Conditional Distributivity Equation for Uninorms With Continuous Underlying Operators

Abstract

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 U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> over U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , where both U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> are uninorms with continuous underlying operators. Then, the almost complete characterization of such a pair (U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) 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 U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ∈ U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> ∪ U <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> . Finally, we also construct a counterexample to illustrate that the obtained results are necessary but not sufficient for general cases with unlimited condition.