Conditional Distributivity Equation for Uninorms With Continuous Underlying Operators

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 U₁ over U₂, where both U₁ and U₂ are 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 ∈ U_max ∪ U_min. Finally, we also construct a counterexample to illustrate that the obtained results are necessary but not sufficient for general cases with unlimited condition.