Fuzzy filter on BI-algebras

In this paper, the concepts of fuzzy filter and fuzzy sub-algebra on BI-algebras are introduced. Then some of their properties are investigated. Fuzzy sub-algebras and fuzzy filters are classified according to level sets and equivalent definitions for fuzzy filters are provided. It has also been proven that in every distributive BI-algebra, every filter is a sub-algebra and we have shown that in a group of BI-algebras, every subset contains 1 is a sub-algebra and every fuzzy subset that satisfies the first condition of the filter is a fuzzy filter. Moreover, it is proved that the set of all fuzzy filters on BI-algebras is a complete lattice. The relation of congruence and quotient BI-algebra produced by fuzzy filters is also investigated.