On some types of fuzzy covering-based rough sets

Fuzzy coverings are a natural extension of the coverings by replacing crisp sets with fuzzy sets. Recently, an excellent introduction to the definition of a fuzzy β-covering is due to Ma and two fuzzy covering-based rough set models are presented. In this paper, by introducing some new definitions of fuzzy β-covering approximation spaces, the properties of fuzzy β-covering approximation spaces and Ma's fuzzy covering-based rough set models are studied. Furthermore, three new types of fuzzy covering-based rough set models as generalizations of Ma's models are first proposed in this paper. First, some properties of fuzzy β-covering and its fuzzy β-neighborhood family are proposed. We present a necessary and sufficient condition for fuzzy β-neighborhood family induced by a fuzzy β-covering to be equal to the fuzzy β-covering itself. Then we study the characterizations of Ma's fuzzy covering-based rough set models and give a necessary and sufficient condition for two fuzzy β-coverings to generate the same fuzzy covering lower approximation or the same fuzzy covering upper approximation. Finally, this paper proposes three new types of fuzzy covering-based rough set models by introducing a new notion of a fuzzy complementary β-neighborhood.