Abstract
Recently, we have examined solutions of the following distributive functional equation I(x,S 1(y,z)) = S 2(I(x,y),I(x,z)), when S 1, S 2 are continuous Archimedean t-conorms and I is an unknown function [5,3]. Earlier, in [1,2], we have also discussed solutions of the following distributive equation I(x,T 1(y,z)) = T 2(I(x,y),I(x,z)), when T 1, T 2 are strict t-norms. In particular, in both cases, we have presented solutions which are fuzzy implications in the sense of Fodor and Roubens. In this paper we continue these investigations for the situation when T 1, T 2 are continuous Archimedean t-norms, thus we give a partial answer for one open problem postulated in [2]. Obtained results are not only theoretical – they can be also useful for the practical problems, since such distributive equations have an important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems.
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Baczyński, M. (2010). On the Distributivity of Fuzzy Implications over Continuous Archimedean Triangular Norms. In: Rutkowski, L., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds) Artificial Intelligence and Soft Computing. ICAISC 2010. Lecture Notes in Computer Science(), vol 6113. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13208-7_1
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DOI: https://doi.org/10.1007/978-3-642-13208-7_1
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