Abstract
In this paper the characterization of idempotent uninorms given in [21] is revisited and some technical aspects are corrected. Examples clarifying the situation are given and the same characterization is translated in terms of symmetrical functions. The particular cases of left-continuity and right-continuity are studied retrieving the results in [7].
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References
Benítez, J.M., Castro, J.L., Requena, I.: Are artificial neural networks black boxes? IEEE Transactions on Neural Networks 8, 1156–1163 (1997)
Benvenuti, P., Mesiar, R.: Pseudo-arithmetical operations as a basis for the general measure and integration theory. Information Sciences 160, 1–11 (2004)
Bustince, H., Mohedano, V., Barrenechea, E., Pagola, M.: Definition and construction of fuzzy DI-subsethood measures. Information Sciences 176, 3190–3231 (2006)
Bustince, H., Pagola, M., Barrenechea, E.: Construction of fuzzy indices from fuzzy DI-subsethood measures: application to the global comparison of images. Information Sciences 177, 906–929 (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)
Czogala, E., Drewniak, J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets and Systems 12, 249–269 (1984)
De Baets, B.: Idempotent uninorms. European J. Oper. Res. 118, 631–642 (1999)
De Baets, B., Fodor, J.: Van Melle’s combining function in MYCIN is a representable uninorm: an alternative proof. Fuzzy Sets and Systems 104, 133–136 (1999)
De Baets, B., Fodor, J.C.: Residual operators of uninorms. Soft Computing 3, 89–100 (1999)
De Baets, B., Fodor, J., Ruiz-Aguilera, D., Torrens, J.: Idempotent uninorms on finite ordinal scales. Int. J. of Uncertainty, Fuzziness and Knowledge-based Systems 17, 1–14 (2009)
De Baets, B., Kwasnikowska, N., Kerre, E.: Fuzzy morphology based on uninorms. In: Seventh IFSA World Congress, Prague, pp. 215–220 (1997)
Drygaś, P.: Discussion of the structure of uninorms. Kybernetika 41, 213–226 (2005)
Drygaś, P.: Remarks about idempotent uninorms. Journal of Electrical Engineering 57, 92–94 (2006)
Fodor, J.C., Yager, R.R., Rybalov, A.: Structure of Uninorms. Int. J. Uncertainty, Fuzziness, Knowledge-based Systems 5, 411–427 (1997)
Gabbay, D., Metcalfe, G.: Fuzzy logics based on [0,1)-continuous uninorms. Arch. Math. Logic 46, 425–449 (2007)
González, M., Mir, A.: Noise Reduction Using Alternate Filters Generated by Fuzzy Mathematical Operators Using Uninorms (ΦMM-U Morphology). In: Proceedings of EUROFUSE 2009, Pamplona, pp. 233–238 (2009)
González, M., Mir, A., Ruiz-Aguilera, D., Torrens, J.: Edge-Images using a Uninorm-Based Fuzzy Mathematical Morphology. Opening and Closing. In: Tavares, J., Jorge, N. (eds.) Advances in Computational Vision and Medical Image Processing, pp. 137–157. Springer, Netherlands (2009)
Klement, E.P., Mesiar, R., Pap, E.: Integration with respect to decomposable measures, based on a conditionally distributive semiring on the unit interval. Int. J. Uncertainty, Fuzziness, Knowledge-Based Systems 8, 701–717 (2000)
Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Kluwer Academic Publishers, London (2000)
Maes, K.C., De Baets, B.: Orthosymmetrical monotone functions. Bulletin of the Belgian Mathematical Society - Simon Stevin 14, 99–116 (2007)
Martín, J., Mayor, G., Torrens, J.: On locally internal monotonic operations. Fuzzy Sets and Systems 137, 27–42 (2003)
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)
Pap, E.: Pseudo-additive measures and their applications. In: Pap, E. (ed.) A Handbook of Measure Theory, pp. 1403–1468. Elsevier, Amsterdam (2002)
Ruiz-Aguilera, D., Torrens, J.: Distributive idempotent uninorms. Int. J. Uncertainty, Fuzziness, Knowledge-based Systems 11, 413–428 (2003)
Ruiz-Aguilera, D., Torrens, J.: Residual implications and co-implications from idempotent uninorms. Kybernetika 40, 21–38 (2004)
Takács, M.: Approximate reasoning in fuzzy systems based on pseudo-analysis and uninorm residuum. Acta Polytechnica Ungarica 1, 49–62 (2004)
Yager, R.R.: Uninorms in fuzzy systems modeling. Fuzzy Sets and Systems 122, 167–175 (2001)
Yager, R.R., Kreinovich, V.: On the relation between two approaches to combining evidence: Ordered abelian groups and uninorms. J. Intell. Fuzzy Syst. 14, 7–12 (2003)
Yager, R.R., Kreinovich, V.: Universal approximation theorem for uninorm-based fuzzy systems modelling. Fuzzy Sets and Systems 140, 331–339 (2003)
Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets and Systems 80, 111–120 (1996)
Yan, P., Chen, G.: Discovering a cover set of ARsi with hierarchy from quantitative databases. Information Sciences 173, 319–336 (2005)
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Ruiz-Aguilera, D., Torrens, J., De Baets, B., Fodor, J. (2010). Some Remarks on the Characterization of Idempotent Uninorms. 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_44
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DOI: https://doi.org/10.1007/978-3-642-14049-5_44
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