Distributive Pseudo Be-Algebras

Abstract

In this paper, we introduce the notion of distributive pseudo BE-algebra and show that the related relation defined on this structure is transitive and prove that every pseudo upper set is a pseudo filter. Also, the pseudo filter generated by a set is define and show that the set of all pseudo filters is distributive complete lattice but it is not complemented. the notion of prime and irreducible subset and prove that every irreducible subset is prime.

Some results on fuzzy congruence relations in pseudo BE-algebras

“…Remark 2.3: If X is a pseudo BE-algebra satisfying x * y = x y, for all x, y ∈ X, then X is a BE-algebra. [2]) In a pseudo BE-algebra X, the following statements hold:…”

“…Definition 2.5: [2] A pseudo BE-algebra X is said to be distributive if it satisfies in the following condition:…”

“…Theorem 2.8: [2] Let (X; * , , 1) be a distributive pseudo BE-algebra. (X, , * , 1) is a distributive pseudo BE-algebra if and only if (X; * , 1) is a BE-algebra (i. e. x * y = x y, for all x, y ∈ X).…”

“…Furthermore, the notion of pseudo upper sets in pseudo BE-algebras introduced and prove that the every pseudo filter is a union of pseudo upper sets. We show that in distributive pseudo BEalgebras normal pseudo filters and pseudo filters are equivalent [2]. L. C. Ciungu introduce the notion of a commutative pseudo BE-algebra and generalize some results proved by A. Walendziak for the case of commutative BE-algebras and proved that the class of commutative BE-algebras is term equivalent to the class of commutative pseudo BCK-algebras [8].…”

Fuzzy congruence relations on pseudo BE-algebras

“…Rezaei et al introduced the notion of distributive pseudo BE-algebras and normal pseudo filters and proved some basic properties. They showed that in distributive pseudo BE-algebras normal pseudo filters and pseudo filters coincide [4]. L.C.…”