Skip to main content
Log in

Obstinate, weak implicative and fantastic filters of non commutative residuated lattices

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

In this paper, we introduce the notions of obstinate filters, weak implicative filters and fantastic filters on a non commutative residuated lattice and study their properties. Some characterizations of these filters are obtained. The relations among these filters and (sub) positive implicative filters are investigated. It is proved that each obstinate filters are sub positive implicative filters but the converse may not be true. Then the conditions under which a sub positive implicative filter is an obstinate filter are obtained. Also, we prove that each sub positive implicative filters are (fantastic) weak implicative filters and the converse may not be true. We establish the condition under which a (fantastic) weak implicative filter is a sub positive implicative filter.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blount, K., Tsinakis, C.: The structure of residuated lattices. Int. J. Algebra Comput. 13, 437–461 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blyth, T.S., Janovitz, M.F.: Residuation Theory, Progamon Press (1972)

  3. Chang, C.C.: Algebraic analysis of many valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ciungu, L.C.: Classes of residuated lattices. Ann. Univ. Cralova, Math. Comp. Sci. Ser. 33,189–207 (2006)

  5. Nola, A.Di., Georgescu, G., Lorgulescu A.: Pseudo BL-algebras: part I, Mult. Val. Logic 8, 673–716 (2002)

  6. Nola, A.Di., Georgescu, G., Iorgulescu, A.: Pseudo BL-algebras: part II, Mult. Val. Logic 8, 717–750 (2002)

  7. Flondor, P., Georgescu, G., Iorgulescu, A.: Pseudo t-norms and pseudo BL-algebras. Soft Comput. 5, 355–371 (2001)

    Article  MATH  Google Scholar 

  8. Georgescu, G., Leustean, L.: Some classes of pseudo BL-algebras. J. Aust. Math. Soc. 73, 127–153 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ghorbani, Sh: Sub posetive implicative filter of residuated filters. World Appl. Sci. J. 12(5), 586–590 (2011)

    Google Scholar 

  10. Ghorbani, Sh.: Vague Filters of Residuated Lattices. J. Discrete Math. (2014)

  11. Ghorbani, Sh, Hasankhani, A.: Fuzzy convex subalgebras of commutative residuated lattices. Iranian J. Fuzzy Syst. 7(2), 41–54 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Hajek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  13. Jipsen, P., Tsinakis, C.: A survey of residuated lattices. In: Martinez, J. (ed.) Ordered Algebraic Structures, pp. 19–56. Kluwer Academic Publishers, Dordrect (2002)

    Chapter  Google Scholar 

  14. Hoo, C.S., Sessa, S.: Implicative and Boolean ideals of MV-algebras. Math. Jap. 39, 215–219 (1994)

    MathSciNet  MATH  Google Scholar 

  15. Kowalski, T., Ono, H.: Residuated lattices, an algebraic glimpse at logic without contraction, Japan Advanced Institute of Science and Technology (2001)

  16. Liu, L.Z., Li, K.T.: Fuzzy Boolean and positive implicative filters of BL-algebras. Fuzzy Sets Syst. 152, 333–348 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, L.Z., Li, K.T.: Boolean filters and positive implicative filters of residuated lattice. Inform. Sci. 177, 5725–5738 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, L.Z.: Research on Residuated lattice-based Logical Algebras. Xian Jiaotong University, Thesis (2005)

  19. Turunen, E.: Boolean deductive systems of BL-algebras. Arch. Math. Logic 40, 467–473 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Turunen, E.: Mathematics Behind Fuzzy Logic. Physica-Verlag, Heidelberg (1999)

    MATH  Google Scholar 

  21. Ward, M., Dilworth, R.P.: Residuated lattices. Trans. Am. Math. Soc. 45, 335–354 (1939)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xu, Y., Qin, K.Y.: On filters of lattice implication algebras. J. Fuzzy Math. 1, 251–260 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Zhanao, X., Yunhua, X., Weihua, L., Huiru, C., Yuejun, L.: Intuitionistic fuzzy filter theory of BL-algebras. Int. J. Mach. Learn. Cybernet. 4(6), 659–669 (2013)

    Article  Google Scholar 

  24. Zhang, J.L., Zhou, H.J.: Fuzzy filter on the residuated lattice. New Math. Nat. Comput. 2, 11–28 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Shokoofeh Ghorbani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghorbani, S. Obstinate, weak implicative and fantastic filters of non commutative residuated lattices. Afr. Mat. 28, 69–84 (2017). https://doi.org/10.1007/s13370-016-0429-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-016-0429-9

Keywords

Mathematics Subject Classification

Navigation