007/s13370-011-0055-5
  27. 27.
    Faisal, Yaqoob, N., Ghareeb, A.: Left regular AG-groupoids in terms of fuzzy interior ideals. Afr. Mat. 1–11 (2012). doi:10.1007/s13370-012-0081-y
  28. 28.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Zadeh, L.A.: The concept of a linguistic variable and its applications to approximate reasoning. Inf. Sci. 8, 199–249 (1975)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Zenab, R.: Some Studies in Fuzzy AG-groupoids. M. Phil Dissertation, Quaid-i-Azam University, Islamabad (2009)Google Scholar
  31. 31.
    Zhan, J., Davvaz, B., Shum, K.P.: Generalized fuzzy hyperideals of hyperrings. Comput. Math. Appl. 56, 1732–1740 (2008)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Muhammad Aslam
    • 1
  • Saleem Abdullah
    • 1
  • Samreen Aslam
    • 1
  1. 1.Department of MathematicsQuaid-i-Azam UniversityIslamabadPaksitan

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