Applications of N -structures in implicative filters of BE-algebras

1 Introduction

A (crisp) set A in a universe X can be defined in the form of its characteristic function μ _A  : X → {0, 1} yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A.

So far most of the generalization of the crisp set have been conducted on the unit interval [0, 1] and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets spread positive information that fit the crisp point {1} into the interval [0, 1].

Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply a mathematical tool.

To attain such an object, Lee et al. [12] introduced a new function which is called a negative-valued function, and constructed N-structures. They applied N-structures to BCK/BCI-algebras, and discussed N-ideals in BCK/BCI-algebras. In 1966, Iseki and Imai [7] and Iseki [8] introduced two classes of abstract algebras: BCK-algebras and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. As a generalization of a BCK-algebra, Kim and Kim [3] introduced the notion of a BE-algebra and filter of a BE-algebra, and investigated several properties. So and Ahn [6] introduced the notion of ideals in a BE-algebra. They considered several descriptions of ideals in a BE-algebra. They defined the upper set in a BE-algebra and derived some relation between ideal theory and upper set in a BE-algebra. Kim and Lee generalized the concept of So and Ahn, to defined extended upper set in a BE-algebra [4]. They provided the relations between filters and extended upper set in a BE-algebra. Ahn et al. introduced the concepts of self-distributive and a transitive BE-algebra and also discussed some properties of filters in a commutative BE-algebra [1]. They discussed some properties of the characterizations of generalized upper set A _n  (u, v) to the structure of ideals and filters in a BE-algebra. The Congruence’s and BE-relations on a BE-algebra was defined by Yon et al. [2]. Recently, Saeid et al. studied filters in BE-algebras [10]. They defined positive filters, normal filters and implicative filters of BE-algebras. Also gave some relation among these types of filters in a BE-algebra. They introduced some interesting results on these filters. Also gave some characterization of a BE-algebra by the properties of these filters [10]. In [13], Liu and Zhang introduced implicative and positive implicative prefilters of EQ-algebras. Zhan et. al. studied the some types of falling shadow fuzzy filters of BL-algebras [14]. Liu studied interval valued T-fuzzy filters and interval-valued T-fuzzy congruences on residuated lattices in [15]. In [16], Saeid and Mohtashamnia, studied, implication BL-algebras. Kang and Jun [9], introduced the notion of an N-ideal of BE-algebra and studied some basic results.

In this paper, we formulate the notion of N-fuzzy filters in a BE-algebras and discussed some connected assets. We define N-fuzzy filter, N-Level filters, N-fuzzy implicative filter and their results. We discuss a characterization of N-fuzzy implicative filters of a BE-algebras in terms of N-fuzzy level filters. We also observe that every N-fuzzy implicative filter of a BE-algebra is a N-fuzzy filter but the converse is not true in general. We give some equivalent conditions of N-fuzzy filter of a BE-algebra to become a N-fuzzy implicative filter. An extension property of N-fuzzy implicative filters is also studied. The properties of homomorphic images of N-fuzzy implicative filters are studied. Finally, we study some properties of the cartesian product of N-fuzzy implicative filters in BE-algebras.

2 Preliminaries

Definition 1. [3] Let K (τ) be a class all algebras of type τ = (2, 0). A system (X ; ∗ , 1) ∈ K (τ) define a BE-Algebra if the following axioms hold:

(∀ x ∈ X) ( x ∗ x = 1),

A relation " ≤ " on a BE-algebra X is defined by (∀ x, y ∈ X) ( x ≤ y ⇔ x ∗ y = 1).

Definition 2. [3] A BE-algebra X is called Self-distributive if x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z) for all x, y, z ∈ X.

Definition 3. [4] A BE-algebra (X ; ∗ , 1) is said to be Transitive if it satisfies:

(∀ x, y, z ∈ X) ( y ∗ z ≤ (x ∗ y) ∗ (x ∗ z)).

Definition 4. [6] Let I be a non-empty subset of a BE-algebra X. Then, I is called an Ideal of X if;

(∀ x ∈ X, s ∈ I) ( x ∗ s ∈ I),

Lemma 1. [6] A non-empty subset I of X is an ideal of X if and only if it satisfies:

1 ∈ I,

Definition 5. Implicative filter F of a BE-algebra X is a non-empty subset of X if the following conditions hold:

1 ∈ F,

Every implicative filter of a BE-algebra X is filter of X .

Denote by F ( X , [ - 1 , 0 ] ) the collection of functions from a set X to [-1, 0]. We say that an element of F ( X , [ - 1 , 0 ] ) is a negative-valued function from X to [-1, 0] (briefly, N-function on X). By an N-structure we mean an ordered pair (X, f) of X and an N-function f on X. Let (X, f) be an N-structure on a non-empty set X. Then, for t ∈ [-1, 0] the set C (f : t) ={ x ∈ X : f (x) ≤ t } is called a closed (f, t)-cut of (X, f).

In what follows, let X denote a BE-algebra and f an N-function on X unless otherwise specified.

Definition 6. [9] For any N-structure (X, f) the non-empty set C ( f ; t ) : = { x ∈ X | f ( x ) ≤ t } is called a closed (f, t)-cut of (X, f), where t ∈ [-1, 0].

Definition 7. [9] By an N-ideal of X we mean an N-structure (X, f) which satisfies the following condition: (∀ t ∈ [-1, 0]) ( C (f ; t) ∈ J (X) ∪ {∅}), where J (X) is a set of all ideal of X.

3 N-fuzzy filters in BE-algebras

In this section we define an N-fuzzy filters in BE-algebras and discuss some basic properties.

Definition 8. Let f be an N-fuzzy set of a BE-algebra X. Then, f is called an N-Fuzzy Filter of X if it satisfies:

(∀ x ∈ X) ( f (1) ≤ f (x)),

Define an N-fuzzy set f on X as follows: f ( y ) = { − 0.8 if y ∈ { 1 , μ } − 0.3 otherwise for all y ∈ X. Then clearly f is an N-fuzzy filter of X.

Theorem 1. Let X be a BE-algebra and f be an N-fuzzy filter of X. Then the following conditions hold:

(∀ x ∈ X) (f (1) ≤ f (x)),

Proof 1. Proof follows from Definition of an N-fuzzy filter of X.

Lemma 2. Let f be an N-fuzzy filter of a BE-algebra X. Then the following conditions hold;

(∀ x, y ∈ X)   ( f (x ∗ y) = f (1) ⇒ f (y) ≤ f (x)),

Proof 2. Straightforward

4 N-fuzzy implicative filters of BE-algebras

In this section we define an N-fuzzy implicative filter of a BE-algebra and study some properties of an N-fuzzy implicative filter of BE-algebra.

Definition 9. Let f be a N-fuzzy filter of a BE-algebra X. Then f _α  = {x ∈ X | f (x) ≤ α} where α∈ [-1, 0] is called an N-level filter of X.

Definition 10. Let f be an N-fuzzy set of a BE-algebra X. Then, f is called an N-fuzzy implicative filter of a BE-algebra if it satisfies the following conditions;

(∀ x ∈ X)   ( f (1) ≤ f (x)),

Theorem 2. Every N-fuzzy filter of a distributive BE-algebra X is an N-fuzzy implicative filter of X.

Proof 3. Suppose that f is a N-fuzzy filter of X. So, f (1) ≤ f (x) for all x ∈ X. Let x, y, z ∈ X. Since f is a N-fuzzy filter, so by definition of N-fuzzy filter, we get f (x ∗ z) ≤ max {f (x ∗ y) ∗ (x ∗ z)) , f (x ∗ y)} = max {f (x ∗ (y ∗ z)) , f (x ∗ y)}, because X is distributive. Therefore, f is an N-fuzzy implicative filter of X.

Theorem 3. If f (y ∗ z) ≤ max {f (x) , f (x ∗ (y ∗ (y ∗ z)))} for all x, y, z ∈ X, then every N-fuzzy filter of a transitive BE-algebra X is an N-fuzzy implicative filter of X.

Proof 4. Suppose that f is an N-fuzzy filter of X satisfying the given condition. Let x, y, z ∈ X. By using (V4) and since X is a transitive BE-algebra, we get f ( x ∗ ( y ∗ z ) = f ( y ∗ ( x ∗ z ) ≥ f ( ( x ∗ y ) ∗ ( x ∗ ( x ∗ z ) ) ) Hence by assumed condition, f ( x ∗ z ) ≤ max { f ( x ∗ y ) , f ( x ∗ y ) ∗ ( x ∗ ( x ∗ z ) ) ) } ≤ max { f ( x ∗ y ) , f ( x ∗ ( y ∗ z ) ) } Therefore, f is an N-fuzzy implicative filter in X.

Theorem 4. Let f be an N-fuzzy filter of a transitive BE-algebra X. Then the following conditions are equivalent;

f is an N-fuzzy implicative filter of X,

Proof 5. (1) ⇒ (2): By (1), Let f be an N-fuzzy implicative filter of X. Let x, y ∈ X. Since x ∗ y ≤ x ∗ (x ∗ y), we get f (x ∗ y) ≥ f (x ∗ (x ∗ y)). Since f is an N-fuzzy filter in X, so

f (x ∗ y) ≤ max {f (x ∗ (x ∗ y)) , f (x ∗ x)}

  = max {f (x ∗ (x ∗ y)) , f (1)}

  = max {f (x ∗ (x ∗ y)) , f (1)}

  = max {f (x ∗ (x ∗ y)) , f (1)}

  = f (x ∗ (x ∗ y))

Hence f (x ∗ y) ≤ f (x ∗ (x ∗ y)).(2) ⇒ (3): Assume that (2) hold. Let x, y, z ∈ X. Since X be transitive, we get y ∗ z ≤ (x ∗ y) ∗ (x ∗ z) and hence we get x ∗ (y ∗ z) ≤ x ∗ ((x ∗ y) ∗ (x ∗ z)). Thus, f (x ∗ (y ∗ z)) ≥ f (x ∗ ((x ∗ y) ∗ (x ∗ z))). Now, we get

f ((x ∗ y) ∗ (x ∗ z))

  = f (x ∗ ((x ∗ y) ∗ z))

  = f (x ∗ (x ∗ ((x ∗ y) ∗ z))

  = f (x ∗ ((x ∗ y) ∗ (x ∗ z))

  ≤ f (x ∗ (y ∗ z))

(3) ⇒ (1): Assume that (3) hold. Let x, y, z ∈ X. Since f is an N-fuzzy filter in X, we have the following

f (x ∗ z)

  ≤ max {f ((x ∗ y) ∗ (x ∗ z)) , f (x ∗ y)}

  ≤ max {f (x ∗ (y ∗ z)) , f (x ∗ y)}.

Therefore f is an N-fuzzy implicative filter in X.

Proposition 1. Let X be a BE-algebra. Then f is an N-fuzzy implicative filter in X if and only if for each α ∈ [-1, 0], the level subset f _α  ≠ φ is an N-fuzzy implicative filter of X.

Proof 6. Suppose that f is an N-fuzzy implicative filter of X. So, f (1) ≤ f (x) for all x ∈ X. In particular, f (1) ≤ f (x) for all x ∈ X. Hence 1 ∈ f _α . Let x ∗ (y ∗ z), x ∗ y ∈ f _α . Then f (x ∗ (y ∗ z)) ≤ α and f (x ∗ y) ≤ α. Since f is a N-fuzzy implicative filter, we have f (x ∗ z) ≤ max {f (x ∗ (y ∗ z)) , f (x ∗ y)} ≤ α. Thus, x ∗ z ∈ f _α . Therefore f _α is a implicative filter in X.

Conversely, suppose that f _α is an implicative filter in X for all α ∈ [-1, 0] with f _α ≠ ∅. Suppose that there exists x _∘ ∈ X such that f (1) ≥ f (x _∘). Then f (1) > α _∘ and 0 ≥ α _∘ > f (x _∘) ≥ -1. Hence x _∘ ∈ f _{α ∘} and f _{α ∘} ≠ ∅. Since f _{α ∘} is an N-fuzzy implicative filter in X, we get 1 ∈ f _{α ∘} and hence f (1) ≤ α _∘, this is a contradiction. Therefore f (1) ≤ f (x) for all x ∈ X. Again, let α ∘ = 1 2 ( f ( 1 ) + f ( x ) ). Let x, y, z ∈ X be such that f (x ∗ (y ∗ z)) = α _q and f (x ∗ y) = α _r . Then x ∗ (y ∗ z) ∈ f _{α q} and x ∗ y ∈ f _{α r} . In general, suppose that α _q  ≤ α _r . Then clearly f _{α r}  ⊆ f _{α q} . Hence x ∗ y ∈ f _{α q} . Since f _{α q} is an N-fuzzy implicative filter in X, we get x ∗ z ∈ f _{α q} . Thus f (x ∗ z) ≤ α _q  = max {α _q , α _r } = max {f (x ∗ (y ∗ z)) , f (x ∗ y)}. this implies that f (x ∗ z) ≤ max {f (x ∗ (y ∗ z)) , f (x ∗ y)}. Therefore f is an N-fuzzy implicative filter of X.

Theorem 5. Let F be an N-fuzzy implicative filter of a BE-algebra X. Then there exist a N-fuzzy implicative filter f of X such that f _α  = F for some α ∈ (-1, - 0.5).

Proof 7. Suppose that N-fuzzy set f of a BE-algebra X is defined by f ( x ) = { α if x ∈ F − 1 otherwise where α is a fixed number (-1 < α < -0.5). Since 1 ∈ F, we get f (1) = α ≤ f (x) for all x ∈ X. Let x, y, z ∈ X. Suppose x ∗ (y ∗ z), x ∗ y ∈ F. Since F is an N-fuzzy implicative filter, we get x ∗ z ∈ F. Then f (x ∗ y) = f (x ∗ (y ∗ z)) = f (x ∗ z) = α. Hence f (x ∗ z) ≤ max {f (x ∗ (y ∗ z)) , f (x ∗ y)}. Suppose x ∗ (y ∗ z) ∉ F and x ∗ y ∉ F. Then f (x ∗ y) = f (x ∗ (y ∗ z)) = -1. Hence f (x ∗ z) ≤ max {f (x ∗ (y ∗ z)) , f (x ∗ y)}. If exactly one of x ∗ (y ∗ z) and x ∗ y is in F, then exactly one of f (x ∗ (y ∗ z)) = -1 or f (x ∗ y) = -1. Hence f (x ∗ z) ≤ max {f (x ∗ (y ∗ z)) , f (x ∗ y)}. By summarizing the proposition 4, we get, f (x ∗ z) ≤ max {f (x ∗ (y ∗ z)) , f (x ∗ y)} ∀ x, y, z ∈ X. Therefore f is a N-fuzzy implicative filter of X. Clearlyf _α  = F.

Theorem 6. Let f be an N-fuzzy implicative filter of a BE-algebra X. Then two N-fuzzy implicative filters f _{α 1} and f _{α 2} (with α₁ > α₂) of f are equal if and only if there is no x ∈ X such that α₁ ≥ f (x) > α₂.

Proof 8. Assume that f _{α 1}  = f _{α 2} for α ₁ > α ₂. Suppose there exists some x ∈ X such that α ₁ ≥ f (x) > α ₂. Then f _{α 1} is a proper subset of f _{α 2} , which is impossible. Hence there is no x ∈ X such that α ₁ ≥ f (x) > α ₂.

Conversely suppose that there is no x ∈ X for which α ₁ ≥ f (x) > α ₂. Since α ₁ > α ₂, we get that f _{α 1} ⊆ f _{α 2} . If x ∈ f _{α 2} , then f (x) ≤ α ₂. Hence by assume condition, we get that f (x)≤ α ₁. Hence x ∈ f _{α 1} and so f _{α 2}  ⊆ f _{α 1} . Therefore f _{α 1}  = f _{α 2} .

Theorem 7. Let f and ν be two N-fuzzy filters of a transitive BE-algebra X with ν ≤ f and f (1) = ν (1). If f is a N-fuzzy implicative filter of X, then so is ν.

Proof 9. Assume that f and ν are two N-fuzzy filters of a transitive BE-algebra X such that ν ≤ f and ν (1) = f (1). Suppose f is a N-fuzzy implicative filter of X. Let x, y, z ∈ X. Since ν ⊆ f and f is a N-fuzzy implicative filter, then by using theorem 4(3) and (V4), we get

ν ((x ∗ y) ∗ (x ∗ ((x ∗ (y ∗ z)) ∗ z)))

  ≤ f ((x ∗ y) ∗ (x ∗ ((x ∗ (y ∗ z)) ∗ z)))

  ≤ f ((x ∗ y) ∗ (x ∗ ((x ∗ (y ∗ z)) ∗ z)))

  ≤ f (x ∗ (y ∗ ((x ∗ (y ∗ z)) ∗ z)))

  = f ((x ∗ ((x ∗ (y ∗ z)) ∗ (y ∗ z)))

  = f ((x ∗ (y ∗ z)) ∗ (x ∗ (y ∗ z)))

  = f (1)

  = ν (1)

ν ((x ∗ y) ∗ (x ∗ ((x ∗ (y ∗ z)) ∗ z))) ≤ ν (1).

Hence,

ν ((x ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ (x ∗ z)))

  = ν ((x ∗ y) ∗ ((x ∗ (y ∗ z)) ∗ (x ∗ z)))

  = ν ((x ∗ y) ∗ (x ∗ ((x ∗ (y ∗ z)) ∗ z)))

  ≤ ν (1)

Hence

ν ((x ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ (x ∗ z))) = ν (1). Since ν is a N-fuzzy filter, by lemma 3(1), we get that

≤ν (1).

By theorem 4(3), it yields that ν is a  N-fuzzy implicative filter in X.

Definition 11. [6] Let (X, ∗ , 1) and (H, ∘ , 1) be two BE-algebras, respectively. A map ψ : X ⟶ H is called a BE-homomorphism if for all x, y ∈ X, ψ (a ∗ b) = ψ (a) ∘ ψ (b). It it clear that if ψ : X ⟶ H is a BE-homomorphism, then .

Definition 12. Let ψ : X → H is a BE-homomorphism of a BE-algebras and f is an N-fuzzy set in H. Then define a mapping f ^ψ  : X → [-1, 0] such that f ^ψ  (x) = f (ψ (x)) for all x ∈ X.

Clearly the above mapping f ^ψ is well-defined and an N-fuzzy set in X.

Theorem 8. Let ψ : X → H is onto homomorphism of BE-algebras and f is a N-fuzzy set in H. Then f is a N-fuzzy implicative filter in H if and only if f ^ψ is a N-fuzzy implicative filter in X.

Proof 10. Suppose that f is a N-fuzzy implicative filter in H. For any x ∈ X, we have f ψ ( 1 ) = f ( ψ ( 1 ) ) = f ( 1 ′ ) ≤ f ( ψ ( x ) ) = f ψ ( x ) . Let x, y, z ∈ X. Then

f ^ψ  (x ∗ z) = f (ψ (x ∗ z))

   = f (ψ (x) ∗ ψ (z))

≤ max { f ( ψ ( x ) ∗ ( ψ ( y ) ∗ ψ ( z ) ) ) , f ( ψ ( x ) ∗ ψ ( y ) ) }

   = max {f (ψ (x ∗ (y ∗ z))) , f (ψ (x ∗ y))}

   = max {f ^ψ  (x ∗ (y ∗ z)) , f ^ψ  (x ∗ y)}.

Hence f ^ψ is a N-fuzzy implicative filter of X.

Conversely, suppose that f ^ψ is a N-fuzzy implicative filter in X. Let x ∈ H. Since ψ is onto, we have ψ (y) = x for all y ∈ X. So, . Let x, y, z ∈ H. Then there exist m, d, h ∈ X such that ψ (m) = x, ψ (d) = y and ψ (h) = z. Hence, we get

   f (x ∗ z) = f (ψ (m) ∗ q (h))

   = f (ψ (m ∗ h))

   = f ^ψ  (m ∗ h)

   ≤ max {f ^ψ  (m ∗ (d ∗ h)) , f ^ψ  (m ∗ d)}

   = max {f (ψ (m ∗ (d ∗ h))) , f (ψ (m ∗ d))}

= max { f ( ψ ( m ) ∗ ( ψ ( d ) ∗ ψ ( h ) ) ) , f ( ψ ( m ) ∗ ψ ( d ) ) }

   = max {f (x ∗ (y ∗ z)) , f ((x ∗ y)}.

Therefore f is a N-fuzzy implicative filter in H.

Define a map ψ : X ⟶ H by ψ ( t ) = 1 if t = 1 , x , a if t = y b if t = z Thus, it is easy to verify that q is an onto BE-homomorphism. Now let define an N-fuzzy set f on H as following: f x = { − 0.9 if x = 1 , a − 0.6 if x = b Then, clearly f is an N-fuzzy implicative filter of H. Now define N-fuzzy set f ^ψ of X from an N-fuzzy implicative filter f of H as:f ^ψ  (t) = f (ψ (t)). f ψ t = − 0.9 if x = 1 , x , y − 0.6 if t = z Thus, clearly f ^ψ is an N-fuzzy implicative filter of X.

5 Cartesian products of N-fuzzy implicative filters

In this part, we discuss some properties of the cartesian products of N-fuzzy implicative filters of BE-algebras. The notion of N-fuzzy relations are extended to the case of N-fuzzy implicative filters of BE-algebras and the properties of homomorphic images of N-fuzzy implicative filters are studied.

Definition 13. A N-fuzzy relation on a set S is a N-fuzzy set f : S × S → [-1, 0].

Definition 14. Let f be a N-fuzzy relation on a set S and ν a N-fuzzy set in S. Then f is a N-fuzzy relation on ν if for all x, y ∈ S f ( x , y ) ≥ max { ν ( x ) , ν ( y ) }

Definition 15. Let f and ν be two N-fuzzy sets in a BE-algebra X. The cartesian product f and h is defined by ( f × h ) ( x , y ) = max { f ( x ) , h ( y ) } for all x, y ∈ X .

From above definitions we have the following Lemma.

Lemma 3. Let f and ν be two N-fuzzy sets in a BE-algebra X. Then the following hold.

f × ν be a N-fuzzy relation on X,

Definition 16. For any two BE-algebras X and Q, define an operation ∗ on X × Q as follows: ( x , y ) ∗ ( x ' , y ' ) = ( x ∗ x ' , y ∗ y ' ) for all x, x^' and y,y^' ∈ Q. Then we can be easily observed that (X × Q ; ∗ , (1, 1)) is a BE-algebra.

Proposition 2. Let f and g be two N-implicative filters of the BE-algebras X and Q, respectively. Then f × g is an N-implicative filter in X × Q.

Proof 11. It can be routinely verified.

Theorem 9. Let f and ν be two N-fuzzy implicative filters of a BE-algebra X. Prove that f × ν is a N-fuzzy implicative filter in X × X.

Proof 12. Let (x, y) ∈ X × X. Since f and ν are N-fuzzy implicative filters in X, we get ( f × ν ) ( 1 , 1 ) = max { f ( 1 ) , ν ( 1 ) } ≤ max { f ( x ) , ν ( y ) } = ( f × ν ) ( x , y ) Let . Put q = x ∗ (y ∗ z) and q ^{′=x′∗(y′} ∗ z′). Clearly . Since f and ν are two N-fuzzy implicative filters in X, Now we get

  = max {^{f(x∗z),ν(x′∗z′})}

≤ max { max { f ( x ∗ y ) , f ( q ) } , max { ν ( x ′ ∗ y ′ ) , ν ( q ′ ) } }

= max { max { f ( x ∗ y ) , ν ( x ′ ∗ y ′ ) } , max { f ( q ) , ν ( q ′ ) } }

  = max {^{(f×ν)(x∗y,x′∗y′}) , (f × ν) (q, q′)}

  = max {(f × ν) (x, x ^{′)∗(y,y′}) , (f × ν) (q, q′)}}

≤ max { ( f × ν ) ( x , x ′ ) ∗ ( y , y ′ ) , ( f × ν ) ( ( x ∗ ( y ∗ z ) , x ′ ∗ ( y ′ ∗ z ′ ) ) }.

Therefore f × ν is a N-fuzzy implicative filter in X × X.

Theorem 10. Let f and ν be two N-fuzzy sets in a BE-algebra X such that f × ν is a N-fuzzy filter of X × X. Then the following axioms hold:

either f (1) ≤ f (x) or ν (1) ≤ ν (x) ∀ x ∈ X,

Proof 13. (1): Suppose f (x) < f (1) and ν (y) < ν (1) for some x, y ∈ X. So, (f × ν) (x, y) = max {f (x) , ν (y)} < max {f (1) , ν (1)} = (f × ν) (1, 1), which is a contradiction. Hence either f (1) ≤ f (x) or ν (1) ≤ ν (x) for all x ∈ X.

(2): Assume that f (1) ≤ f (x) for all x ∈ X. Suppose f (x) < f (1) and ν (y) < ν (1) for some x, y ∈ X. Then (f × ν) (1, 1) = max {f (1) , ν (1)} = f (1). Hence (f × ν) (x, y) = max {f (x) , ν (y)} < f (1) = (f × ν) (1, 1) which is a contradiction. Therefore (2) holds.

(3): It can be obtained in a similar fashion.

(4): Since, by (1), either f (1) ≤ f (x) or ν (1) ≤ ν (x) for all x ∈ X. Without loss of generality f (1) ≤ f (x) for all x ∈ X. From (2), we can get either f (1) ≤ f (x) or ν (1) ≤ ν (x) for all x ∈ X.

Case 1: Suppose f (x) ≥ ν (1) for all x ∈ X. Then (f × ν) (x, 1) = max {f (x) , ν (1)} = f (x) for all x ∈ X. Since f × ν is a N-fuzzy implicative filter in X × X, we get f (x) = max {f (x) , ν (1)} = (f × ν) (x, 1). Also f ( x ∗ z ) = max { f ( x ∗ z ) , ν ( 1 ) } = ( f × ν ) ( x ∗ z , 1 ) = ( f × ν ) ( x ∗ z , x ∗ 1 ) = ( f × ν ) ( ( x , x ) ∗ ( z , 1 ) ) ≤ max { ( f × ν ) ( ( x , x ) ∗ ( y , 1 ) ) , ( f × ν ) ( ( x , x ) ∗ ( y , 1 ) ∗ ( z , 1 ) ) ) } = max { ( f × ν ) ( x ∗ y , x ∗ 1 ) ) , ( f × ν ) ( x ∗ ( y ∗ z ) , x ∗ ( 1 ∗ 1 ) ) } = max { max { f ( x ∗ y ) , ν ( x ∗ 1 ) } , max { f ( x ∗ ( y ∗ z ) , ν ( x ∗ ( 1 ∗ 1 ) ) } } = max { max { f ( x ∗ y ) , ν ( 1 ) } , max { f ( x ∗ ( y ∗ z ) , ν ( 1 ) } } = max { f ( x ∗ y ) , f ( x ∗ ( y ∗ z ) ) } Therefore f is a N-fuzzy implicative filter in X. Case 2: Suppose ν (1) ≤ ν (x) for all x ∈ X. Suppose ν (1) ≤ f (x) for all x ∈ X. Then it leads to case 1, from which we can conclude that f is a N-fuzzy implicative filter. Suppose f (t) < ν (1) for some t ∈ X. Then f (1) ≤ f (t) < ν (1). Since ν (1) ≤ ν (x) for all x ∈ X, it yields that f (1) < ν (x) for all x ∈ X. Hence ν ( x ∗ z ) = max { f ( 1 ) , ν ( x ∗ z ) } = ( f × ν ) ( 1 , x ∗ z ) = ( f × ν ) ( x ∗ 1 , x ∗ z ) = ( f × ν ) ( ( x , x ) ∗ ( 1 , z ) ) ≤ max { ( f × ν ) ( ( x , x ) ∗ ( 1 , y ) ) , ( f × ν ) ( ( x , x ) ∗ ( ( 1 , y ) ∗ ( 1 ∗ z ) ) ) } = max { ( f × ν ) ( x ∗ 1 , x ∗ y ) , ( f × ν ) ( ( x ∗ ( ( 1 ∗ 1 ) , x ∗ ( y ∗ z ) ) } = max { ( f × ν ) ( 1 , x ∗ y ) , ( f × ν ) ( 1 , x ∗ ( y ∗ z ) ) } = max { ν ( x ∗ y ) } , ν ( x ∗ ( y ∗ z ) ) } Therefore ν is a N-fuzzy implicative filter in X.

Definition 17. Let ν be a N-fuzzy set in a BE-algebra X. Then the strongest  N-fuzzy relation f _ν is a  N-fuzzy relation X define by f ν ( x , y ) = max { ν ( x ) , ν ( y ) } for all x, y ∈ X.

Theorem 11. Let ν be a N-fuzzy set in a BE-algebra X and f _ν be the strongest N-fuzzy relation on X. Then ν is a N-fuzzy implicative filter in X if and only if f _ν is a N-fuzzy implicative filter of X × X.

Proof 14. Suppose that ν is a N-fuzzy implicative filter of X. For each (x, y) ∈ X × X, we have f ν ( x , y ) = max { ν ( x ) , ν ( y ) } ≥ max { ν ( 1 ) , ν ( 1 ) } = f ν ( 1 , 1 ) . Let   (x, x′) , (y, y′),(z, z′) ∈ X × X. Then we have f ν ( ( x , x ′ ) ∗ ( z , z ′ ) ) = f ν ( ( x ∗ z ) , ( x ′ ∗ z ′ ) ) = max { ν ( x ∗ z ) , ν ( x ′ ∗ z ′ ) } ≤ max { max { ν ( x ∗ z ) , ν ( t ) } , max { ν ( x ′ ∗ z ′ ) , ν ( t ′ ) } } where } t =x* (y* z) and t^′ =x^′ * (y^′ * z^′) = max { max { ν ( x ∗ y ) , ν ( x ′ ∗ y ′ ) } , max { ν ( t ) , ν ( t ′ ) ) } } = max { f ν ( x ∗ y , x ′ ∗ y ′ ) } , f ν ( x ∗ ( y ∗ z ) , x ′ ∗ ( y ′ ∗ z ′ ) ) } = max { f ν ( ( x , x ′ ) ∗ ( y , y ′ ) ) , f ν ( ( x , x ′ ) ∗ ( ( y , y ′ ) ∗ ( z , z ′ ) ) ) } Therefore f_ν is a N-fuzzy implicative filter in X× X.

Conversely, suppose that f_ν is a N-fuzzy implicative filter in X× X. So we have ν ( 1 ) = max { ν ( 1 ) , ν ( 1 ) } = f ν ( 1 , 1 ) ≤ f ν ( x , y ) = max { ν ( x ) , ν ( y ) } for all x,y∈ X. Hence, it yields that ν (1)≤ ν (x)for all x∈ X. Let (x,x^′),(y,y^′),(z,z^′)∈ X× X. Then we have ν ( x ∗ z ) = max { ν ( x ∗ z ) , ν ( 1 ) } = f ν ( x ∗ z , 1 ) = f ν ( x ∗ z , x ∗ 1 ) = f ν ( ( x , x ) ∗ ( z , 1 ) ) ≤ max { f ν ( ( x , z ) ∗ ( y , 1 ) ) , f ν ( ( x , z ) ∗ ( ( y , 1 ) ∗ ( z , 1 ) ) ) } = max { f ν ( x ∗ y , z ∗ 1 ) ) , f ν ( x ∗ ( y ∗ z ) , z ∗ ( 1 ∗ 1 ) ) } = max { f ν ( x ∗ y , 1 ) , f ν ( x ∗ ( y ∗ z ) , 1 ) } = max { max { ν ( x ∗ y ) , ν ( 1 ) } , max { ν ( x ∗ ( y ∗ z ) ) , ν ( 1 ) } = max { ν ( x ∗ y ) , ν ( x ∗ ( y ∗ z ) ) } Therefore ν is a N-fuzzy implicative filter in X.

6 Conclusion

BE-algebra is a type of logical algebra like BCK/BCI/BCH-algebras. A BE-algebra is a another generalization of BCK/BCI/BCH-algebras. In this article we applied the N-structure to a filter of a BE-algebra to introduced the concept of N-fuzzy filters in BE-algebras. We studied the notion of N-fuzzy implicative filter. We discuss a characterization of N-fuzzy implicative filters of BE-algebras in terms of N-fuzzy filters and discuss some algebraic properties and results. Some properties of ideals of BE-algebras, transitive and distributive BE-algebra, N-fuzzy filters used in the results of N-fuzzy implicative filter.

In future we will study the following topics: We will generalize the present concept by using point N-structure to N-structure. We will define N-fuzzy soft filters in BE-algebras and Rough filter inBE-algebras.