Residuated Lattice Fuzzy Filter And Generalized Fuzzy Filters,

In this paper, the properties of fuzzy filters in the residuated lattices are investigated, and some equivalent conditions and conclusions are derived. On this basis, it is discussed that how to generate a fuzzy filter by using of a fuzzy set. Then, some properties of fuzzy positive implicative filters and fuzzy Boolean filters are obtained. The relation between them is investigated. The concept of fuzzy congruence relations is introduced. There is a bijection between the set of fuzzy filters and the set of fuzzy congruence relations. We prove that the fuzzy quotient sets induced by fuzzy filters are still residuated lattices, and the fuzzy quotient sets induced by fuzzy Boolean filters are Boolean algebra. Finally, the notions of "belong" and "quasicoincidence" are extended to residuated lattices. The concepts of (∈,∈vq)-fuzzy filter and (∈,∈vq)-fuzzy positive implicative filter in residuated lattices are introduced. The characterizations of these generalized fuzzy filters are derived, and the relations among these generalized fuzzy filters are discussed.