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Explicit Descriptions of Associative Sugeno Integrals

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

Abstract

The associativity property, usually defined for binary functions, can be generalized to functions of a given fixed arity n ≥ 1 as well as to functions of multiple arities. In this paper, we investigate these two generalizations in the case of Sugeno integrals over bounded distributive lattices and present explicit descriptions of the corresponding associative functions. We also show that, in this case, both generalizations of associativity are essentially the same.

Keywords

  • bounded distributive lattice
  • Sugeno integral
  • associativity
  • idempotency
  • functional equation

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References

  1. Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Studies in Fuziness and Soft Computing. Springer, Berlin (2007)

    MATH  Google Scholar 

  2. Couceiro, M.: On the Lattice of Equational Classes of Boolean Functions and its Closed Intervals. J. Mult.-Valued Logic Soft Comput. 14(1-2), 81–104 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Couceiro, M., Marichal, J.-L.: Polynomial Functions over Bounded Distributive Lattices, http://arxiv.org/abs/0901.4888

  4. Couceiro, M., Marichal, J.-L.: Characterizations of Discrete Sugeno Integrals as Polynomial Functions over Distributive Lattices. Fuzzy Sets and Systems 161, 694–707 (2009)

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Couceiro, M., Marichal, J.-L.: Associative Polynomial Functions over Bounded Distributive Lattices. Order (to appear), http://arxiv.org/abs/0902.2323

  6. Dörnte, W.: Untersuchengen über einen Verallgemeinerten Gruppenbegriff. Math. Z. 29, 1–19 (1928)

    CrossRef  MATH  Google Scholar 

  7. Dudek, W.A.: Varieties of Polyadic Groups. Filomat 9, 657–674 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Dudek, W.A.: On Some Old and New Problems in n-ary Groups. Quasigroups and Related Systems 8, 15–36 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Dudek, W.A., Glazek, K., Gleichgewicht, B.: A Note on the Axiom of n-Groups. Coll. Math. Soc. J. Bolyai 29, 195–202 (1977)

    MathSciNet  Google Scholar 

  10. Fodor, J.C.: An Extension of Fung-Fu’s Theorem. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 4(3), 235–243 (1996)

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Glazek, K.: Bibliography of n-Groups (Polyadic Groups) and Some Group-Like n-ary Systems. In: Proc. of the Symposium on n-ary Structures, Macedonian Academy of Sciences and Arts, Skopje, pp. 253–289 (1982)

    Google Scholar 

  12. Glazek, K., Gleichgewicht, B.: Remarks on n-Groups as Abstract Algebras. Colloq. Math. 17, 209–219 (1967)

    MathSciNet  MATH  Google Scholar 

  13. Goodstein, R.L.: The Solution of Equations in a Lattice. Proc. Roy. Soc. Edinburgh Sect. A 67, 231–242 (1965/1967)

    MathSciNet  MATH  Google Scholar 

  14. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Encyclopedia of Mathematics and Its Applications, vol. 127. Cambridge University Press, Cambridge (2009)

    CrossRef  MATH  Google Scholar 

  15. Hosszú, M.: On the Explicit Form of n-Group Operations. Publ. Math. Debrecen 10, 88–92 (1963)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  17. Marichal, J.-L.: Aggregation Operators for Multicriteria Decision Aid. Ph.D. thesis, Institute of Mathematics, University of Liège, Liège, Belgium (December 1998)

    Google Scholar 

  18. Marichal, J.-L.: Weighted Lattice Polynomials. Discrete Math. 309(4), 814–820 (2009)

    MathSciNet  CrossRef  MATH  Google Scholar 

  19. Monk, J.D., Sioson, F.M.: On the General Theory of m-Groups. Fund. Math. 72, 233–244 (1971)

    MathSciNet  MATH  Google Scholar 

  20. Post, E.L.: Polyadic Groups. Trans. Amer. Math. Soc. 48, 208–350 (1940)

    MathSciNet  CrossRef  MATH  Google Scholar 

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Couceiro, M., Marichal, JL. (2010). Explicit Descriptions of Associative Sugeno Integrals. 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_48

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  • DOI: https://doi.org/10.1007/978-3-642-14055-6_48

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14054-9

  • Online ISBN: 978-3-642-14055-6

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