Characterization of a Class of Fuzzy Implications Satisfying the Law of Importation With Respect to Uninorms With Continuous Underlying Operators

The law of importation, given by the equality (x^y)→z ≈ (x→(y→z)), is a tautology in classical logic and has been proved to be widely used in approximate reasoning and image processing. Some open problems of fuzzy implication dealing with the law of importation were suggested on 8th international conference on Fuzzy Set Theory and Applications (FSTA 2006). In this paper, we partially solve one open problem associated with this property. Specifically, we mainly devote ourselves to solving the general form of the law of importation I(U(x, y), z) = I(x, I(y, z)), where I is a fuzzy implication and U is a conjunctive uninorm with a continuous underlying tnorm and a continuous underlying t-conorm. Along this study, given a fixed uninorm with continuous underlying operators, all fuzzy implications that satisfy the law of importation with respect to this uninorm, and having an α-section that is a continuous negation, are characterized.