Some functional equations connected to the distributivity laws for fuzzy implications and triangular conorms

Recently in some considerations connected with the distributivity laws of fuzzy implications over triangular norms and conorms, the following functional equation appeared ƒ(min(x + y, a)) = min(ƒ(x) + ƒ(y), b), (1) where a; b are finite or infinite nonnegative constants (see [1]). In [2] we considered a generalized version of this equation in the case when both a and b are finite, namely the equation ƒ(m1(x + y)) = m2(ƒ(x) + ƒ(y)), where m1, m2 are functions defined on some finite intervals of ℝ satisfying additional assumptions. In this article we enhance the results from [2], [3] and consider generalized versions of the equation (1) in the cases when a or b is infinite. We show that some well known solutions of several functional equations, that we presented earlier in [1], [4], can be obtained as corollaries of these new facts.