Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERAL TYPES OF (α,β)-FUZZY IDEALS OF HEMIRINGS

  • Jun, Y.B. (Department of Mathematics Education, Gyeongsang National University) ;
  • Dudek, W.A. (Institute of Mathematics and Computer Science, Wroclaw University of Technology Wyb.) ;
  • Shabir, M. (Department of Mathematics, Quaid-i-Azam University) ;
  • Kang, Min-Su (Department of Mathematics, Hanyang University)
  • Received : 2010.07.26
  • Accepted : 2010.08.26
  • Published : 2010.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

W. A. Dudek, M. Shabir and M. Irfan Ali discussed the properties of (${\alpha},{\beta}$)-fuzzy ideals of hemirings in [9]. In this paper, we discuss the generalization of their results on (${\alpha},{\beta}$)-fuzzy ideals of hemirings. As a generalization of the notions of $({\alpha},\;\in{\vee}q)$-fuzzy left (right) ideals, $({\alpha},\;\in{\vee}q)$-fuzzy h-ideals and $({\alpha},\;\in{\vee}q)$-fuzzy k-ideals, the concepts of $({\alpha},\;\in{\vee}q_m)$-fuzzy left (right) ideals, $({\alpha},\;\in{\vee}q_m)$-fuzzy h-ideals and $({\alpha},\;\in{\vee}q_m)$-fuzzy k-ideals are defined, and their characterizations are considered. Using a left (right) ideal (resp. h-ideal, k-ideal), we construct an $({\alpha},\;\in{\vee}q_m)$-fuzzy left (right) ideal (resp. $({\alpha},\;\in{\vee}q_m)$-fuzzy h-ideal, $({\alpha},\;\in{\vee}q_m)$-fuzzy k-ideal). The implication-based fuzzy h-ideals (k-ideals) of a hemiring are considered.

Keywords

References

  1. J. Ahsan, Semirings characterized by their fuzzy ideals, J. Fuzzy Math. 6 (1998) 181-192.
  2. J. Ahsan, K. Saifullah, M. F. Khan, Fuzzy semirings, Fuzzy Sets and Systems 60 (1993) 309-320. https://doi.org/10.1016/0165-0114(93)90441-J
  3. A. W. Aho, J. D. Ullman, Languages and Computation, Addison-Wesley, Reading, MA, 1979.
  4. S. K. Bhakat, (${\in}{\vee}q$)-level subset, Fuzzy Sets and Systems 103 (1999), 529-533. https://doi.org/10.1016/S0165-0114(97)00158-9
  5. S. K. Bhakat, (${\in},{\in}{\vee}q$)-fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and Systems 112 (2000), 299-312. https://doi.org/10.1016/S0165-0114(98)00029-3
  6. 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
  7. 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
  8. S. K. Bhakat and P. Das, Fuzzy subrings and ideals redefined, Fuzzy Sets and Systems 81 (1996), 383-393. https://doi.org/10.1016/0165-0114(95)00202-2
  9. W. A. Dudek, M. Shabir, M. Irfan Ali, ($\alpha,\beta$)-fuzzy ideals of hemirings, Comput. Math. Appl. 58 (2009), 310-321. https://doi.org/10.1016/j.camwa.2009.03.097
  10. F. Feng, Y. B. Jun, Z. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008) 2621-2628. https://doi.org/10.1016/j.camwa.2008.05.011
  11. S. Ghosh, Fuzzy k-ideals of semirings, Fuzzy Sets and Systems 95 (1998) 103-108. https://doi.org/10.1016/S0165-0114(96)00306-5
  12. K. Glazck. A guide to literature on semirings and their applications in mathematics and information sciences: With complete bibliography. Kluwer Acad. Publ., Dordrecht, 2002.
  13. J. S. Golan, Semirings and their Applications. Kluwer Acad. Publ., 1999.
  14. U. Hebisch. H. J. Weinert. Semirings. Algebraic Theory and Application in the Computer Science, World Scientific, 1998.
  15. 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
  16. Y. B. Jun, On ($\alpha,\beta$)-fuzzy subalgebras of BCK BCI-algebras, Bull. Korean Math. Soc. 42(4) (2005) 703-711. https://doi.org/10.4134/BKMS.2005.42.4.703
  17. Y. B. Jun. Fuzzy subalgebras of type ($\alpha,\beta$) in BCK BCI-algebras, Kyungpook Math. J. 47 (2007) 403-410.
  18. Y. B. Jun, Note on "($\alpha,\beta$)-fuzzy ideals of hemirings", Comput. Math. Appl. 59 (2010) 2582-2586. https://doi.org/10.1016/j.camwa.2010.01.017
  19. Y. B. Jun, M. A. Ozurk, S. Z. Song, On fuzzy h-ideals in hemirings. Inform. Sci. 162 (2004) 211-226. https://doi.org/10.1016/j.ins.2003.09.007
  20. 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
  21. 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
  22. M. S. Ying, On standard models of fuzzy modal logics, Fuzzy Sets and Systems 26 (1988) 357-363. https://doi.org/10.1016/0165-0114(88)90128-5
  23. J. Zhan. W. A. Dudek, Fuzzy h-ideals of hemirings, Inform. Sci. 177 (2007) 876-886. https://doi.org/10.1016/j.ins.2006.04.005

Cited by

  1. New types of fuzzy bi-ideals in ordered semigroups vol.21, pp.S1, 2012, https://doi.org/10.1007/s00521-012-0843-3
  2. Ordered semigroups characterized by ( $${ \in,\in \vee q}_{k}$$ )-fuzzy generalized bi-ideals vol.21, pp.S1, 2012, https://doi.org/10.1007/s00521-011-0731-2
  3. Characterizations of hemirings by (∈,∈∨qk)-fuzzy ideals vol.61, pp.4, 2011, https://doi.org/10.1016/j.camwa.2010.12.056