On the distributivity of fuzzy implications over representable uninorms

Recently, many works have appeared dealing with the distributivity of fuzzy implications over t-norms, t-conorms and uninorms. These equations have a very important role to play in efficient inferencing in approximate reasoning, especially fuzzy control systems. In this work we deal with the following two distributive equations I ( x , U 1 ( y , z ) ) = U 2 ( I ( x , y ) , I ( x , z ) ) and I ( U 1 ( x , y ) , z ) = U 2 ( I ( x , z ) , I ( y , z ) ) , where U1, U2 are given representable uninorms and I is an unknown function, in particular a fuzzy implication. Using the obtained characterizations we show that the first equation does not hold when U1, U2 are both representable uninorms and I is a continuous fuzzy implication. Therefore, we present solutions of the first equation which are fuzzy implications with continuous vertical sections. For the second equation the situation is quite different—when U1, U2 are both representable uninorms, then there are no solutions which are fuzzy implications. As a byproduct result we have obtained solutions of the additive Cauchy functional equation for an unknown function f : [ − ∞ , ∞ ] → [ − ∞ , ∞ ] with different meaning of additions ∞ + ( − ∞ ) and ( − ∞ ) + ∞ .