On fuzzy implicative ideals in BL-algebras

Document Type : Original Article

Author

Department of Mathematics, University of Bojnord, Bojnord, Iran

Abstract

In this paper, the concept of fuzzy implicative ideal in BL-algebras is introduced and several properties of it are stated. Using the concept of level subsets, some characterizations of fuzzy implicative ideals are proved. Also, it is proved that the concepts of fuzzy implicative ideal and fuzzy Boolean ideal in BL-algebras are coincide. Moreover, it is shown that a BL-algebra L is a Boolean algebra if and only if any fuzzy ideal of L is a fuzzy implicative ideal. Finally, it is proved that the homomorphic image and preimage of fuzzy implicative ideals are fuzzy implicative ideal.

Keywords

References

[1] C.C. Chang, Algebraic analysis of many valued logics, Transactions of the American Mathematical
Society, 88 (1958), 467–490.
[2] A. Di Nola, L. Leustean, Compact representations of BL-algebras, Department of Computer
Science, University Aarhus. BRICS Report Series, (2002).
[3] P. Hájek, Metamathematics of fuzzy logic, Klower Academic Publishers, Dordrecht, (1999).
[4] C. Lele, J.B. Nganou, MV-algebras derived from ideals in BL-algebras, Fuzzy Sets and Systems,
218 (2013), 103–113.
[5] C. Lele, J.B. Nganou, Psedo-addition and fuzzy ideals in BL-algebras, Annals of Fuzzy Mathematics
and Informations, 8(2) (2014), 193–207.
[6] B.L. Meng, X.L. Xin, On fuzzy ideals of BL-algebras, The Scientific World Journal, 2014,
Article ID 757382, 12 pages, (2014).
[7] A. Paad, Fuzzy n-fold integral and fuzzy n-fold Boolean ideals in BL-algebras, Annals of Fuzzy
Mathematics and Informatics, 11(1) (2016), 119–133.
[8] E. Turunen, BL-algebras and basic fuzzy logic, Mathware Soft Computing, 6 (1999), 49–61.
[9] Y. Yang, X. Xin, On characterization of BL-algebras via implicative ideals, Italian Journal of
Pure and Applied Mathematics, 37 (2017), 493–506.
[10] X.H. Zhang, Y.B. Jun, M.I. Doh, On fuzzy filters and fuzzy ideals of BL-algebras, Fuzzy
Systems and Mathematics, 20(3) (2006), 1604–1616.