Counting Finite Residuated Lattices

  • Radim Belohlavek
  • Vilem Vychodil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4529)


We study finite residuated lattices with up to 11 elements. We present an algorithm for generating all non-isomorphic finite residuated lattices with a given number of elements. Furthermore, we analyze selected properties of all the lattices generated by our algorithm and present summarizing statistics.


Fuzzy Logic Residuated Lattice Table Entry Lattice Reducts Truth Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bartušek, T., Navara, M.: Program for generating fuzzy logical operations and its use in mathematical proofs. Kybernetika 38(3), 235–244 (2002)MathSciNetGoogle Scholar
  2. 2.
    Belohlavek, R.: Fuzzy Relational Systems: Foundations and Principles. Plenum Publishers, New York (2002)MATHGoogle Scholar
  3. 3.
    Belohlavek, R., Vychodil, V.: Scales behind computational intelligence: exploring properties of finite lattices. To appear in Proc. 1st IEEE Symposium on Foundations of Comput. Intelligence – FOCI ’07, Honolulu, Hawaii, USA, 1–5 Apr. (2007)Google Scholar
  4. 4.
    Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124, 271–288 (2001)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Esteva, F., Godo, L., Hájek, P., Navara, M.: Residuated fuzzy logics with an involutive negation. Arch. Math. Logic 39, 103–124 (2000)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Gerla, G.: Fuzzy Logic. Mathematical Tools for Approximate Reasoning. Kluwer, Dordrecht (2001)MATHGoogle Scholar
  7. 7.
    Goguen, J.A.: The logic of inexact concepts. Synthese 18(-9), 325–373 (1968)Google Scholar
  8. 8.
    Gratzer, G.A.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998)Google Scholar
  9. 9.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998)MATHGoogle Scholar
  10. 10.
    Hájek, P.: On very true. Fuzzy Sets and Systems 124, 329–333 (2001)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Heitzig, J., Reinhold, J.: Counting Finite Lattices. Algebra Universalis 48(1), 43–53 (2002)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Höhle, U.: On the fundamentals of fuzzy set theory. J. Math. Anal. Appl. 201, 786–826 (1996)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Kure, M.: Computer-aided study of finite posets. UP Olomouc, MSc. thesis (2004)Google Scholar
  14. 14.
    Klir, G.J., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Theory and Applications. Prentice-Hall, Englewood Cliffs (1995)MATHGoogle Scholar
  15. 15.
    Miller, G.T.: The magical number seven, plus or minus two: some limits on our capacity for processing information. The Psychological Review 63, 81–97 (1956)CrossRefGoogle Scholar
  16. 16.
    Pavelka, J.: On fuzzy logic I, II, III. Z. Math. Logik Grundlagen Math. 25, 45–52, 119–134, 447–464 (1979)Google Scholar
  17. 17.
    Vychodil, V.: Truth-depressing hedges and BL-logic. Fuzzy Sets and Systems 157(15), 2074–2090 (2006)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Radim Belohlavek
    • 1
    • 2
  • Vilem Vychodil
    • 2
  1. 1.Dept. Systems Science and Industrial Engineering, Binghamton University—SUNY, Binghamton, NY 13902USA
  2. 2.Dept. Computer Science, Palacky University, Olomouc, Tomkova 40, CZ-779 00 OlomoucCzech Republic

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