Advertisement

Aggregation of Fuzzy Relations and Preservation of Transitivity

  • Susanne Saminger
  • Ulrich Bodenhofer
  • Erich Peter Klement
  • Radko Mesiar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4342)

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.

Keywords

Aggregation Operator Fuzzy Relation Triangular Norm Fuzzy Equivalence Relation Binary Fuzzy Relation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aczél, J.: Lectures on Functional Equations and their Applications. Academic Press, New York (1966)zbMATHGoogle Scholar
  2. 2.
    Alsina, C., Frank, M., Schweizer, B.: Associative Functions: Triangular Norms and Copulas. World Scientific Publishing Company, Singapore (2006)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bezdek, J.C., Harris, J.D.: Fuzzy partitions and relations: An axiomatic basis for clustering. Fuzzy Sets and Systems 1, 111–127 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 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. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bodenhofer, U.: Representations and constructions of similarity-based fuzzy orderings. Fuzzy Sets and Systems 137(1), 113–136 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 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. 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. 9.
    Bodenhofer, U., Küng, J.: Fuzzy orderings in flexible query answering systems. Soft Computing 8(7), 512–522 (2004)zbMATHCrossRefGoogle Scholar
  10. 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. 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. 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. 13.
    Bouchon-Meunier, B. (ed.): Aggregation and Fusion of Imperfect Information. Physica-Verlag, Heidelberg (1998)zbMATHGoogle Scholar
  14. 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. 15.
    Calvo, T., Mesiar, R.: Weighted means based on triangular conorms. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 9(2), 183–196 (2001)zbMATHMathSciNetGoogle Scholar
  16. 16.
    De Baets, B., Mesiar, R.: Pseudo-metrics and T-equivalences. J. Fuzzy Math. 5(2), 471–481 (1997)zbMATHMathSciNetGoogle Scholar
  17. 17.
    De Baets, B., Mesiar, R.: T-partitions. Fuzzy Sets and Systems 97, 211–223 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 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. 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. 20.
    Höhle, U., Blanchard, N.: Partial ordering in L-underdeterminate sets. Inform. Sci. 35, 133–144 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)zbMATHGoogle Scholar
  22. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Position paper III: continuous t-norms. Fuzzy Sets and Systems 145, 439–454 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. U.S.A. 8, 535–537 (1942)CrossRefMathSciNetGoogle Scholar
  26. 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. 27.
    Mesiar, R., Saminger, S.: Domination of ordered weighted averaging operators over t-norms. Soft Computing 8, 562–570 (2004)zbMATHCrossRefGoogle Scholar
  28. 28.
    Petry, F.E., Bosc, P.: Fuzzy Databases: Principles and Applications. International Series in Intelligent Technologies. Kluwer Academic Publishers, Boston (1996)zbMATHGoogle Scholar
  29. 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. 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. 31.
    Saminger, S., De Baets, B., De Meyer, H.: On the dominance relation between ordinal sums of conjunctors. Kybernetika 42(3), 337–350 (2006)MathSciNetGoogle Scholar
  32. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 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. 34.
    Sarkoci, P.: Dominance is not transitive on continuous triangular norms. Aequationes Mathematicae (submitted, 2006)Google Scholar
  35. 35.
    Schweizer, B., Sklar, A.: Statistical metric spaces. Pacific J. Math. 10, 313–334 (1960)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Schweizer, B., Sklar, A.: Associative functions and statistical triangle inequalities. Publ. Math. Debrecen 8, 169–186 (1961)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York (1983)zbMATHGoogle Scholar
  38. 38.
    Silvert, W.: Symmetric summation: A class of operations on fuzzy sets. IEEE Trans. Systems Man Cybernet. 9, 657–659 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Tardiff, R.M.: Topologies for probabilistic metric spaces. Pacific J. Math. 65, 233–251 (1976)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Yager, R.R.: On ordered weighted averaging aggregation operators in multicriteria decisionmaking. IEEE Trans. Systems Man Cybernet 18, 183–190 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Yager, R.R., Filev, D.P.: Essentials of Fuzzy Modelling and Control. J. Wiley & Sons, New York (1994)Google Scholar
  42. 42.
    Zadeh, L.A.: Similarity relations and fuzzy orderings. Inform. Sci. 3, 177–200 (1971)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Susanne Saminger
    • 1
  • Ulrich Bodenhofer
    • 2
  • Erich Peter Klement
    • 1
  • Radko Mesiar
    • 3
    • 4
  1. 1.Department of Knowledge-Based Mathematical SystemsJohannes Kepler UniversityLinzAustria
  2. 2.Institute of BioinformaticsJohannes Kepler UniversityLinzAustria
  3. 3.Department of Mathematics and Descriptive GeometryFaculty of Civil Engineering, Slovak University of TechnologyBratislavaSlovakia
  4. 4.Institute for Research and Applications of Fuzzy ModelingUniversity of OstravaOstrava 1Czech Republic

Personalised recommendations