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

Abstract 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.