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

Abstract

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.

Keywords

BE-algebra

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),

(∀ x ∈ X) ( x ∗ 1 =1),

(∀ x ∈ X) ( 1 ∗ x = x),

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

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),

(∀ x ∈ X, s, q ∈ I) ( (s ∗ (q ∗ x)) ∗ x ∈ I).

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

1 ∈ I,

(∀ x, z ∈ X) ( ∀y ∈ I) ( x ∗ (y ∗ z) ∈ I ⇒ x ∗ z ∈ 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,

x ∗ (y ∗ z) ∈ F and x ∗ y ∈ F imply y ∗ z ∈ F for all x, y, z ∈ X.

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 $C (f; t) : = {x \in X | f (x) \leq t}$ 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)),

(∀ x, y ∈ X) ( f (y) ≤ min {f (x) , f (x ∗ y)}).

Example 1. Let X = {1, μ, m, ω, 0} be a set with a multiplication table given by;

*	1	µ	m	ω	0
1	1	µ	m	ω	0
µ	1	1	m	ω	m
m	1	µ	1	m	µ
ω	1	µ	1	1	µ
0	1	1	1	m	1

Define an $N$ -fuzzy set f on X as follows: $f (y) = {\begin{cases} - 0.8 & if y \in {1, μ} \\ - 0.3 & otherwise \end{cases}$ 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)),

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

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)),

(∀ x, y ∈ X) ( y ≤ x ⇒ 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)),

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

Example 2. Let X = {1, h, 0, m, g} be a BE-algebra with a multiplication table given by;

*	1	h	0	m	g
1	1	h	0	m	g
h	1	1	0	m	0
0	1	h	1	0	h
m	1	h	1	1	h
g	1	1	1	0	1

Define a

N

-fuzzy set on X as follows:

h (y) = {\begin{cases} - 0.9 & if y \in {1, h} \\ - 0.2 & otherwise \end{cases}

for all y ∈ X. This shows that f is an

N

-fuzzy implicative filter of 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 $\begin{matrix} f (x * (y * z) & = & f (y * (x * z) \\ \geq & f ((x * y) * (x * (x * z))) \end{matrix}$ Hence by assumed condition, $\begin{matrix} f (x * z) \leq max {f (x * y), f (x * y) * (x * (x * z)))} \\ \leq max {f (x * y), f (x * (y * z))} \end{matrix}$ 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,

(∀ x, y ∈ X) ( f (x ∗ y) ≤ f (x ∗ (x ∗ y))),

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

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)}

= 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 $α_{\circ} = \frac{1}{2} (f (1) + f (x))$ . 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 $f (x) = {\begin{cases} α & if x \in F \\ - 1 & otherwise \end{cases}$ 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 ∗ (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 $\begin{matrix} f^{ψ} (1) & = & f (ψ (1)) \\ = & f (1^{'}) \leq f (ψ (x)) \\ = & f^{ψ} (x) . \end{matrix}$ Let x, y, z ∈ X. Then

f ^ψ (x ∗ z) = f (ψ (x ∗ z))

= f (ψ (x) ∗ ψ (z))

$\leq max {\begin{matrix} f (ψ (x) * (ψ (y) * ψ (z))), \\ f (ψ (x) * ψ (y)) \end{matrix}}$

= 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))}

$= max {\begin{matrix} f (ψ (m) * (ψ (d) * ψ (h))), \\ f (ψ (m) * ψ (d)) \end{matrix}}$

= max {f (x ∗ (y ∗ z)) , f ((x ∗ y)}.

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

Example 3. Let X ={ 1, x, y, z } and H ={ 1, a, b } be two non-empty sets with following Cayley tables

*	1	x	y	z
1	1	x	y	z
x	1	1	y	z
y	1	x	1	z
z	1	1	1	1

o	1	a	b
1	1	a	b
a	1	1	b
b	1	1	1

Then, clearly (X, ∗ , 1) and (H, ∘ , 1) are BE-algebras.

Define a map ψ : X ⟶ H by $ψ (t) = \{\begin{array}{c} 1 if t = 1, x, \\ a if t = y \\ b if t = z \end{array}$ 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: $f (x) = {\begin{array}{c} - 0.9 if x = 1, a \\ - 0.6 if x = b \end{array}$ 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)). $f^{ψ} (t) = \{\begin{array}{c} - 0.9 if x = 1, x, y \\ - 0.6 if t = z \end{array}$ 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 $f (x, y) \geq max {ν (x), ν (y)}$

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 $(f \times h) (x, y) = max {f (x), h (y)}$ 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,

(f × ν) _α = f_α × ν_α for all α ∈ [-1, 0].

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 $\begin{matrix} (f \times ν) (1, 1) & = & max {f (1), ν (1)} \\ \leq & max {f (x), ν (y)} \\ = & (f \times ν) (x, y) \end{matrix}$ 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′})}

$\leq max {\begin{matrix} max {f (x * y), f (q)}, \\ max {ν (x^{'} * y^{'}), ν (q^{'})} \end{matrix}}$

$= max {\begin{matrix} max {f (x * y), ν (x^{'} * y^{'})}, \\ max {f (q), ν (q^{'})} \end{matrix}}$

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

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

$\leq max {\begin{matrix} (f \times ν) (x, x^{'}) * (y, y^{'}), \\ (f \times ν) ((x * (y * z), x^{'} * (y^{'} * z^{'})) \end{matrix}}$ .

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,

if f (1) ≤ f (x) for all x ∈ X, then either f (1) ≤ ν (x) or ν (1) ≤ ν (x),

if ν (1) ≤ ν (x) for all x ∈ X, then either v (1) ≤ f (x) or f (1) ≤ f (x),

there is f otherwise ν is a $N$ -fuzzy implicative filter of 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 $\begin{matrix} f (x * z) \\ = max {f (x * z), ν (1)} \\ = (f \times ν) (x * z, 1) \\ = (f \times ν) (x * z, x * 1) \\ = (f \times ν) ((x, x) * (z, 1)) \\ \leq max {\begin{matrix} (f \times ν) ((x, x) * (y, 1)), \\ (f \times ν) ((x, x) * (y, 1) * (z, 1))) \end{matrix}} \\ = max {\begin{matrix} (f \times ν) (x * y, x * 1)), \\ (f \times ν) (x * (y * z), x * (1 * 1)) \end{matrix}} \\ = max {\begin{matrix} max {f (x * y), ν (x * 1)}, \\ max {f (x * (y * z), ν (x * (1 * 1))} \end{matrix}} \\ = max {\begin{matrix} max {f (x * y), ν (1)}, \\ max {f (x * (y * z), ν (1)} \end{matrix}} \\ = max {f (x * y), f (x * (y * z))} \end{matrix}$ 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 $\begin{matrix} ν (x * z) \\ = max {f (1), ν (x * z)} \\ = (f \times ν) (1, x * z) \\ = (f \times ν) (x * 1, x * z) \\ = (f \times ν) ((x, x) * (1, z)) \\ \leq max {\begin{matrix} (f \times ν) ((x, x) * (1, y)), \\ (f \times ν) ((x, x) * ((1, y) * (1 * z))) \end{matrix}} \\ = max {\begin{matrix} (f \times ν) (x * 1, x * y), \\ (f \times ν) ((x * ((1 * 1), x * (y * z)) \end{matrix}} \\ = max {\begin{matrix} (f \times ν) (1, x * y), \\ (f \times ν) (1, x * (y * z)) \end{matrix}} \\ = max {ν (x * y)}, ν (x * (y * z))} \end{matrix}$ 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 $\begin{matrix} f_{ν} (x, y) & = & max {ν (x), ν (y)} \\ \geq & max {ν (1), ν (1)} \\ = & f_{ν} (1, 1) . \end{matrix}$ Let (x, x′) , (y, y′),(z, z′) ∈ X × X. Then we have $\begin{matrix} f_{ν} ((x, x^{'}) * (z, z^{'})) \\ = f_{ν} ((x * z), (x^{'} * z^{'})) \\ = max {ν (x * z), ν (x^{'} * z^{'})} \end{matrix}$ $\begin{matrix} \leq max {\begin{matrix} max {ν (x * z), ν (t)}, \\ max {ν (x^{'} * z^{'}), ν (t^{'})} \end{matrix}} \end{matrix}$ where } t =x* (y* z) and t^′ =x^′ * (y^′ * z^′) $\begin{matrix} = max {\begin{matrix} max {ν (x * y), ν (x^{'} * y^{'})}, \\ max {ν (t), ν (t^{'}))} \end{matrix}} \\ = max {\begin{matrix} f ν (x * y, x^{'} * y^{'})}, \\ f_{ν} (x * (y * z), x^{'} * (y^{'} * z^{'})) \end{matrix}} \\ = max {\begin{matrix} f_{ν} ((x, x^{'}) * (y, y^{'})), \\ f_{ν} ((x, x^{'}) * ((y, y^{'}) * (z, z^{'}))) \end{matrix}} \end{matrix}$ 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 $\begin{matrix} ν (1) & = & max {ν (1), ν (1)} \\ = & f_{ν} (1, 1) \leq f_{ν} (x, y) \\ = & max {ν (x), ν (y)} \end{matrix}$ 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 $\begin{matrix} ν (x * z) & = & max {ν (x * z), ν (1)} \\ = & f_{ν} (x * z, 1) \\ = & f_{ν} (x * z, x * 1) \\ = & f_{ν} ((x, x) * (z, 1)) \\ \leq & max {\begin{matrix} f_{ν} ((x, z) * (y, 1)), \\ f_{ν} ((x, z) * ((y, 1) * (z, 1))) \end{matrix}} \\ = & max {\begin{matrix} f_{ν} (x * y, z * 1)), \\ f_{ν} (x * (y * z), z * (1 * 1)) \end{matrix}} \\ = & max {f_{ν} (x * y, 1), f_{ν} (x * (y * z), 1)} \\ = & max {\begin{matrix} max {ν (x * y), ν (1)}, \\ max {ν (x * (y * z)), ν (1) \end{matrix}} \\ = & max {ν (x * y), ν (x * (y * z))} \end{matrix}$ 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.

References

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

Abstract

Abstract

Keywords

1 Introduction

2 Preliminaries

3 N -fuzzy filters in BE-algebras

4 N -fuzzy implicative filters of BE-algebras

5 Cartesian products of N -fuzzy implicative filters

6 Conclusion

References

3 $N$ -fuzzy filters in BE-algebras

4 $N$ -fuzzy implicative filters of BE-algebras

5 Cartesian products of $N$ -fuzzy implicative filters