Redefined fuzzy implicative filters

Using the belongs to relation (∈) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (θ, ψ)-fuzzy implicative filters where θ, ψ are any two of {∈, q, ∈ ∨ q, ∈ ∧ q} with θ ≠ ∈ ∧ q is introduced, and related properties are discussed. Relations between (∈ ∨ q, ∈ ∨ q)-fuzzy implicative filters and (∈, ∈ ∨ q)-fuzzy implicative filters are investigated, and conditions for an (∈, ∈ ∨ q)-fuzzy implicative filter to be an (∈, ∈)-fuzzy implicative filter are provided. Characterizations of (∈, ∈ ∨ q)-fuzzy implicative filters are given, and conditions for a fuzzy set to be a (q, ∈ ∨ q)-fuzzy implicative filter are provided.