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Triangular Norms and Conorms on the Set of Discrete Fuzzy Numbers

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Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Methods (IPMU 2010)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 80))

Abstract

In this paper, we study different methods to construct triangular operations (t-norms and t-conorms) on the bounded distributive lattice \(\mathcal{A}_1^{L}\), of discrete fuzzy numbers whose support is a subset of consecutive natural numbers on a finite chain L of consecutive natural numbers. Moreover, we propose a method to compare two t-norms (t-conorms) defined on \(\mathcal{A}_1^{L}\).

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References

  1. De Baets, B., Mesiar, R.: Triangular norms on product lattices. Fuzzy Sets and Systems 104, 61–75 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Casasnovas, J., Riera, J.V.: On the addition of discrete fuzzy numbers. WSEAS Transactions on Mathematics, 549–554 (2006)

    Google Scholar 

  3. Casasnovas, J., Riera, J.V.: Discrete fuzzy numbers defined on a subset of natural numbers. In: Theoretical Advances and Applications of Fuzzy Logic and Soft Computing: Advances in Soft Computing, vol. 42, pp. 573–582. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  4. Casasnovas, J., Riera, J.V.: Maximum and minimum of discrete fuzzy numbers. In: Frontiers in Artificial Intelligence and Applications: artificial intelligence research and development, vol. 163, pp. 273–280. IOS Press, Amsterdam (2007)

    Google Scholar 

  5. Casasnovas, J., Riera, J.V.: Lattice properties of discrete fuzzy numbers under extended min and max. In: Proceedings IFSA-EUSFLAT, Lisbon, pp. 647–652 (2009)

    Google Scholar 

  6. Casasnovas, J., Riera, J.V.: Extension of discrete t-norms and t-conorms to discrete fuzzy numbers. In: Fifth international summer school on aggregation operators (AGOP 2009), Palma de Mallorca, pp. 77–82 (2009)

    Google Scholar 

  7. Deschrijver, G., Cornelis, C., Kerre, E.E.: On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Trans. Fuzzy Systems 12, 45–61 (2004)

    Article  MATH  Google Scholar 

  8. Goguen, J.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  10. Klir, G., Bo, Y.: Fuzzy sets and fuzzy logic (Theory and applications). Prentice-Hall, Englewood Cliffs (1995)

    MATH  Google Scholar 

  11. Mayor, G., Torrens, J.: Triangular norms on discrete settings. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, pp. 189–230. Elsevier, Amsterdam (2005)

    Chapter  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Voxman, W.: Canonical representations of discrete fuzzy numbers. Fuzzy Sets and Systems 54, 457–466 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, G., Wu, C., Zhao, C.: Representation and Operations of discrete fuzzy numbers. Southeast Asian Bulletin of Mathematics 28, 1003–1010 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Zhang, D.: Triangular norms on partially ordered sets. Fuzzy Sets and Systems 153, 195–209 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Balearic Islands, 07122, Palma de Mallorca, Spain

    Jaume Casasnovas & J. Vicente Riera

Authors
  1. Jaume Casasnovas
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  2. J. Vicente Riera
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Editor information

Editors and Affiliations

  1. Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Eyke Hüllermeier

  2. Department of Knowledge Processing and Language Engineering, Otto-von-Guericke University of Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Rudolf Kruse

  3. Fakultät für Elektrotechnik und Informationstechnik, Technische Universität Dortmund, 44221, Dortmund, Germany

    Frank Hoffmann

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Casasnovas, J., Riera, J.V. (2010). Triangular Norms and Conorms on the Set of Discrete Fuzzy Numbers. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Methods. IPMU 2010. Communications in Computer and Information Science, vol 80. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14055-6_72

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  • DOI: https://doi.org/10.1007/978-3-642-14055-6_72

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14054-9

  • Online ISBN: 978-3-642-14055-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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