Distributivity and Conditional Distributivity for Uninorms With Continuous Underlying Operators Over a Given Continuous t-Norm

This article focuses on distributivity and conditional distributivity for uninorms over a given continuous t-norm, closely related to the open problem recalled by Klement in the Linz2000 closing session. This problem has been solved for the well-known classes of uninorms except the class of uninorms with the continuous underlying operators. In this article, the complete characterizations of uninorms with continuous underlying operators (conditionally) distributive over a given continuous t-norm are presented. From the obtained results, we deduce that conditional distributivity is equivalent to distributivity for a pair of such operators.