On the Distributivity of Fuzzy Implications over Continuous Archimedean Triangular Norms

Abstract

AbstractRecently, 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.KeywordsFuzzy connectivesFuzzy implicationDistributivity EquationsT-normCombs Methods