On the distributivity of fuzzy implications over continuous archimedean triangular norms

Authors:
Michał Baczyński
Affiliations:
Institute of Mathematics, University of Silesia, Katowice, Poland
Venue:
ICAISC'10 Proceedings of the 10th international conference on Artificial intelligence and soft computing: Part I
Year:
2010

Citing 14
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Abstract

Recently, we have examined solutions of the following distributive functional equation I(x, S1(y, z)) = S2(I(x, y), I(x, z)), when S1, S2 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, T1(y, z)) = T2(I(x, y), I(x, z)), when T1, T2 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 T1, T2 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.