Distributivity Conditions of Idempotent Uninorms and Two Special Kinds of Aggregation Functions.

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.