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fuzzy connectives; distributivity functional equations; T-norms; T-conorms; 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. And then the full characterizations of these equations are obtained under the assumption that the unary function is non-decreasing and distributive over special aggregation operators, for examples, continuous t-norms, continuous t-conorms and two classes of uninorms. Along this way of thinking, in this paper, we will investigate and fully characterize the following unary distributive functional equation $f(U(x,y))=U(f(x),f(y))$, where $f\colon[0,1]\rightarrow[0,1]$ is an unknown function but unnecessarily non-decreasing, a uninorm $U\in{\mathcal U}_{\min}$ has a continuously underlying t-norm $T_U$ and a continuously underlying t-conorm $S_U$. Our investigation shows that the key point is a transformation from this functional equation to the several known ones. Moreover, this equation has also non-monotone solutions completely different with already obtained ones. Finally, our results extend the previous ones about the Cauchy-like equation $f(A(x,y))=B(f(x),f(y))$, where $A$ and $B$ are some continuous t-norm or t-conorm.
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