Non-dual fuzzy connections

Abstract.

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.

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References

  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)

    Article  Google 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 appear

  5. 5.

    Bělohlávek, R.: Fuzzy closure operators I. J. Math. Anal. Appl. 262, 473–489 (2001)

    Article  MathSciNet  Google Scholar 

  6. 6.

    Bělohlávek, R.: Fuzzy closure operators II. Soft Comput. 7 (1), 53–64 (2002)

    Article  Google Scholar 

  7. 7.

    Bělohlávek, R.: Concept equations. To appear in Journal of Logic and Computation

  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. Submitted

  10. 10.

    Bělohlávek, R.: Fuzzy relational systems: foundations and principles. Kluwer, 2002

  11. 11.

    Bělohlávek, R., Funiková, T.: Fuzzy interior operators. Submitted

  12. 12.

    Birkhoff, G.: Lattice Theory. AMS Coll. Publ. 25, Providence, RI, 1967

  13. 13.

    Bodenhofer, U.: A Unified Framework of Opening and Closure Operators with Respect to Arbitrary Fuzzy Relations. Soft Comput. 7, 220–227 (2003)

    Article  MATH  Google Scholar 

  14. 14.

    Ganter, B., Wille, R.: Formal concept analysis. Mathematical Foundations, Springer-Verlag, Berlin, 1999

  15. 15.

    Georgescu, G., Popescu, A.: Non-commutative fuzzy Galois connections. Soft Comput. 7 (7), 458–467 (2003)

    MATH  Google Scholar 

  16. 16.

    Gerla, G.: Graded consequence relations and fuzzy closure operator. J. Appl. Non- Classical Logics 6 (4), 369–379 (1996)

    MATH  Google Scholar 

  17. 17.

    Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–147 (1967)

    Article  MATH  Google 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, 1998

  20. 20.

    Hájek, P.: Fuzzy logic with non-commutative conjunctions. Journal of Logic and Computation 13, 469–479 (2003)

    MathSciNet  Google 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)

    MATH  Google Scholar 

  23. 23.

    Popescu, A.: A general approach to fuzzy concepts. In Math. Log. Quart. 50 (3), 1–17 (2004)

    MATH  Google Scholar 

  24. 24.

    Turunen, E.: Mathematics behind fuzzy logic. Physica-Verlag, Heidelberg, 1999

  25. 25.

    Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. I.Rival, Oredered Sets, Reidel, Dordrecht, Boston, 1982, pp. 445–470

  26. 26.

    Zadeh: Fuzzy sets. Inform. Control 8, 338–353 (1965)

    Google Scholar 

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Correspondence to George Georgescu.

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Georgescu, G., Popescu, A. Non-dual fuzzy connections. Arch. Math. Logic 43, 1009–1039 (2004). https://doi.org/10.1007/s00153-004-0240-4

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Key words or phrases:

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