Interval-valued Intuitionistic Fuzzy Filters of BL-algebras

In order to study the interval-valued intuitionistic fuzzy filter theory of BL-algebras, the notions of interval-valued intuitionistic fuzzy filters, lattice filters, prime filters, Boolean filters, implicative filters, positive implicative filters, ultra filters and obstinate filters are introduced, respectively, and their properties are investigated. In interval-valued intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy filters, Boolean filters, ultrafilters are proved to be equivalent to lattice filters, implicative filters, obstinate filters, respectively. Each interval-valued intuitionistic fuzzy Boolean filter is an interval-valued intuitionistic fuzzy positive implicative filter, but the converse may not be true in BL-algebras. The conditions under an interval-valued intuitionistic fuzzy positive implicative filter being an interval-valued intuitionistic fuzzy Boolean filter are constructed. Finally, the interval- valued intuitionistic fuzzy ultra filter is proved to be equivalent to the interval-valued intuitionnistic fuzzy obstinate filter in BL-algebras.