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.
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