Fuzzy extended filters on residuated lattices

The notion of fuzzy extended filters is introduced on residuated lattices, and its essential properties are investigated. By defining an operator ⇝ between two arbitrary fuzzy filters in terms of fuzzy extended filters, two results are immediately obtained. We show that (1) the class of all fuzzy filters on a residuated lattice forms a complete Heyting algebra, and its classical version is equivalent to the one introduced in Kondo (Soft Comput 18(3):427–432, 2014 ), which is defined with respect to (crisp) generated filters of singleton sets; (2) the connection between fuzzy extended filters and fuzzy generated filters is built, with which three other classes generating complete Heyting algebras, respectively, are presented. Finally, by the aid of fuzzy t -filters, we also develop the characterization theorems of the special algebras and quotient algebras via fuzzy extended filters.