On fuzzy filters of pseudo BL-algebras

The aim of this paper is to introduce the notion of (∈,∈∨ q)-fuzzy (implicative, MV and G-) filters of pseudo-BL algebras and to investigate some of their related properties. Some characterization theorems of these generalized fuzzy (implicative) filters are derived. The relationship among these generalized fuzzy filters of pseudo-BL algebras is considered. It is proved that an (∈,∈ ∨q)-fuzzy filter of a pseudo-BL algebra is an (∈,∈ ∨ q)-fuzzy implicative filter if and only if it is both an (∈,∈ ∨ q)-fuzzy MV -filter and an (∈,∈ ∨ q)-fuzzy G-filter. Finally, we consider the concept of implication-based fuzzy implicative filters of pseudo-BL algebras, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.