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.

中文翻译：

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模糊真值的满足卷积和联合卷积的卷积运算的分配律


2 型模糊集 (T2FS) 由 Zadeh 于 1975 年引入，作为 1 型模糊集 (T1FS) 的扩展。 T2FS的隶属度等级是区间[0, 1]内的T1FS，称为模糊真值（即从[0, 1]到[0, 1]的函数）。最近，T2FS 上的（卷积）操作已成为研究热点，例如 2 型 t 范数、2 型聚合操作、2 型含义等。因此，这些卷积运算之间的分配律已成为一个有趣且自然的研究领域。在本文中，我们分别研究了满足卷积和连接卷积（模糊真值的两种基本运算）上的卷积运算的分布规律。首先，我们考虑分配律是否在所有模糊真值集合的各个子集中成立，例如集值 T2F​​S、单例、区间值 T2F​​S 以及所有正态凸模糊真值集合。然后，我们讨论分配律对于某些特定的卷积运算是否成立。最后，我们给出了join-convolution相对于mete-convolution具有分布性的充要条件，同样给出了mete-convolution相对于join-convolution具有分布性的充要条件。