On the distributive equation for t-representable t-norms generated from nilpotent and strict t-norms

Abstract

Recently, in [4], we have discussed the following distributive equation of implications I(x, T1(y, z)) = T2(I(x, y), I(x, z)) over t-representable t-norms, generated from strict t-norms, in interval-valued fuzzy sets theory. In this work we continue these investigations, but with the assumption that T1 is generated from nilpotent t-norms, while T2 is generated from strict t-norms. As a byproduct result we show all solutions for the following functional equation f(min(u1 +v1, a),min(u2 +v2, a)) = f(u1, u2) + f(v1, v2) related to this case.