New Results on the Distributive Laws of Uninorms Over Overlap Functions
Hui Liu, Bin Zhao
Abstract
Recently, Qiao studied the distributive laws of uninorms over overlap functions when the uninorms were one of the usual classes <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> , the family of representable uninorms or uninorms continuous in $(0,1)^{2}$ and obtained some partially characterizations. This article will continue to consider the characterizations of this kind of distributivity equations when the uninorms lie in the family of uninorms continuous in $(0,1)^{2}$. First of all, we investigate the distributive laws of continuous t-norms over overlap functions and give their full characterizations. Then, we give the necessary and sufficient conditions for the solutions of the distributivity equations when uninorms are continuous in $(0,1)^{2}$.