New Types of Fuzzy Filter on Lattice Implication Algebras

Abstract Extending the belongs to (∈) relation and quasi-coincidence with(q) relation between fuzzy points and a fuzzy subsets, the concept of (α, β)-fuzzy filters and (α, β)-fuzzy filters of lattice implication algebras are introduced, where α, β ∈ {∈h , qδ, ∈h ∨qδ, ∈h ∧qδ}, α, β ∈ {∈h, qδ, ∈h ∨ qδ, ∈h ∧ qδ} but α ∈h ∧qδ, α ∈h ∧ qδ, respectively, and some related properties are investigated. Some equivalent characterizations of these generalized fuzzy filters are derived. Finally, the relations among these generalized fuzzy filters are discussed. Special attention to (∈h, ∈h ∨qδ)-fuzzy filter and (∈h, ∈h ∨ qδ)-fuzzy filter are generalizations of (∈, ∈ ∨q)-fuzzy filter and (∈, ∈ ∨ q)-fuzzy filter, respectively.