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General Representation Theorems for Fuzzy Weak Orders

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Theory and Applications of Relational Structures as Knowledge Instruments II

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4342))

Abstract

The present paper gives a state-of-the-art overview of general representation results for fuzzy weak orders. We do not assume that the underlying domain of alternatives is finite. Instead, we concentrate on results that hold in the most general case that the underlying domain is possibly infinite. This paper presents three fundamental representation results: (i) score function-based representations, (ii) inclusion-based representations, (iii) representations by decomposition into crisp linear orders and fuzzy equivalence relations.

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References

  1. Bandler, W., Kohout, L.J.: Fuzzy power sets and fuzzy implication operators. Fuzzy Sets and Systems 4, 183–190 (1980)

    Article  MathSciNet  Google Scholar 

  2. Bandler, W., Kohout, L.J.: Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and artificial systems. In: Wang, S.K., Chang, P.P. (eds.) Fuzzy Sets: Theory and Application to Policy Analysis and Information Systems, pp. 341–367. Plenum Press, New York (1980)

    Google Scholar 

  3. Bodenhofer, U.: Representations and constructions of strongly linear fuzzy orderings. In: Proc. EUSFLAT-ESTYLF Joint Conference, Palma de Mallorca, pp. 215–218 (September 1999)

    Google Scholar 

  4. Bodenhofer, U.: A similarity-based generalization of fuzzy orderings preserving the classical axioms. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 8(5), 593–610 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bodenhofer, U.: Representations and constructions of similarity-based fuzzy orderings. Fuzzy Sets and Systems 137(1), 113–136 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bodenhofer, U., Küng, J.: Fuzzy orderings in flexible query answering systems. Soft Computing 8(7), 512–522 (2004)

    Article  MATH  Google Scholar 

  7. Cantor, G.: Beiträge zur Begründung der transfiniten Mengenlehre. Math. Ann. 46, 481–512 (1895)

    Article  Google Scholar 

  8. De Baets, B., Fodor, J., Kerre, E.E.: Gödel representable fuzzy weak orders. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 7(2), 135–154 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. De Baets, B., Mesiar, R.: Pseudo-metrics and T-equivalences. J. Fuzzy Math. 5(2), 471–481 (1997)

    MATH  MathSciNet  Google Scholar 

  10. De Baets, B., Mesiar, R.: Metrics and T-equalities. J. Math. Anal. Appl. 267, 331–347 (2002)

    Article  MathSciNet  Google Scholar 

  11. Fodor, J., Ovchinnikov, S.V.: On aggregation of T-transitive fuzzy binary relations. Fuzzy Sets and Systems 72, 135–145 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)

    MATH  Google Scholar 

  13. Gottwald, S.: Fuzzy Sets and Fuzzy Logic. Vieweg, Braunschweig (1993)

    Google Scholar 

  14. Gottwald, S.: A Treatise on Many-Valued Logics. In: Studies in Logic and Computation, Research Studies Press, Baldock (2001)

    Google Scholar 

  15. Hájek, P.: Metamathematics of Fuzzy Logic. Trends in Logic, vol. 4. Kluwer Academic Publishers, Dordrecht (1998)

    MATH  Google Scholar 

  16. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Trends in Logic, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)

    MATH  Google Scholar 

  17. Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A.: Foundations of Measurement. Academic Press, San Diego (1971)

    MATH  Google Scholar 

  18. Ovchinnikov, S.V.: Similarity relations, fuzzy partitions, and fuzzy orderings. Fuzzy Sets and Systems 40(1), 107–126 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ovchinnikov, S.V.: An introduction to fuzzy relations. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets. The Handbooks of Fuzzy Sets, vol. 7, pp. 233–259. Kluwer Academic Publishers, Boston (2000)

    Google Scholar 

  20. Ovchinnikov, S.V.: Numerical representation of transitive fuzzy relations. Fuzzy Sets and Systems 126, 225–232 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Roberts, F.S.: Measurement Theory. Addison-Wesley, Reading (1979)

    MATH  Google Scholar 

  22. Rosenstein, J.G.: Linear Orderings. Pure and Applied Mathematics, vol. 98. Academic Press, New York (1982)

    MATH  Google Scholar 

  23. Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fund. Math. 16, 386–389 (1930)

    MATH  Google Scholar 

  24. Valverde, L.: On the structure of F-indistinguishability operators. Fuzzy Sets and Systems 17(3), 313–328 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  25. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  26. Zadeh, L.A.: Similarity relations and fuzzy orderings. Inform. Sci. 3, 177–200 (1971)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Bioinformatics, Johannes Kepler University, A-4040, Linz, Austria

    Ulrich Bodenhofer

  2. Dept. of Applied Mathematics, Biometrics, and Process Control, Ghent University, Coupure links 653, B-9000, Gent, Belgium

    Bernard De Baets

  3. Institute of Intelligent Engineering Systems, Budapest Tech, H-1034, Budapest, Hungary

    János Fodor

Authors
  1. Ulrich Bodenhofer
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  2. Bernard De Baets
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  3. János Fodor
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Editor information

Editors and Affiliations

  1. Faculty of Philosophy, Tilburg University, P.O. Box 90153, 5000, Tilburg, LE, The Netherlands

    Harrie de Swart

  2. National Institute of Telecommunications, Warsaw,  

    Ewa Orłowska

  3. Institute for Software Technology, Department of Computing Science, Universität der Bundeswehr München, 85577, Neubiberg, Germany

    Gunther Schmidt

  4. Faculté Polytechnique de Mons, Rue de Houdain 9, B-7000, Mons, Belgium

    Marc Roubens

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© 2006 Springer-Verlag Berlin Heidelberg

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Bodenhofer, U., De Baets, B., Fodor, J. (2006). General Representation Theorems for Fuzzy Weak Orders. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds) Theory and Applications of Relational Structures as Knowledge Instruments II. Lecture Notes in Computer Science(), vol 4342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11964810_11

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  • DOI: https://doi.org/10.1007/11964810_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-69223-2

  • Online ISBN: 978-3-540-69224-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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