Archive for Mathematical Logic

, Volume 43, Issue 8, pp 1009–1039 | Cite as

Non-dual fuzzy connections

  • George Georgescu
  • Andrei Popescu


The lack of double negation and de Morgan properties makes fuzzy logic unsymmetrical. This is the reason why fuzzy versions of notions like closure operator or Galois connection deserve attention for both antiotone and isotone cases, these two cases not being dual. This paper offers them attention, comming to the following conclusions:

– some kind of hardly describable ‘‘local preduality’’ still makes possible important parallel results;

– interesting new concepts besides antitone and isotone ones (like, for instance, conjugated pair), that were classically reducible to the first, gain independency in fuzzy setting.

Key words or phrases:

Duality Isotone structure Fuzzy set theory Galois connection conjugated pair Closure operator 


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  1. 1.
    Bělohlávek, R.: Fuzzy Galois Connections. Math. Logic Quart. 45 (4), 497–504 (1999)Google Scholar
  2. 2.
    Bělohlávek, R.: Lattices of fixed points of Fuzzy Galois Connections. Math. Logic Quart. 47 (1), 111–116 (2001)CrossRefGoogle Scholar
  3. 3.
    Bělohlávek, R.: Similarity relations in concept lattices. J. Logic Comput. 10 (6), 823–845 (2000)Google Scholar
  4. 4.
    Bělohlávek, R.: Concept lattices and order in fuzzy logic. Annals of Pure and Appl. Logic. To appearGoogle Scholar
  5. 5.
    Bělohlávek, R.: Fuzzy closure operators I. J. Math. Anal. Appl. 262, 473–489 (2001)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bělohlávek, R.: Fuzzy closure operators II. Soft Comput. 7 (1), 53–64 (2002)CrossRefGoogle Scholar
  7. 7.
    Bělohlávek, R.: Concept equations. To appear in Journal of Logic and ComputationGoogle Scholar
  8. 8.
    Bělohlávek, R.: Logical precision in concept lattices. J. Logic Comput. 12 (6), 137–148 (2002)Google Scholar
  9. 9.
    Bělohlávek, R.: Fuzzy closure operators induced by similarity. SubmittedGoogle Scholar
  10. 10.
    Bělohlávek, R.: Fuzzy relational systems: foundations and principles. Kluwer, 2002Google Scholar
  11. 11.
    Bělohlávek, R., Funiková, T.: Fuzzy interior operators. SubmittedGoogle Scholar
  12. 12.
    Birkhoff, G.: Lattice Theory. AMS Coll. Publ. 25, Providence, RI, 1967Google Scholar
  13. 13.
    Bodenhofer, U.: A Unified Framework of Opening and Closure Operators with Respect to Arbitrary Fuzzy Relations. Soft Comput. 7, 220–227 (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Ganter, B., Wille, R.: Formal concept analysis. Mathematical Foundations, Springer-Verlag, Berlin, 1999Google Scholar
  15. 15.
    Georgescu, G., Popescu, A.: Non-commutative fuzzy Galois connections. Soft Comput. 7 (7), 458–467 (2003)zbMATHGoogle Scholar
  16. 16.
    Gerla, G.: Graded consequence relations and fuzzy closure operator. J. Appl. Non- Classical Logics 6 (4), 369–379 (1996)zbMATHGoogle Scholar
  17. 17.
    Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–147 (1967)CrossRefzbMATHGoogle Scholar
  18. 18.
    Goguen, J.A.: The Logic of inexact concepts. Synthese 19, 325–373 (1968-69)Google Scholar
  19. 19.
    Hájek, P.: Metamathematics of fuzzy logic. Kluwer, 1998Google Scholar
  20. 20.
    Hájek, P.: Fuzzy logic with non-commutative conjunctions. Journal of Logic and Computation 13, 469–479 (2003)MathSciNetGoogle Scholar
  21. 21.
    Hájek, P.: Observations on non-commutative logic. Soft Computing 8, 38–43 (2003)Google Scholar
  22. 22.
    Ore, O.: Galois Connexions. Trans. AMS 55, 493–513 (1994)zbMATHGoogle Scholar
  23. 23.
    Popescu, A.: A general approach to fuzzy concepts. In Math. Log. Quart. 50 (3), 1–17 (2004)zbMATHGoogle Scholar
  24. 24.
    Turunen, E.: Mathematics behind fuzzy logic. Physica-Verlag, Heidelberg, 1999Google Scholar
  25. 25.
    Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. I.Rival, Oredered Sets, Reidel, Dordrecht, Boston, 1982, pp. 445–470Google Scholar
  26. 26.
    Zadeh: Fuzzy sets. Inform. Control 8, 338–353 (1965)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institute of MathematicsBucharestRomania
  2. 2.Fundamentals of Computer ScienceFaculty of MathematicsBucharestRomania

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