Abstract
In this paper smooth aggregation functions on a finite scale are studied and characterized as solutions of a functional equation analogous to the Frank functional equation. The particular cases of quasi-copulas and copulas are also characterized through a similar functional equation. Previous characterizations of these kind of operations through special matrices are used jointly with the new ones to derive some invariant properties on quasi-copulas and copulas on finite scales.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aguiló, I., Suñer, J., Torrens, J.: Matrix representation of discrete quasi-copulas. Fuzzy Sets and Systems 159, 1658–1672 (2008)
Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practicioners. Springer, Berlin (2007)
Calvo, T., Mayor, G., Mesiar, R. (eds.): Aggregation operators. New trends and applications, Studies in Fuzziness and Soft Computing, vol. 97. Physica-Verlag, Heidelberg (2002)
De Baets, B., Fodor, J., Ruiz-Aguilera, D., Torrens, J.: Idempotent uninorms on finite ordinal scales. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17, 1–14 (2009)
Fodor, J.C.: Smooth associative operations on finite ordinal scales. IEEE Trans. on Fuzzy Systems 8, 791–795 (2000)
Frank, M.J.: On the simultaneous associativity of F(x,y) and x + y - F(x,y). Aequationes Math. 19, 194–226 (1979)
Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E.: Aggregation functions. Encyclopedia of Mathematics and its Applications, vol. 127. Cambridge University Press, Cambridge (2009)
Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Kluwer Academic Publishers, London (2000)
Klement, E.P., Mesiar, R., Pap, E.: Invariant copulas. Kybernetika 38, 275–285 (2002)
Kolesárová, A.: 1-Lipschitz aggregation operators and quasi-copulas. Kybernetika 39, 615–629 (2003)
Kolesárová, A., Mayor, G., Mesiar, R.: Weighted ordinal means. Information Sciences 177, 3822–3830 (2007)
Mas, M., Mayor, G., Torrens, J.: t-Operators and uninorms on a finite totally ordered set. International Journal of Intelligent Systems 14, 909–922 (1999)
Mas, M., Monserrat, M., Torrens, J.: On left and right uninorms on a finite chain. Fuzzy Sets and Systems 146, 3–17 (2004)
Mayor, G., Suñer, J., Torrens, J.: Copula-like operations on finite settings. IEEE Transactions on Fuzzy Systems 13, 468–477 (2005)
Mayor, G., Torrens, J.: Triangular norms in discrete settings. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, pp. 189–230. Elsevier, Amsterdam (2005)
Robbins, D.P., Rumsey, H.: Determinants and alternating-sign matrices. Advances in Math. 62, 169–184 (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mas, M., Monserrat, M., Torrens, J. (2010). Smooth Aggregation Functions on Finite Scales. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds) Computational Intelligence for Knowledge-Based Systems Design. IPMU 2010. Lecture Notes in Computer Science(), vol 6178. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14049-5_41
Download citation
DOI: https://doi.org/10.1007/978-3-642-14049-5_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14048-8
Online ISBN: 978-3-642-14049-5
eBook Packages: Computer ScienceComputer Science (R0)