Conditional Distributivity Of Continuous Triangular Norms Over 2-uninorms

Conditional distributivity of aggregation functions,which has received wide attention from the researchers,is vital for many different fields,for example,integration theory,utility theory and so on.This article is mainly devoted to dealing with the conditional distributivity of continuous t-norms over 2-uninorms,and we give the complete characterization of all pairs(T,H)fulfilling this property.Due to the wide application of conditional distributivity and the importance of aggregation operators,the 2-uninorms considered need not to have two special local neutral elements.As the first step for relaxing the restriction of the local neutral elements,we assume a sensible assumption:the 2-uninorms attached with some appropriate continuity conditions.The obtained results demonstrate that although distributivity and conditional distributivity are two different research topic,the following two perspectives show that they are closely related.One is that if the conditional distributivity equation has a solution,then so has the distributivity equation and vice versa.Another is that if every point in(e(?),1)is an idempotent element of H,then distributivity and conditional distributivity on this topic are equivalent,otherwise,they are not equivalent.