Cauchy-like functional equation based on a class of uninorms

Commuting is an important property in any two-step information merging procedure where the results should not depend on the order in which the single steps are performed. In the case of bisymmetric aggregation operators with the neutral elements, Saminger, Mesiar and Dubois, already reduced characterization of commuting n-ary operators to resolving the unary distributive functional equations, but only some sufficient conditions of unary functions distributive over two particular classes of uninorms are given out. Along this way of thinking, in this paper, we will investigate and fully characterize the following functional equation f(U(x, y)) = U(f(x), f(y)), where f : [0,1] → [0,1] is an unknown function, a uninorm U ε U_min has a continuous underlying t-norm T_U and a continuous underlying t-conorm S_U- Our investigation shows the key point is a transformation from this functional equation to the several known ones. Moreover, this equation has non-monotone solutions different completely with those obtained ones.