On the distributive equation of implication based on a continuous t-norm and a continuous Archimedean t-conorm

In order to avoid combinatorial rule explosion in fuzzy reasoning, in this work we explore the distributive equation of implication I(T(x, y), z) = S(I(x, z), I(y, z)). In detail, by means of the sections of I, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication I(T (x, y), z) = S(I(x, z), I(y, z)), when T is a continuous but not Archimedean triangular norm, S is a continuous and Archimedean triangular conorm and I is an unknown function. This obtained characterizations indicate that there are no continuous solutions, for the previous functional equation, satisfying the boundary conditions of implications. However, under the assumptions that I is continuous except the point (1, 1), we get its complete characterizations.