On the distributivity of fuzzy implications over representable uninorms

摘要

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, v), I(x, z))and I(U_1(x, y), z)= U_2(I(x, z), I(y, z)), where U_1, U_2 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 U_1,U_2 are both representable uninorms and / 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 U_1, U_2 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 (-∞) + ∞.