Generalized Fuzzy Filters Of BL-algebras

When we study the semantic theory of a non-classical mathematical logic(especially, discussing the logical systems),in most case,it is necessary to examine the structures of relevant algebraic systems,such as Lukasiewicz continuous value logic and MV-algebras, form system L and Ro-algebra,basic logic system BL and BL-algebras etc.From this view, it is similar to that between classical logic and Boolean algebra.Therefore,it is important to study inner relationships originate from various logic algebraic systems,not only for studying algebraic structures itself,but also for revealing the inherent relations between various logic systems.Based on the above theory,we introduce the notions of(∈,∈∨q)- fuzzy Boolean (implicative,positive implicative and fantastic)filters of BL-algebras and investigate some related properties.Some characterizations of these generalized fuzzy filters are derived by means of their level subsets.Finally,the relations among these generalized fuzzy filters are discussed.It is proved that a fuzzy set of a BL-algebra is an(∈,∈∨q)-fuzzy implicative(Boolean)filter if and only if it is both an (∈,∈∨q)- fuzzy positive implicative filter and an(∈,∈∨q)- fuzzy fantastic filter.The obtained results can be applied to other logic algebras.