Weak Implicative Filters of Be-Algebras

Abstract

The concept of weak implicative filters is introduced in BEalgebras. Some characterizations of weak implicative filters are derived in terms of filters of a BE-algebra. Fuzzification is applied to the class of weak implicative filters. Some properties of fuzzy weak implicative filters are studied with respect to fuzzy relations and homomorphisms. The notion of triangular normed fuzzy weak implicative filters is introduced in BEalgebras and their properties are studied.

Fuzzy Implicative Filters of Be-Algebras Based on the Theory of Falling Shadows

“…Definition 2.4. [7] A BE-algebra (X, * , 1) is said to be self-distributive if x * (y * z) = (x * y) * (x * z) for all x, y, z ∈ X.…”

“…Theorem 2.1. [7] A fuzzy set µ of a BE-algebra X is a fuzzy filter in X if and only if it satisfies the following conditions:…”

“…Definition 2.8. [7] A fuzzy set µ in a BE-algebra X is called a fuzzy implicative filter of X if it satisfies (G1) and…”

“…Proposition 2.4. [7] Let µ be a fuzzy set in a BE-algebra X. Then µ is a fuzzy implicative filter of X if and only if its nonempty level subset µ α is an implicative filter of X for all α ∈ [0, 1].…”

“…Proposition 2.5. [7] Let µ and ν be two fuzzy filters of a transitive BE-algebra X with µ ≤ ν and µ(1) = ν(1). If µ is a fuzzy implicative filter of X, then so is ν.…”