Quotient approximation spaces under fuzzy β−coverings

The quotient space is the main research object in the process of transforming a fine-grained space into a coarse-grained space. However, in order to achieve better application results, we are relatively lacking in effective rough operator models to realize the regular approximation transformations between the original space and its quotient space. In response to this situation, we will attempt to adopt a completely new approach to define novel and effective fuzzy β-covering rough set operator models on the original space and its quotient space, thereby establishing a better transformation relationships and related applications. Firstly, we define an induced fuzzy quotient mapping by the natural quotient mapping. This allowed the series of concepts and related conclusions of crisp subsets under the natural mapping to be extended to the fuzzy subsets under the induced fuzzy quotient mapping. Then the quotient fuzzy β-covering approximation space is defined, and novel fuzzy β-covering rough set operator models are built. We not only study their interesting transformation properties, but also compare the new models with the existing ones. Furthermore, a wealth of examples are provided to explain each new concept and demonstrate the simple applications of the main results. More importantly, the profound topological essence and significance revealed lies in that, under the induced fuzzy quotient mapping, the transformations of the lower approximation and the upper approximation exhibit exactly the same behavior as the transformations of the interior and closure under a topological continuous mapping. Finally, the shortcomings of this paper are summarized and the prospects for future research are presented.