Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

(∈, ∈ ∨qk)-FUZZY IDEALS IN LEFT REGULAR ORDERED $\mathcal{LA}$-SEMIGROUPS

  • Yousafzai, Faisal (School of Mathematical Sciences, University of Science and Technology of China) ;
  • Khan, Asghar (Department of Mathematics, Abdul Wali Khan University) ;
  • Khan, Waqar (School of Mathematical Sciences, University of Science and Technology of China) ;
  • Aziz, Tariq (Department of Mathematics, COMSATS Institute of Information Technology)
  • Received : 2013.05.09
  • Accepted : 2013.09.09
  • Published : 2013.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

We generalize the idea of (${\in}$, ${\in}{\vee}q_k$)-fuzzy ordered semi-group and give the concept of (${\in}$, ${\in}{\vee}q_k$)-fuzzy ordered $\mathcal{LA}$-semigroup. We show that (${\in}$, ${\in}{\vee}q_k$)-fuzzy left (right, two-sided) ideals, (${\in}$, ${\in}{\vee}q_k$)-fuzzy (generalized) bi-ideals, (${\in}$, ${\in}{\vee}q_k$)-fuzzy interior ideals and (${\in}$, ${\in}{\vee}q_k$)-fuzzy (1, 2)-ideals need not to be coincide in an ordered $\mathcal{LA}$-semigroup but on the other hand, we prove that all these (${\in}$, ${\in}{\vee}q_k$)-fuzzy ideals coincide in a left regular class of an ordered $\mathcal{LA}$-semigroup. Further we investigate some useful conditions for an ordered $\mathcal{LA}$-semigroup to become a left regular ordered $\mathcal{LA}$-semigroup and characterize a left regular ordered $\mathcal{LA}$-semigroup in terms of (${\in}$, ${\in}{\vee}q_k$)-fuzzy one-sided ideals. Finally we connect an ideal theory with an (${\in}$, ${\in}{\vee}q_k$)-fuzzy ideal theory by using the notions of duo and (${\in}{\vee}q_k$)-fuzzy duo.

Keywords

References

  1. Faisal, N. Yaqoob and A. Ghareeb, Left regular AG-groupoids in terms of fuzzy interior ideals, Afrika Mathematika, DOI: 10.1007/s13370-012-0081-y.
  2. S.K. Bhakat and P. Das, On the denition of a fuzzy subgroup, Fuzzy Sets and Systems, 51 (1992), 235-241. https://doi.org/10.1016/0165-0114(92)90196-B
  3. S.K. Bhakat and P. Das, (${\in},\;{\in}\;{\vee}q$)-fuzzy subgroups, Fuzzy Sets and Systems, 80 (1996), 359-368. https://doi.org/10.1016/0165-0114(95)00157-3
  4. B. Davvaz, Fuzzy R-subgroups with threshholds of near-rings and implication operators, Soft Comput., 12 (2008), 875-879. https://doi.org/10.1007/s00500-007-0255-y
  5. Y.B. Jun, Generalizations of (${\in},\;{\in}\;{\vee}q$)-fuzzy subalgebras in BCK/BCI-algebras, Comput. Math. Appl., 58 (2009), 1383-1390. https://doi.org/10.1016/j.camwa.2009.07.043
  6. Y.B. Jun, A. Khan and M. Shabir, Ordered semigroups characterized by their (${\in},\;{\in}\;{\vee}q$)-fuzzy bi-ideals, Bull. Malaysian Math. Sci. Soc., 2(3) (2009), 391-408.
  7. 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
  8. M.A. Kazim and M. Naseeruddin, On almost semigroups, Aligarh. Bull. Math., 2 (1972), 1-7.
  9. M. Khan and Faisal, On fuzzy ordered Abel-Grassmann's groupoids, J. Math. Res., 3 (2011), 27-40.
  10. V. Murali, Fuzzy points of equivalent fuzzy subsets, Inform. Sci., 158 (2004), 277-288. https://doi.org/10.1016/j.ins.2003.07.008
  11. Q. Mushtaq and S.M. Yusuf, On LA-semigroups, Aligarh. Bull. Math., 8 (1978), 65-70.
  12. Q. Mushtaq and S.M. Yusuf, On LA-semigroup dened by a commutative inverse semigroup, Math. Bech., 40 (1988), 59-62.
  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. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517. https://doi.org/10.1016/0022-247X(71)90199-5
  15. M. Shabir, Y.B. Jun and Y. Nawaz, Semigroups characterized by (${\in},\;{\in}\;{\vee}qk$)-fuzzy bi-ideals, Computers and Mathematics with Applications, 60 (2010), 1473-1493. https://doi.org/10.1016/j.camwa.2010.06.030
  16. N. Stevanovic and P.V. Protic, Composition of Abel-Grassmann's 3-bands, Novi Sad. J. Math., 2(34) (2004), 175-182.
  17. L. A. Zadeh, Fuzzy sets, Inform. Control., 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X

Cited by

  1. Existence of non-associative algebraic hyper-structures and related problems vol.26, pp.5-6, 2015, https://doi.org/10.1007/s13370-014-0259-6