On the distributivity of fuzzy implications over continuous and Archimedean triangular conorms

Recently, we have examined the solutions of the following distributive functional equation I(x,S1(y,z))=S2(I(x,y),I(x,z)), when S1, S2 are either both strict or nilpotent t-conorms and I is an unknown function. In particular, between these solutions, we have presented functions which are fuzzy implications. In this paper we continue these investigations for the situation when S1, S2 are continuous and Archimedean t-conorms, i.e., we consider in detail the situation when S1 is a strict t-conorm and S2 is a nilpotent t-conorm and vice versa. Towards this end, we firstly present solutions of two functional equations related to the additive Cauchy functional equation. Using obtained results we show that the above distributive equation does not hold when S1, S2 are continuous and Archimedean t-conorms and I is a continuous fuzzy implication. Further, we present the solutions I which are non-continuous fuzzy implications. Obtained results are not only theoretical but also useful for the practical problems, since such equations have an important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems.