Some kinds of $(\overline{\in},\overline{\in}\vee \overline{q})$-fuzzy filters of $BL$-algebras

The concepts of $(\overline{\in},\overline{\in} \vee \overline{q})$-fuzzy (implicative, positive implicative and fantastic) filters of $BL$-algebras are introduced and some related properties are investigated. Some characterizations of these generalized fuzzy filters are derived. In particular, we describe the relationships among ordinary fuzzy (implicative, positive implicative and fantastic) filters, $(\in,\ivq)$-fuzzy (implicative, positive implicative and fantastic) filters and $(\overline{\in},\overline{\in} \vee \overline{q})$-fuzzy (implicative, positive implicative and fantastic) filters of $BL$-algebras. Finally, we prove that a fuzzy set $F$ on a $BL$-algebra $L$ is an $(\overline{\in},\overline{\in} \vee \overline{q})$-fuzzy implicative filter of $L$ if and only if it is both $(\overline{\in},\overline{\in} \vee \overline{q})$-fuzzy positive implicative filter and an $(\overline{\in},\overline{\in} \vee \overline{q})$-fuzzy fantastic filter.