Weak Fuzzy Equivalence and Equality Relations

  • Branimir Šešelja
  • Andreja Tepavčević
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5909)


Weak fuzzy (lattice valued) equivalences and equalities are introduced by weakening the reflexivity property. Every weak fuzzy equivalence relation on a set determines a fuzzy set on the same domain. In addition, its cut relations are crisp equivalences on the corresponding cut subsets. Analogue properties of weak fuzzy equalities are presented. As an application, fuzzy weak congruence relations and fuzzy identities on algebraic structures are investigated.

Keywords and phrases

lattice-valued fuzzy set lattice-valued fuzzy relation block cut fuzzy equivalence fuzzy equality fuzzy identity 

AMS Mathematics Subject Classification (2000)

primary 03B52 03E72 secondary 06A15 


  1. 1.
    De Baets, B., Mesiar, R.: T-partitions. Fuzzy Sets and Systems 97, 211–223 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic/Plenum Publishers, New York (2002)zbMATHGoogle Scholar
  3. 3.
    Bělohlávek, R., Vychodil, V.: Algebras with fuzzy equalities. Fuzzy Sets and Systems 157, 161–201 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bodenhofer, U., Demirci, M.: Strict Fuzzy Orderings with a given Context of Similarity. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 16(2), 147–178 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, Heidelberg (1981)zbMATHGoogle Scholar
  6. 6.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1992)Google Scholar
  7. 7.
    Demirci, M.: Vague Groups. J. Math. Anal. Appl. 230, 142–156 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Demirci, M.: Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations part I, part II and part III. Int. J. General Systems 32(3), 123–155, 157–175, 177–201 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Di Nola, A., Gerla, G.: Lattice valued algebras. Stochastica 11, 137–150 (1987)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Goguen, J.A.: L-fuzzy Sets. J. Math. Anal. Appl. 18, 145–174 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Höhle, U.: Quotients with respect to similarity relations. Fuzzy Sets and Systems 27, 31–44 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Malik, J.N., Mordeson, D.S., Kuroki, N.: Fuzzy Semigroups. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  13. 13.
    Montes, S., Couso, I., Gil, P.: Fuzzy δ − ε-partition. Information Sciences 152, 267–285 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Murali, V.: Fuzzy equivalence relations. Fuzzy Sets and Systems 30, 155–163 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Šešelja, B., Tepavčević, A.: Fuzzy identities. In: Proceedings of FUZZ-IEEE 2009, pp. 1660–1663 (2009)Google Scholar
  16. 16.
    Šešelja, B., Tepavčević, A.: On Generalizations of Fuzzy Algebras and Congruences. Fuzzy Sets and Systems 65, 85–94 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Šešelja, B., Tepavčević, A.: Fuzzy groups and collections of subgroups. Fuzzy Sets and Systems 83, 85–91 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Tepavčević, A., Vujić, A.: On an application of fuzzy relations in biogeography. Information Sciences 89(1-2), 77–94 (1996)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Branimir Šešelja
    • 1
  • Andreja Tepavčević
    • 1
  1. 1.Department of Mathematics and InformaticsUniversity of Novi SadSerbia

Personalised recommendations