Fuzzy filters of BL-algebras

The aim of this paper is to introduce the notions of fuzzy filters, fuzzy prime filters and the cosets of a fuzzy filter in BL-algebras and investigate some of their properties. The fuzzy filter generated by a fuzzy set is discussed. Some characterizations of fuzzy filters and fuzzy prime filters are derived. The extension theorem of fuzzy prime filters and the fuzzy prime filters theorem are established. Finally, we prove that the algebra L/f which is the set of all cosets of f is a BL-algebra, and is isomorphic to the BL-algebra L/ff(1), where ff(1) = {x ∈ L ∣ f (x) = f (1)}. Moreover, each BL-algebra L is a subdirect product of linearly ordered BL-algebras L/fi (i ∈ Γ), where fi (i ∈ Γ) are fuzzy prime filters of L.