Galois Connections with Truth Stressers: Foundations for Formal Concept Analysis of Object-Attribute Data with Fuzzy Attributes

  • Radim Bělohlávek
  • Taťána Funioková
  • Vilém Vychodil
Part of the Advances in Soft Computing book series (AINSC, volume 33)

Abstract

Galois connections appear in several areas of mathematics and computer science, and their applications. A Galois connection between sets X and Y is a pair 〈 ↑, ↓〉 of mappings ↑ assigning subcollections of Y to subcollections of X, and ↓ assigning subcollections of X to subcollections of Y. By definition, Galois connections have to satisfy certain conditions. Galois connections can be interpreted in the following manner: For subcollections A and B of X and Y, respectively, A↑ is the collection of all elements of Y which are in a certain relationship to all elements from A, and B↓ is the collection of all elements of X which are in the relationship to all elements in B. From the very many examples of Galois connections in mathematics, let us recall the following. Let X be the set of all logical formulas of a given language, Y be the set of all structures (interpretations) of the same language. For AX and BY, let A↑ consist of all structures in which each formula from A is true, let B↓ denote the set of all formulas which are true in each structure from B. Then, ↑ and ↓ is a Galois connection.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnauld A., Nicole P.: La logique ou l’art de penser. 1662. Also in German: Die Logik oder die Kunst des Denkens. Darmstadt, 1972.Google Scholar
  2. 2.
    Baaz M.: Infinite-valued Gödel logics with 0–1 projections and relativizations. GÖDEL’ 96 — Logical Foundations of Mathematics, Computer Sciences and Physics, Lecture Notes in Logic vol. 6, Springer-Verlag 1996, 23–33.MathSciNetGoogle Scholar
  3. 3.
    Barbut M.: Note sur l’algèbre des techniques d’analyse hiérarchique. Appendice de l’Analyse hiérarchique, M. Matalon, Paris, Gauthier-Villars, 1965, 125–146.Google Scholar
  4. 4.
    Barbut M., Monjardet B.: L’ordre et la classification, algèbre et combinatoire, tome II. Paris, Hachette, 1970.Google Scholar
  5. 5.
    Bělohlávek R.: Fuzzy concepts and conceptual structures: induced similarities. In Proc. Joint Conf. Inf. Sci.’98, Vol. I, pages 179–182, Durham, NC, 1998.Google Scholar
  6. 6.
    Bělohlávek R.: Fuzzy Galois connections. Math. Logic Quarterly 45,4 (1999), 497–504.Google Scholar
  7. 7.
    Bělohlávek R.: Lattices of fixed points of fuzzy Galois connections. Math. Logic Quarterly 47,1 (2001), 111–116.CrossRefGoogle Scholar
  8. 8.
    Bělohlávek R.: Similarity relations in concept lattices. J. Logic and Computation Vol. 10 No. 6(2000), 823–845.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bělohlávek R.: Fuzzy closure operators. J. Math. Anal. Appl. 262(2001), 473–489.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Bělohlávek R.: Fuzzy closure operators II. Soft Computing 7(2002) 1, 53–64.CrossRefGoogle Scholar
  11. 11.
    Bělohlávek R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer, Academic/Plenum Publishers, New York, 2002.Google Scholar
  12. 12.
    Bělohlávek R.: Fuzzy closure operators induced by similarity. Fundamenta Informaticae 58(2)(2003), 79–91.MathSciNetGoogle Scholar
  13. 13.
    Bělohlávek R.: Concept lattices and order in fuzzy logic. Ann. Pure Appl. Logic (to appear, 22 pp.).Google Scholar
  14. 14.
    Bělohlávek R., Sklenář V., Zacpal J.: Crisply generated fuzzy concepts: reducing the number of concepts in formal concept analysis (submitted).Google Scholar
  15. 15.
    Birkhoff G.: Lattice Theory, 3-rd edition. AMS Coll. Publ. 25, Providence, R.I., 1967.Google Scholar
  16. 16.
    Burusco A., Fuentes-Gonzales R.: The study of L-fuzzy concept lattice. Mathware & Soft Computing 3(1994), 209–218.Google Scholar
  17. 17.
    Dostál M., Vychodil V.: Evolutionary approach to automated deduction (in preparation).Google Scholar
  18. 18.
    Ganter B., Wille R.: Formal concept analysis. Mathematical Foundations. Springer-Verlag, Berlin, 1999.MATHGoogle Scholar
  19. 19.
    Goguen J. A.: L-fuzzy sets. J. Math. Anal. Appl. 18(1967), 145–174.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gottwald S.: A Treatise on Many-Valued Logics. Research Studies Press, Baldock, Hertfordshire, England, 2001.MATHGoogle Scholar
  21. 21.
    Hájek P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998.MATHGoogle Scholar
  22. 22.
    Hájek P.: On very true. Fuzzy sets and systems 124(2001), 329–333.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Höhle U.: On the fundamentals of fuzzy set theory. J. Math. Anal. Appl. 201(1996), 786–826.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Klement E. P., Mesiar R., Pap E.: Triangular Norms. Kluwer, Dordrecht, 2000.MATHGoogle Scholar
  25. 25.
    Novák, V., Perfileva I., Močkoř J.: Mathematical Principles of Fuzzy Logic. Kluwer, Dordrecht, 1999.MATHGoogle Scholar
  26. 26.
    Ore O.: Galois connections. Trans. AMS 55(1944), 493–513.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Pollandt S.: Fuzzy Begriffe. Springer-Verlag, Berlin/Heidelberg, 1997.MATHGoogle Scholar
  28. 28.
    Takeuti G., Titani S.: Globalization of intuitionistic set theory. Annal of Pure and Applied Logic 33(1987), 195–211.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Wille R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival I.: Ordered Sets. Reidel, Dordrecht, Boston, 1982, 445–470.Google Scholar
  30. 30.
    Zadeh L. A.: Fuzzy sets. Inf. Control 8(3)(1965), 338–353.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Radim Bělohlávek
    • 1
    • 2
  • Taťána Funioková
    • 1
  • Vilém Vychodil
    • 1
  1. 1.Dept. Computer SciencePalacky UniversityOlomoucCzech Republic
  2. 2.Inst. Research and Applications of Fuzzy ModelingUniversity of OstravaOstravaCzech Republic

Personalised recommendations