Abstract
A fuzzy covering is a natural generalization of coverings. This paper shows that the families of all definable fuzzy sets induced by a fuzzy covering-based rough set have lattice structures. There may be some superfluous elements in the universal set and the fuzzy covering for the families of all definable sets. We define class reduction, element reduction, and bireduction to remove these elements respectively. For the family of all lower definable fuzzy sets, it is shown that the class reduction coincides with covering reduction, and the element reduction can be transformed into a reduction in information systems. Bireduction considers class reduction and element reduction, and it can be achieved by class and element reductions. Moreover, our results also hold for coverings because a covering is a special fuzzy covering.
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