Cauchy-like functional equations for uninorms continuous in (0, 1)2.

Commutativity is an important property in two-step information merging procedure. It is shown that the result obtained from the procedure should not depend on the order in which signal steps are performed. In the case of a bisymmetric aggregation operator with the neutral element, Saminger et al. have provided a full characterization of commutative n-ary operator by means of unary distributive functions. Further, characterizations of these unary distributive functions can be viewed as resolving a kind of the Cauchy-like equations f(x⊕y)=f(x)⊕f(y), where f:[0,1]→[0,1] is a monotone function, ⊕ is a bisymmetric aggregation operator with the neutral element. In this paper, we are still devoted to investigating and fully characterizing the Cauchy-like equation f(U(x,y))=U(f(x),f(y)), where f:[0,1]→[0,1] is an unknown function but not necessarily monotone, U is a uninorm continuous in (0,1)2. These results show the key technology is how to find a transformation from this equation into several known cases. Moreover, this equation has completely different and non-monotone solutions in comparison with the obtained results.