Skip to main content
SpringerLink
Log in

L-E-Fuzzy Lattices

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

A new definition of a fuzzy lattice ( L-E-fuzzy lattice) as a particular fuzzy algebraic structure is introduced in the framework of fuzzy equalities and fuzzy identities. The membership values structure is a complete lattice. An L-E-fuzzy lattice is defined on a bi-groupoid M, as its fuzzy sub-bi-groupoid μ equipped with a fuzzy equality E, fulfilling fuzzy lattice identities. It is proved that the new notion is a generalization of known lattice-valued structures. Basic properties of the introduced new fuzzy lattices are presented. In particular, it is proved that the quotients of cuts of μ over the corresponding cuts of E are classical lattices. By a suitable example, it is shown how the new introduced structures can be applied.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Goguen, J.A.: \(L\)-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  2. Di Nola, A., Gerla, G.: Lattice valued algebras. Stochastica 11, 137–150 (1987)

    MathSciNet  MATH  Google Scholar 

  3. Šešelja, B., Tepavčević, A.: Partially ordered and relational valued algebras and congruences. Rev. Res. Fac. Sci. Math. Ser. 23, 273–287 (1993)

    MATH  Google Scholar 

  4. B. Šešelja, A. Tepavčević, L-E-Fuzzy Lattices. In: Proceedings of the of iFUZZY, p. 108. Kaohsiung, Taiwan, 26–28 Nov 2014

  5. Höhle, U., Šostak, A.P.: Axiomatic Foundations of Fixed-basis Fuzzy Topology. Springer, Dordrecht (1999)

    Book  Google Scholar 

  6. Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic/Plenum Publishers, New York (2002)

    Book  Google Scholar 

  7. Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall PTR, Upper Saddle River, NJ (1995)

    MATH  Google Scholar 

  8. Höhle, U.: Quotients with respect to similarity relations. Fuzzy Sets Syst. 27, 31–44 (1988)

    Article  MATH  Google Scholar 

  9. M. Demirci, Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations part I: fuzzy functions and their applications, part II: vague algebraic notions, part III: constructions of vague algebraic notions and vague arithmetic operations, Int. J. Gen. Syst. 32 (3) (2003) 123–155, 157–175, 177–201

  10. Bělohlávek, R., Vychodil, V.: Algebras with fuzzy equalities. Fuzzy Sets Syst. 157, 161–201 (2006)

    Article  MATH  Google Scholar 

  11. B. Šešelja, A. Tepavčević, Fuzzy Identities. In: Proceedings of the 2009 IEEE International Conference on Fuzzy Systems 1660–1664

  12. Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: Fuzzy identities with application to fuzzy semigroups. Inf. Sci. 266, 148–159 (2014)

    Article  Google Scholar 

  13. Budimirović, B., Budimirović, V., Šešelja, B., Tepavčević, A.: Fuzzy equational classes are Fuzzy varieties. Iran. J. Fuzzy Syst. 10, 1–18 (2013)

    Google Scholar 

  14. Tepavčević, A., Trajkovski, G.: L-fuzzy lattices: an introduction. Fuzzy Sets Syst. 123, 209–216 (2001)

    Article  MATH  Google Scholar 

  15. Jun-Fang Zhang, A Novel Definition of Fuzzy Lattice Based on Fuzzy Set, The Scientific World Journal Volume: 2013. Article ID 678586, (2013). doi:10.1155/2013/678586

  16. Zhang, Q., Xie, W., Fan, L.: Fuzzy complete lattices. Fuzzy Sets and Syst. 160, 2275–2291 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Demirci, M.: A theory of vague lattices based on many-valued equivalence relations I: general representation results. Fuzzy Sets Syst. 151, 437–472 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Demirci, M.: A theory of vague lattices based on many-valued equivalence relations II: complete lattices. Fuzzy Sets Syst. 151, 473–489 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ajmal, N., Thomas, K.V.: Fuzzy lattices. Inf. Sci. 79, 271–291 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zimmermann, H.J.: Fuzzy Set Theory and its Applications. Kluwer, Boston (2011)

    Google Scholar 

  21. B. Budimirović, V. Budimirović, B. Šešelja, A. Tepavčević, Fuzzy equational classes, Fuzzy Systems (FUZZ-IEEE), IEEE International Conference, pp. 1–6 (2012)

  22. Engesser, K., Gabbay, D.M., Lehmann, D. (eds.): Handbook of Quantum Logic and Quantum Structures: Quantum Structures. Elsevier, Amsterdam (2011)

    Google Scholar 

  23. Ganter, B., Wille, R.: Formal Concept Analysis, vol. 284. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  24. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1981)

    Book  MATH  Google Scholar 

  25. Filep, L.: Study of fuzzy algebras and relations from a general viewpoint. Acta Math. Acad. Paedagog. Nyházi 14, 49–55 (1998)

    MathSciNet  MATH  Google Scholar 

  26. Šešelja, B., Tepavčević, A.: On generalizations of fuzzy algebras and congruences. Fuzzy Sets Syst. 65, 85–94 (1994)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

Research supported by Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 174013 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, Grant “Ordered structures and applications”.

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovića 4, 21000, Novi Sad, Serbia

    Branimir Šešelja & Andreja Tepavčević

Authors
  1. Branimir Šešelja
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreja Tepavčević
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andreja Tepavčević.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Šešelja, B., Tepavčević, A. L-E-Fuzzy Lattices. Int. J. Fuzzy Syst. 17, 366–374 (2015). https://doi.org/10.1007/s40815-015-0057-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-015-0057-9

Keywords

Mathematics Subject Classification

Search

Navigation