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Aggregation of Fuzzy Relations and Preservation of Transitivity

  • Conference paper
Book cover Theory and Applications of Relational Structures as Knowledge Instruments II

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

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Abstract

This contribution provides a comprehensive overview on the theoretical framework of aggregating fuzzy relations under the premise of preserving underlying transitivity conditions. As such it discusses the related property of dominance of aggregation operators. After a thorough introduction of all necessary and basic properties of aggregation operators, in particular dominance, the close relationship between aggregating fuzzy relations and dominance is shown. Further, principles of building dominating aggregation operators as well as classes of aggregation operators dominating one of the basic t-norms are addressed. In the paper by Bodenhofer, Küng and Saminger, also in this volume, the interested reader finds an elaborated (real world) example, i.e., an application of the herein contained theoretical framework.

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References

  1. Aczél, J.: Lectures on Functional Equations and their Applications. Academic Press, New York (1966)

    MATH  Google Scholar 

  2. Alsina, C., Frank, M., Schweizer, B.: Associative Functions: Triangular Norms and Copulas. World Scientific Publishing Company, Singapore (2006)

    Book  MATH  Google Scholar 

  3. Bezdek, J.C., Harris, J.D.: Fuzzy partitions and relations: An axiomatic basis for clustering. Fuzzy Sets and Systems 1, 111–127 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bodenhofer, U.: A Similarity-Based Generalization of Fuzzy Orderings. Schriftenreihe der Johannes-Kepler-Universität Linz, vol. C 26, Universitätsverlag Rudolf Trauner (1999)

    Google Scholar 

  5. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Bodenhofer, U., Bogdanowicz, P., Lanzerstorfer, G., Küng, J.: Distance-based fuzzy relations in flexible query answering systems: Overview and experiences. In: Düntsch, I., Winter, M. (eds.) Proc. 8th Int. Conf. on Relational Methods in Computer Science, St. Catharines, ON, Brock University, pp. 15–22 (February 2005)

    Google Scholar 

  8. Bodenhofer, U., Küng, J.: Enriching vague queries by fuzzy orderings. In: Proc. 2nd Int. Conf. in Fuzzy Logic and Technology (EUSFLAT 2001), Leicester, UK, pp. 360–364 (September 2001)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  10. Bodenhofer, U., Küng, J., Saminger, S.: Flexible query answering using distance-based fuzzy relations. In: de Swart, H., Orlowska, E., Roubens, M., Schmidt, G. (eds.) Theory and Applications of Relations Structures as Knowledge Instruments II. LNCS (LNAI). Springer, Heidelberg (2006)

    Google Scholar 

  11. Bosc, P., Buckles, B., Petry, F., Pivert, O.: Fuzzy databases: Theory and models. In: Bezdek, J., Dubois, D., Prade, H. (eds.) Fuzzy Sets in Approximate Reasoning and Information Systems, pp. 403–468. Kluwer Academic Publishers, Boston (1999)

    Google Scholar 

  12. P. Bosc, L. Duval, and O. Pivert. Value-based and representation-based querying of possibilistic databases. In G. Bordogna and G. Pasi, editors, Recent Issues on Fuzzy Databases, pages 3–27. Physica-Verlag, Heidelberg, 2000.

    Google Scholar 

  13. Bouchon-Meunier, B. (ed.): Aggregation and Fusion of Imperfect Information. Physica-Verlag, Heidelberg (1998)

    MATH  Google Scholar 

  14. Calvo, T., Kolesárová, A., Komorníková, M., Mesiar, R.: Aggregation operators: properties, classes and construction methods. In: Calvo, T., Mayor, G., Mesiar, R. (eds.) Aggregation Operators. New Trends and Applications, pp. 3–104. Physica-Verlag, Heidelberg (2002)

    Google Scholar 

  15. Calvo, T., Mesiar, R.: Weighted means based on triangular conorms. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 9(2), 183–196 (2001)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  17. De Baets, B., Mesiar, R.: T-partitions. Fuzzy Sets and Systems 97, 211–223 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dujmovic, J.J.: Weighted conjunctive and disjunctive means and their application in system evaluation. Univ. Beograd Publ. Electrotech. Fak, 147–158 (1975)

    Google Scholar 

  19. Höhle, U.: Fuzzy equalities and indistinguishability. In: Proc. 1st European Congress on Fuzzy and Intelligent Technologies, Aachen, vol. 1, pp. 358–363 (1993)

    Google Scholar 

  20. Höhle, U., Blanchard, N.: Partial ordering in L-underdeterminate sets. Inform. Sci. 35, 133–144 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  21. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)

    MATH  Google Scholar 

  22. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper I: Basic analytical and algebraic properties. Fuzzy Sets and Systems 143, 5–26 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper II: general constructions and parameterized families. Fuzzy Sets and Systems 145, 411–438 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper III: continuous t-norms. Fuzzy Sets and Systems 145, 439–454 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. U.S.A. 8, 535–537 (1942)

    Article  MathSciNet  Google Scholar 

  26. Mesiar, R., De Baets, B.: New construction methods for aggregation operators. In: Proc. 8th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems, Madrid, vol. 2, pp. 701–706 (2000)

    Google Scholar 

  27. Mesiar, R., Saminger, S.: Domination of ordered weighted averaging operators over t-norms. Soft Computing 8, 562–570 (2004)

    Article  MATH  Google Scholar 

  28. Petry, F.E., Bosc, P.: Fuzzy Databases: Principles and Applications. International Series in Intelligent Technologies. Kluwer Academic Publishers, Boston (1996)

    MATH  Google Scholar 

  29. Rosado, A., Kacprzyk, J., Ribeiro, R.A., Zadrozny, S.: Fuzzy querying in crisp and fuzzy relational databases: An overview. In: Proc. 9th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems, vol. 3, pp. 1705–1712 (July 2002)

    Google Scholar 

  30. Saminger, S.: Aggregation in Evaluation of Computer-Assisted Assessmen. In: Schriftenreihe der Johannes-Kepler-Universität Linz, vol. C 44. Universitätsverlag Rudolf Trauner (2005)

    Google Scholar 

  31. Saminger, S., De Baets, B., De Meyer, H.: On the dominance relation between ordinal sums of conjunctors. Kybernetika 42(3), 337–350 (2006)

    MathSciNet  Google Scholar 

  32. Saminger, S., Mesiar, R., Bodenhofer, U.: Domination of aggregation operators and preservation of transitivity. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 10(suppl.), 11–35 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  33. Saminger, S., Sarkoci, P., De Baets, B.: The dominance relation on the class of continuous t-norms from an ordinal sum point of view. In: de Swart, H., Orlowska, E., Roubens, M., Schmidt, G. (eds.) Theory and Applications of Relations Structures as Knowledge Instruments II. LNCS (LNAI). Springer, Heidelberg (2006)

    Google Scholar 

  34. Sarkoci, P.: Dominance is not transitive on continuous triangular norms. Aequationes Mathematicae (submitted, 2006)

    Google Scholar 

  35. Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313–334 (1960)

    MATH  MathSciNet  Google Scholar 

  36. Schweizer, B., Sklar, A.: Associative functions and statistical triangle inequalities. Publ. Math. Debrecen 8, 169–186 (1961)

    MATH  MathSciNet  Google Scholar 

  37. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York (1983)

    MATH  Google Scholar 

  38. Silvert, W.: Symmetric summation: A class of operations on fuzzy sets. IEEE Trans. Systems Man Cybernet. 9, 657–659 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  39. Tardiff, R.M.: Topologies for probabilistic metric spaces. Pacific J. Math. 65, 233–251 (1976)

    MATH  MathSciNet  Google Scholar 

  40. Yager, R.R.: On ordered weighted averaging aggregation operators in multicriteria decisionmaking. IEEE Trans. Systems Man Cybernet 18, 183–190 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  41. Yager, R.R., Filev, D.P.: Essentials of Fuzzy Modelling and Control. J. Wiley & Sons, New York (1994)

    Google Scholar 

  42. 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. Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, A-4040, Linz, Austria

    Susanne Saminger & Erich Peter Klement

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

    Ulrich Bodenhofer

  3. Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, SK-813 68, Bratislava, Slovakia

    Radko Mesiar

  4. Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, CZ-701 03, Ostrava 1, Czech Republic

    Radko Mesiar

Authors
  1. Susanne Saminger
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  2. Ulrich Bodenhofer
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  3. Erich Peter Klement
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  4. Radko Mesiar
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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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Saminger, S., Bodenhofer, U., Klement, E.P., Mesiar, R. (2006). Aggregation of Fuzzy Relations and Preservation of Transitivity. 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_9

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  • DOI: https://doi.org/10.1007/11964810_9

  • 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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