Abstract For a given universal algebra, the set of all its fuzzy subsets is endowed with two structures: a structure of algebra called the fuzzy algebra (or the algebra of fuzzy subsets) and a structure of bounded lattice called the lattice of fuzzy subsets. In this paper, first of all, we construct some subuniverses of this fuzzy algebra and give a way to generate such subuniverses. We characterize the subuniverse generated by a fuzzy point of this fuzzy algebra. Some subalgebras containing the subalgebra generated by a finite set of fuzzy points of the fuzzy algebra are specified. We also describe some subuniverses of the initial universal algebra induced by those of its fuzzy algebra, and vice versa. We then give some properties of fuzzy subalgebras of this universal algebra, describe some fuzzy subalgebras generated by others fuzzy subalgebras, and give a partial characterization of universal algebras in which the set of all fuzzy subalgebras is a subuniverse of their fuzzy algebras. After that, we characterize some properties which can be transfered between the codomain lattice of fuzzy subsets (the lattice of truth values) and the lattice of fuzzy subsets. Later, we show that there exists some subsets of the set of all fuzzy subsets which can be both subuniverses of the fuzzy algebra and sublattices of the lattice of fuzzy subsets: that means, these subsets are endowed with a double structure like the set of all fuzzy subsets of the given universal algebra; and finally, we introduce the residuation of some of them by defining the residual operations in a certain way.

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关于泛代数模糊子集的格和代数

摘要 对于给定的泛代数，其所有模糊子集的集合被赋予两种结构：一种称为模糊代数（或模糊子集代数）的代数结构和一种称为模糊子集格子的有界格结构。在本文中，首先，我们构造了这个模糊代数的一些子宇宙，并给出了生成这些子宇宙的方法。我们刻画了由这个模糊代数的一个模糊点生成的子宇宙。指定了一些包含由模糊代数的有限模糊点集生成的子代数的子代数。我们还描述了初始通用代数的一些子宇宙，由模糊代数的那些子宇宙，反之亦然。然后我们给出了这个泛代数的模糊子代数的一些性质，描述了一些由其他模糊子代数产生的模糊子代数，并给出泛代数的部分表征，其中所有模糊子代数的集合是它们的模糊代数的子宇宙。之后，我们表征了一些可以在模糊子集的共域点阵（真值点阵）和模糊子集点阵之间转移的属性。后来，我们证明了所有模糊子集的集合中存在一些子集，它们既可以是模糊代数的子宇宙，也可以是模糊子集格的子格：这意味着，这些子集被赋予了双重结构，就像所有模糊子集的集合一样。给定通用代数的模糊子集；最后，我们通过以某种方式定义残差操作来介绍其中的一些残差。