Cauchy-Like Functional Equation Based on Continuous T-Conorms and Representable 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, have 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. Indeed, it is very difficult to get the full characterizations of these equations because they are bound up with the famous Cauchy functional equation, which has so far not been completely solved. Along this way of thinking, in this paper, we will respectively investigate and fully characterize the following two functional equations f(U(x,y))=S(f(x), f(y)) and f(S(x,y))=U(f(x),f(y)), where f:[0,1]→ [0,1] is an unknown function, U is a representable uninorm, and S is a continuous t-conorm. These results are an important step toward obtaining complete characterization of the other unary distributive functional equations previously mentioned. Our results show that commuting is suitable for only the second equation but not for the first one. This is because solutions of the first equation are all piecewise constant functions, while those of the second equation are formulas with parameters. In addition, the two equations both have nonmonotone solutions completely different from those ones obtained.