LAUSR

In this article, we put forward the concepts of fuzzy soft $$\beta $$β-minimal descriptions and fuzzy soft $$\beta $$β-maximal descriptions and construct four types of fuzzy soft $$\beta $$β-neighborhoods. Secondly, we define five kinds of fuzzy soft $$\beta $$β-coverings-based fuzzy rough sets and investigate the relationships among them. Furthermore, we investigate under what conditions two different fuzzy soft $$\beta $$β-coverings induce the same lower (upper) approximation operators. Then, we introduce the concepts of intersection and union reducible elements. Finally, we put forward the algorithms with respect to the first and the fifth types of fuzzy soft $$\beta $$β-coverings-based fuzzy rough sets, respectively. Through comparison, we find that although these two models are different, the obtained results are the same and the complexity of the algorithm based on the fifth type model is easier than the one based on the first model.