The Distributive Laws of Convolution Operations Over Meet-Convolution and Join-Convolution on Fuzzy Truth Values

Type-2 fuzzy sets (T2FSs) were introduced by Zadeh in 1975 as an extension of type-1 fuzzy sets (T1FSs). The membership grades of T2FSs are T1FSs in interval [0, 1], which are referred to as fuzzy truth values (that is, functions from [0, 1] to [0, 1]). Recently, (convolution) operations on T2FSs have become a hot research topic, such as type-2 t-norms, type-2 aggregation operations, type-2 implications, and so on. So, the distributive laws between these convolution operations have become an interesting and natural research area. In this article, we investigate the distributive laws of convolution operations over meet-convolution and join-convolution (two basic operations on fuzzy truth values), respectively. At first, we consider whether the distributive laws hold in various subsets of the set of all fuzzy truth values, such as set-valued T2FSs, singletons, interval-valued T2FSs, and the set of all normal convex fuzzy truth values. Then, we discuss whether the distributive laws hold for some specific convolution operations. Finally, we give a necessary and sufficient condition under which join-convolution is distributive over meet-convolution, and similarly give a necessary and sufficient condition under which meet-convolution is distributive over join-convolution.