On the Distributivity of Fuzzy Implications over Continuous Archimedean Triangular Norms

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.