SOME PARTITION ON UNIFORM STRUCTURE OF BE-ALGEBRAS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we investigate some relation on a uniform structure of BE-algebras X . Then we prove that for a filter F of X , the set {UF [x] | x ∈ X} is a partition of X .


INTRODUCTION
The concepts of BE-algebras was first introduced by Iseki and Tanaka [3]. In 2007, Kim and Kim [5] introduced and investigated the notion of BE-algebras as a dualization of a generalization of BCK-algebras. Ahn and So [1] introduced ideals and upper sets in BE-algebras and investigated some properties of ideals. In 2008, Walendziak [10] introduced the notion of commutative BE-algebras and discussed several properties of commutative BE-algebras. In 2009, Kim and Lee [4] generalized the notion of upper sets and introduced the concept of extended upper sets. Algebra and topology are fundamental domains of mathematics. Many of the most important objects of mathematics represent a blend of algebraic objects and topological structures. In 2017, Mehrshad and Golzarpoor [7] studied some properties of uniform topology and topological BE-algebras and compare these topologies. Shahdadi and Kouhestani [9] defined (left, right, semi) topological BE-algebras and showed that for each cardinal number α there is at least a topological BE-algebra of order α. Albaracin and Velela [2] studied the topology generated by the family of subsets determined by the right application of BE-ordering of a BEalgebra and investigated some of its properties. In this paper, we investigate some properties of uniform structure on BE-algebras.

PRELIMINARIES
Some essential notations and definitions of BE-algebras and ordinary senses in this work has been introduced in this section.

Definition 2.2. [9]
A BE-algebra is a non empty set X with a constant 1 and a binary operation * satisfying the following axioms, for all x, y, z ∈ X Definition 2.3. [9] Let (X; * ; 1) be a BE-algebra, and let F be a non-empty subset of X. Then F is said to be a filter of X if the following axioms are satisfies, for all x, y, z ∈ X Then (X;*, 1) is a BE-algebra and F 1 := {1, a, b} is a filter of X, but Definition 2.5.
[6] Let ∼ be a binary relation on a set X. Then ∼ is called (iii) transitive if for all x, y, z ∈ X, x∼y and y∼z implies x∼z.
(iv) compatible if for all w, x, y, z ∈ X, w∼x and y∼z implies w * y∼x * z.
We said to be ∼ is an equivalent relation if ∼ is reflexive, symmetric and transitive. A compatible equivalence on X is called a congruence on X In 2010, Yong Ho Yon [11] has been introduced a relation as follow : we define the binary relation ∼ I on X in the following way: x ∼ I y iff x * y ∈ I and y * x ∈ I for all x, y ∈ X Theorem 2.6. [11] If I be a filter of a BE-algebra X. Then ∼ I is a congruence relation on X.
Let X be a set and P be a collection of nonempty subsets of X. Then P is called a partition of X if the following properties are satisfied :

UNIFORM TOPOLOGY ON BE-ALGEBRAS
In this section , M. Mohamadhasani and M. Haveshki [8] in 2010, introduce the notion on a BE-algebra and investigates some of its properties as follow : Definition 3.1. Let (X;*, 1) be a BE-algebra and U,V ⊆ X × X define Let (X;*, 1) be a BE-algebra and K ⊆ X × X ,we said to be (X, K ) is a uniform structure if it satisfies the following axioms : Clearly / 0 and the set X belong to T ,also that T is closed under arbitrary union and finite intersection.
Definition 3.5. Let (X, K ) be a uniform structure. Then the topology ⊆ G} is called a uniform topology on X induced by K Then (X; * , 1) is a BE-algebra.
Clearly F = {1, a, c} is a filter of X and let Λ = {F} By theorem 3.3 we have Open neighborhoods are: Another example : for a filter J = {1, b} in X we have    for all x ∈ X, thus x∈X U F [x] = X . Therefore P is a partition of X.