Fuzzy filters on equality algebras with applications

Abstract

In this paper, we introduce the notion of fuzzy filters on equality algebras and study a fuzzy filter generated by a fuzzy set. We also solve the open problem which was presented by Kadji, Lele and Tonga in [Soft Computing, 21 (2017) 1913-1922]. In addition, we denote the set of all cosets of a fuzzy filter f by E/f, and prove that the structure (E/f, ∧ , ∼ , f₁) is an equality algebra, and is isomorphic to the (E/f_f(1), ∧ , ∼ , f_f(1)), where f_f(1) = {x ∈ E|f (x) = f (1)}. At the meantime, we define fuzzy congruences on equality algebras and show that there is one-to-one correspondence between fuzzy filters and fuzzy congruences. Finally, we investigate some topological properties of uniform topology induced by the special family of extreme fuzzy filters.