Filters in Commutative Be-Algebras

Abstract

Abstract. The notions of terminal sections of BE-algebras are introduced. Characterizations of a commutative BE-algebra are provided.

Rough Set Theory Applied to Fuzzy Filters in Be-Algebras

“…We recall some definitions and results discussed in [1,2,3,8]. An algebra (X; * , 1) of type (2, 0) is called a BE-algebra if (BE1) x * x = 1 for all x ∈ X; (BE2) x * 1 = 1 for all x ∈ X; (BE3) 1 * x = x for all x ∈ X; (BE4) x * (y * z) = y * (x * z) for all x, y, z ∈ X.…”

On Fuzzy Positive Implicative Filters inBE-Algebras

“…([6]) Let X be a BE -algebra. X is said to be commutative if the following identity holds:

( x ∗ y )∗ y = ( y ∗ x )∗ x ; that is, x ∨ y = y ∨ x , where x ∨ y = ( y ∗ x )∗ x , for all x , y ∈ X .

…”

Commutative and Bounded BE-algebras

“…Proof. From Theorem 3.6 in [14], we know that ( ; * , 1) is a semilattice with respect to ∨. Then we need to show that the set { , } for all , ∈ has a greatest lower bound.…”

“…Also they generalized the notion of upper sets in BE-algebras and discussed some properties of the characterizations of generalized upper sets related to the structure of ideals in transitive and self-distributive BE-algebras. In [14], Ahn et al introduced the notion of terminal section of BE-algebras and provided the characterization of the commutative BEalgebras.…”