On phi-fuzzy rough sets: Representation, special cases and induced topology

Consider L = (L, *, 1) be a complete residuated lattice. For an arbitrary function. defined from the set of all singletons to that of all L-sets, Chen and Li introduced a type of L-fuzzy rough sets in 2007, called phi-fuzzy rough sets in this paper. The well-known R-fuzzy rough sets, where R is an L-fuzzy relation, can be regarded as a special phi-fuzzy rough sets. In this paper, we prove that phi-fuzzy rough sets can be represented by a family of R-fuzzy rough sets. Then we define some special phi of being serial, reflexive, symmetric, transitive and Euclidean, and discuss the corresponding phi-fuzzy rough sets, respectively. At last, we study the induced L-topology by phi when it is reflexive or reflexive and transitive.