Abstract All fuzzy covering-based rough set models are constructed under a corresponding fuzzy covering approximation space (FCAS). The fuzzy β-covering approximation space (β-FCAS) is a generalization of the FCAS through replacing the value 1 with a parameter β. In other words, the β-FCAS is the basis of studying fuzzy covering-based rough sets and their applications. Therefore, it is necessary to study some questions in the fuzzy covering approximation space, such as problems of reduction, relationships among some basic concepts and relationships between two fuzzy covering approximation spaces. In this article, we investigate the questions in the β-FCAS further. Firstly, the definitions of I-irreducible element and I-reduct are presented, which can be seen as the complement of the existing notions. Then, relationships among some concepts in the β-FCAS are investigated, such as the relationship between fuzzy β-minimal description and β-reduct, and the relationship between fuzzy β-covering and its I-reduct. Thirdly, inspired by some concepts in the β-FCAS, we present some new concepts between two β-FCASs and their properties. Based on these new concepts, we present some conditions such that two fuzzy β-coverings have the same reduct (or I-reduct). Finally, based on the results above, seven derived β-FCASs are investigated further and a corresponding lattice of them is proposed.

中文翻译：

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模糊 β 覆盖近似空间

摘要 所有基于模糊覆盖的粗糙集模型都是在相应的模糊覆盖逼近空间（FCAS）下构建的。模糊 β 覆盖近似空间 (β-FCAS) 是 FCAS 通过用参数 β 替换值 1 的泛化。换句话说，β-FCAS 是研究基于模糊覆盖的粗糙集及其应用的基础。因此，有必要研究模糊覆盖近似空间中的一些问题，如约简问题、一些基本概念之间的关系以及两个模糊覆盖近似空间之间的关系。在本文中，我们进一步研究了 β-FCAS 中的问题。首先给出I-不可约元和I-约元的定义，可以看作是对现有概念的补充。然后，研究了β-FCAS中一些概念之间的关系，如模糊β-极小描述与β-约简的关系，模糊β-覆盖与其I-约简的关系。第三，受 β-FCAS 中一些概念的启发，我们在两个 β-FCAS 及其特性之间提出了一些新概念。基于这些新概念，我们提出了一些条件，使得两个模糊 β 覆盖具有相同的归约（或 I 归约）。最后，基于上述结果，进一步研究了七个派生的 β-FCAS，并提出了相应的晶格。我们提出了两个 β-FCAS 及其特性之间的一些新概念。基于这些新概念，我们提出了一些条件，使得两个模糊 β 覆盖具有相同的归约（或 I 归约）。最后，基于上述结果，进一步研究了七个派生的 β-FCAS，并提出了相应的晶格。我们提出了两个 β-FCAS 及其特性之间的一些新概念。基于这些新概念，我们提出了一些条件，使得两个模糊 β 覆盖具有相同的归约（或 I 归约）。最后，基于上述结果，进一步研究了七个派生的 β-FCAS，并提出了相应的格子。