On some types of reduction for families of fuzzy sets

A covering is a family of sets. This paper studies the generalization of rough set models via a generalization of the concept of neighborhoods when considering a general family of sets instead of a partition of a set. Given the notion of neighborhood, we propose four different rough set models based on a family of sets and study the reduction problem for such sets. The reduction relationship between rough sets and formal contexts is established. Three reduction algorithms are obtained and applied to identify all reducts. We extend these results to fuzzy sets. Using a family of fuzzy sets to replace a fuzzy covering or a fuzzy β covering, we define set-based fuzzy rough sets and consider three types of reduction. Finally, we give reduction algorithms that are required to identify all reducts.