Distributivity equations of extended aggregation functions in type-2 fuzzy sets

In this paper, we completely characterize the distributivity between the extended conjunctive aggregation functions and the extended disjunctive aggregation functions. We give two necessary and sufficient conditions for the distributivity of the extended aggregation functions and the convexity of fuzzy truth values. Most of the current research focuses on the distributivity of the extended t-norm (resp. t-conorm), overlap function (resp. grouping function) and uninorm (resp. nullnorm), and so on. It is well-known that a conjunctive (resp. disjunctive) aggregation function is a more general aggregation function than a t-norm (resp. t-conorm). Therefore, the obtained results show that the distributivity equation is more general than the previous ones.