Abstract
The notion of fuzzy extended filters is introduced on residuated lattices, and its essential properties are investigated. By defining an operator \(\rightsquigarrow \) 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.
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