Recently, some authors studied the distributive equations for continuous t-norms and some families of usual classes of uninorms over overlap functions in Refs. 25 and 32, but lacked a complete characterization of the distributivity on idempotent uninorms and overlap or grouping functions widely used in image processing. As a supplement to the previous results, in this article we fully characterize the distributivity equations of idempotent uninorms over these two functions by virtue of the associated functions of idempotent uninorms. Moreover, we also discuss the distributivity conditions of above two special functions over idempotent uninorms, yet find in this case that the associated functions of idempotent uninorms must satisfy particular conditions and those two functions usually are a class of special aggregation functions with a constant domain whose value equals to the neutral element of the idempotent uninorm.

The law of importation, given by the equality$(x\wedge y) \longrightarrow z\equiv (x\longrightarrow (y\longrightarrow z))$, 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 article, 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 t-norm 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$\alpha$-section that is a continuous negation, are characterized.

Our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory and focused also by Klement in the Linz Seminar 2000 closing session. This paper is mainly devoted to solving the functional equations of conditional distributivity of U1over U2, where both U1and U2are uninorms with continuous underlying operators. Then, the almost complete characterization of such a pair (U1, U2) of uninorms is given except for only a small fraction of the unit square. Next, we give the complete characterization of the previous pair under some appropriate restriction, that is, the second operator U2∈ Umax∪ Umin. Finally, we also construct a counterexample to illustrate that the obtained results are necessary but not sufficient for general cases with unlimited condition.