Hesitant Fuzzy Filters in BE-algebras
- DOI
- 10.1080/18756891.2016.1144157How to use a DOI?
- Keywords
- BE-algebra; Hesitant fuzzy (implicative) filter; Hesitant level subset; γ-inclusive
- Abstract
In this paper, we introduce the notion of hesitant fuzzy (implicative) filters and get some results on BE-algebras and show that every hesitant fuzzy implicative filter is a hesitant fuzzy filter but not the converse. Finally, we state and prove the relationship between hesitant fuzzy (implicative) filters and γ-inclusive sets.
- Copyright
- © 2016. the authors. Co-published by Atlantis Press and Taylor & Francis
- Open Access
- This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).
1. Introduction
H. S. Kim and Y. H. Kim introduced the notion of a BE-algebra as a generalization of a dual BCK-algebra [3]. A. Borumand Saeid et al. defined some types of filters in BE-algebras and showed the relationship between them [2]. B. L. Meng give a procedure which generated a filter by a subset in a transitive BE-algebra [5]. Recently, A. Walendziak introduced the notion of a normal filter in BE-algebras and showed that there is a bijection between congruence relations and filters in commutative BE-algebras [13].
Fuzzy sets were introduced in 1965 by Zadeh [15] and then fuzzification ideas have been applied to other algebraic structures such as groups and BL-algebras. Fuzzy sets and its extensions have provided successful results dealing with uncertainty in different problems. Worldwide, there has been a rapid growth in interest in applications of fuzzy sets and some generalization of this is discussed by authors such as intuitionistic fuzzy sets, interval-valued fuzzy sets, type-n fuzzy sets and fuzzy multisets. Also, another generalization of this theory was proposed by Torra and Narukawa [11] and Torra [10]. The relationships among hesitant fuzzy sets and other generalizations of fuzzy sets such as intuitionistic fuzzy sets, type-2 fuzzy sets and fuzzy multi sets were discussed. They showed that the envelope of a hesitant fuzzy set is an intuitionistic fuzzy set. Also, they proved that the operations they proposed are consistent with the ones of intitionistic fuzzy sets when applied to the envelopes of hesitant fuzzy sets. Then some researchers who have defined divers concepts, extensions, aggregation operators and measures to handle with hesitant information. It may be mentioned that hesitant fuzzy sets can reflect the humans hesitancy more objectively than the other classical extensions of fuzzy sets and suit the modeling of quantitative settings. We can try to manage those situation, where a set of values are possible in the definition process of the membership of an element with this theory.
In this paper, we introduce the notion of hesitant fuzzy (implicative) filters and get some useful properties. In fact, we show that in self distributive BE-algebras two concepts of hesitant fuzzy implicative filter and hesitant fuzzy filter are equivalent. Also, the notion of γ-inclusive set which denoted by iA(hA;γ) is defined.
2. Preliminaries
In this section, we cite the fundamental definitions that will be used in the sequel:
Definition 1. (Kim and Kim [3])
By a BE-algebra we shall mean an algebra (X; *, 1) of type (2,0) satisfying the following axioms:
- •
(BE1) x * x = 1,
- •
(BE2) x * 1 = 1,
- •
(BE3) 1 * x = x,
- •
(BE4) x * (y * z) = y * (x * z), for all x,y,z ∈ X.
From now on X is a BE-algebra, unless otherwise is stated. We introduce a relation “≤” on X by x ≤ y if and only if x * y = 1. A BE-algebra X is said to be a self distributive if x * (y * z) = (x * y) * (x * z), for all x,y,z ∈ X. A BE-algebra X is said to be commutative if satisfies:
Proposition 1. (Walendziak [13])
If X is a commutative BE-algebra, then for all x,y ∈ X,
We note that “≤” is reflexive by (BE1). If X is self distributive then relation “≤” is a transitive ordered set on X, because if x ≤ y and y ≤ z, then
Hence x ≤ z. If X is commutative then by Proposition 1, relation “≤” is antisymmetric. Hence if X is a commutative self distributive BE-algebra, then relation “≤” is a partial ordered set on X.
Proposition 2. (Kim and Kim [3])
In a BE-algebra X, the following holds:
- (i)
x * (y * x) = 1,
- (ii)
y * ((y * x) * x) = 1, for all x,y ∈ X.
A subset F of X is called a filter of X if it satisfies: (F1) 1 ∈ F, (F2) x ∈ F and x * y ∈ F imply y ∈ F. Define
Definition 2. (Borumand and Rezaei [2])
A nonempty subset F of X is called an implicative filter if satisfies the following conditions:
- (IF1)
1 ∈ F,
- (IF2)
x * (y * z) ∈ F and x * y ∈ F imply that x * z ∈ F, for all x,y,z ∈ X.
If we replace x of the condition (IF2) by the element 1, then it can be easily observed that every implicative filter is a filter. However, every filter is not an implicative filter as shown in the following example.
Example 1.
Let X = {1,a,b} be a BE-algebra with the following table:
| * | 1 | a | b |
| 1 | 1 | a | b |
| a | 1 | 1 | a |
| b | 1 | a | 1 |
Then F = {1,a} is a filter of X, but it is not an implicative filter, since 1 * (a * b) = 1 * a = a ∈ F and 1 * a = a ∈ F but 1 * b = b ∉ F.
Definition 3.
Let (X1; *, 1) and (X2; ○, 1′)) be two BE-algebras. Then a mapping f : X1 → X2 is called a homomorphism if f(x1 * x2) = f(x1) ○ f(x2), for all x1,x2 ∈ X1. It is clear that if f : X1 → X2 is a homomorphism, then f(1) = 1′.
Definition 4. (Rezaei and Borumand [6])
A fuzzy set μ of X is called a fuzzy filter if satisfies the following conditions:
- (FF1)
μ(1) ≥ μ(x),
- (FF2)
μ(y) ≥ min{μ(x * y), μ(x)}, for all x,y ∈ X.
Definition 5. (Rao [9])
A fuzzy set μ of X is called a fuzzy implicative filter of X if satisfies the following conditions:
- (FIF1)
μ(1) ≥ μ(x),
- (FIF2)
μ(x * z) ≥ min{μ(x * (y * z)), μ(x * y)}, for all x,y,z ∈ X.
If we replace x of the condition (FIF2) by the element 1, then it can be easily observed that every fuzzy implicative filter is a fuzzy filter. However, every fuzzy filter is not a fuzzy implicative filter as shown in the following example.
Example 2. (Rao [9])
Let X = {1,a,b,c,d} be a BE-algebra with the following table:
| * | 1 | a | b | c | d |
| 1 | 1 | a | b | c | d |
| a | 1 | 1 | b | c | b |
| b | 1 | a | 1 | b | a |
| c | 1 | a | 1 | 1 | a |
| d | 1 | 1 | 1 | b | 1 |
Then it can be easily verified that (X; *, 1) is a BE-algebra. Define a fuzzy set μ on X as follows:
Then clearly μ is a fuzzy filter of X, but it is not a fuzzy implicative filter of X, since
Definition 6. (Torra [10])
Let X be a reference set. Then a hesitant fuzzy set HFS A in X is represented mathematical as:
So, we can define a set of fuzzy sets an HFS by union of their membership functions.
Definition 7. (Torra [10])
Let A = {μ1,μ2,…,μn} be a set of n membership functions. The HFS that is associated with A, hA, is defined as
It is remarkable that this definition is quite suitable to decision making, when experts have to assess a set of alternatives. In such a case, A represents the assessments of the experts for each alternative and hA the assessments of the set of experts. However, note that it only allows to recover those HFSs whose memberships are given by sets of cardinality less than or equal to n.
For convenience, Xia and Xu in [14] named the set h = hA(x) as a hesitant fuzzy element HFE. The family of all hesitant fuzzy elements defined on X by HFE(X).
Definition 8. (Verma and Dev Sharma [12])
Let h,h1,h2 ∈ HFE(X) and λ ∈ [0,1]. Then the operations complement, union and intersection are defined as follows:
- (i)
hc = {1 − γ : γ ∈ h},
- (ii)
h1 ⊔ h2 = {max(γ1,γ2) : γ1 ∈ h1,γ2 ∈ h2},
- (iii)
h1 ⊓ h2 = {min(γ1,γ2) : γ1 ∈ h1,γ2 ∈ h2},
- (iv)
h1 ⊕ h2 = {γ1 + γ2 − γ1γ2 : γ1 ∈ h1,γ2 ∈ h2},
- (v)
h1 ⊗ h2 = {γ1γ2 : γ1 ∈ h1,γ2 ∈ h2},
- (vi)
λh = {1 − (1 − γ)λ : γ ∈ h}.
3. Hesitant Fuzzy filters
In what follows, we introduce binary operation “⊑” as follow:
It is obvious that ⊑ is a partial order set on ρ([0,1]).
Definition 9.
Hesitant fuzzy set A of X is called a hesitant fuzzy filter if satisfies the following conditions:
- •
(HFF1) hA(x) ⊑ hA(1),
- •
(HFF2) hA(x) ⊓ hA(x * y) ⊑ hA(y), for all x,y ∈ X.
Denote the set of all hesitant fuzzy filters of X by HFF(X).
Note. If |hA(x)| = 1, for all x ∈ X, then hA is a fuzzy filter. In fact in this case we put “≤≔⊑” and “min ≔ ⊓”.
Example 3.
Let X = {1,a,b} be a BE-algebra with the following table:
| * | 1 | a | b |
| 1 | 1 | a | b |
| a | 1 | 1 | b |
| b | 1 | 1 | 1 |
Let t1 = {0.2,0.3}, t2 = {0.4,0.5} and t3 = {0.6,0.8}. Define A as hA(1) = t3, hA(a) = t2 and hA(b) = t1. Then A is a hesitant fuzzy filter.
Proposition 3.
Let A ∈ HFF(X) and x,y,z,ai ∈ X for i = 1,…,n. Then
- (i)
if x ≤ y, then hA(x) ⊑ hA(y),
- (ii)
hA(x) ⊑ hA(y * x),
- (iii)
hA(x) ⊓ hA(y) ⊑ hA(x * y),
- (iv)
hA(x) ⊑ hA((x * y) * y),
- (v)
hA(x) ⊓ hA(y) ⊑ hA((x * (y * z)) * z),
- (vi)
if hA(y) ⊓ hA((x * y) * z) ⊑ hA(z * x), then hA is antitonic (i.e. if x ≤ y, then hA(y) ⊑ hA(x)),
- (vii)
if z ∈ A(x,y), then hA(x) ⊓ hA(y) ⊑ hA(z),
- (viii)
if
Proof.
- (i).
Let x ≤ y. Then x * y = 1 and so by using (BE2) and Definition 9 (HFF2), we have
- (ii).
Since x ≤ (y * x), by using (i) we have hA(x) ⊑ hA(y * x).
- (iii).
By using (ii) we have
- (iv).
It follows from Definition 9,
- (v).
From (iv) we have
- (vi).
Let x ≤ y, that is, x * y = 1.
- (vii).
Let z ∈ A(x,y). Then x * (y * z) = 1. Hence
- (viii).
The proof is by induction on n. By (vii) it is true for n = 1, 2. Assume that it satisfies for n = k, that is,
for all a1,…,ak, x ∈ X.
Suppose that
Since A is a hesitant fuzzy filter of X, we have
In the following example shows that if hA ∈ HFF(X), then
Example 4.
In Example 3, we have
Theorem 4.
Let h,h1,h2 ∈ HFF(X) and λ ∈ [0,1]. Then
- (i)
h1 ⊔ h2 ∈ HFF(X),
- (ii)
h1 ⊓ h2 ∈ HFF(X),
- (iii)
h1 ⊕ h2 ∈ HFF(X),
- (iv)
h1 ⊗ h2 ∈ HFF(X),
- (v)
λh ∈ HFF(X).
Proof.
- (i).
Assume that h1,h2 ∈ HFF(X) and x ∈ X. Then
Now, we have
Therefore, h1 ⊔ h2 ∈ HFF(X).
- (ii).
Assume that h1,h2 ∈ HFF(X) and x ∈ X. Then
Now, we have
Therefore, h1 ⊓ h2 ∈ HFF(X).
- (iii).
Assume that h1,h2 ∈ HFF(X) and x ∈ X. Then (h1 ⊕ h2) (x)
Now, we have
Therefore, h1 ⊕ h2 ∈ HFF(X).
- (iv).
Assume that h1,h2 ∈ HFF(X) and x ∈ X. Then
Now, we have
Therefore, h1 ⊗ h2 ∈ HFF(X).
- (v).
Assume that h ∈ HFF(X), x ∈ X and λ ∈ [0,1].
Now, we have
Therefore, λh ∈ HFF(X).
Lemma 5.
If {hi}i∈Λ ∈ HFF(X), then ⊓i∈Λhi, is too.
Proof.
Straightforward.
Since the set HFF(X) is closed under arbitrary intersections, we have the following theorem.
Theorem 6.
(HFF(X); ⊑) is a complete lattice, but it is not a Boolean algebra.
Proof.
By Theorem 4 and Lemma 5, the proof is obvious. Example 4 shows that it is not a Boolean algebra.
Theorem 7.
Let A ∈ HFF(X). Then the set
Proof.
Obviously, 1 ∈ XhA. Let x,x * y ∈ XhA. Then hA(x) = hA(x * y) = hA(1). Now, by Definition 9, we have
Hence hA(y) = hA(1). Therefore, y ∈ XhA.
Let γ ∈ ρ([0,1]). For a hesitant fuzzy filter A of X, γ-inclusive set which denoted by iA(hA;γ) is defined as follows:
It is obvious that if β ⊑ γ, then iA (hA;γ) ⊑ iA(hA;β), for all γ,β ∈ ρ([0,1]).
Example 5.
In Example 3, γ ≔ {0.1,0.4}, we have iA(hA;γ) = {1,a}.
Theorem 8.
Let A ∈ HFS(X). The following are equivalent:
- (i)
A ∈ HFF(X),
- (ii)
(∀γ ∈ ρ([0,1])) iA(hA;γ) ≠ ∅ imply iA(hA;γ) is a filter of X.
Proof.
(i) ⇒ (ii). Let x,y ∈ X be such that x,x * y ∈ iA(hA;γ), for any γ ∈ ρ([0,1]).
Then γ ⊑ hA(x) and γ ⊑ hA(x * y). Hence
Since hA is a hesitant fuzzy filter, we have y ∈ iA(hA;γ).
(ii) ⇒ (i). Let iA(hA;γ) be a filter of X, for any γ ∈ ρ([0,1]) with iA(hA;γ) ≠ ∅. Put hA(x) = γ, for any x ∈ X. Then x ∈ iA(hA;γ). Since iA(hA;γ) is a filter of X, we have 1 ∈ iA(hA;γ) and so hA(x) = γ ⊑ hA(1).
Now, for any x,y ∈ X, let hA(x * y) = γx*y and hA(x) = γx. Put γ = γx*y ⊓ γx. Then x,x * y ∈ iA(hA;γ), so y ∈ iA(hA;γ). Hence γ ⊑ hA(y) and so
Therefore, A ∈ HFF(X).
Theorem 9.
Let A ∈ HFF(X). Then for all a,b ∈ X and γ ∈ ρ([0,1])
Proof.
Assume that A ∈ HFF(X). Let a,b ∈ X be such that a,b ∈ iA (hA;γ). Then γ ⊑ hA(a) and γ ⊑ hA(b). Let c ∈ A(a,b). Hence a * (b * c) = 1. Now, by Proposition 3 (v), we have
Then c ∈ iA(hA;γ). Therefore, A(a,b) ⊆ iA(hA;γ)).
Corollary 10.
Let A ∈ HFF(X). Then for all γ ∈ ρ([0,1]))
Proof.
It is sufficient prove that
For this, assume that x ∈ iA(hA;γ). Since x * (1 * x) = 1, we have x ∈ A(x,1). Hence
Theorem 11.
Let A ∈ HFS(X). Define a Hesitant fuzzy set
Proof.
Let A ∈ HFF(X) and x,y ∈ X. If x * y,x ∈ iA(hA;γ), then y ∈ iA(hA;γ) by Theorem 8 (ii). Hence
If x * y ∉ iA(hA;γ) or x ∉ iA(hA;γ), then
Therefore, A* ∈ HFF(X).
4. Hesitant Fuzzy implicative filters
Definition 10.
Hesitant fuzzy set A of X is called a hesitant fuzzy implicative filter if satisfies the following conditions:
- •
(HFIF1) hA(x) ⊑ hA(1),
- •
(HFIF2) hA(x * (y * z)) ⊓ hA(x * y) ⊑ hA(x * z), for all x,y,z ∈ X.
Denote the set of all hesitant fuzzy implicative filters on X by HFIF(X). It can seen that every hesitant fuzzy implicative filter is a hesitant fuzzy filter.
Note. If |hA(x)| = 1, for all x ∈ X, then hA is an implicative fuzzy filter. In fact in this case we put “≤≔⊑” and “min ≔ ⊓”.
Example 6.
Let X = {1,a,b,c} with the following table:
| * | 1 | a | b | c |
| 1 | 1 | a | b | c |
| a | 1 | 1 | b | c |
| b | 1 | a | 1 | c |
| c | 1 | 1 | b | 1 |
Then (X; *, 1) is a BE–algebra. Let t1 = {0.6,0.9}, t2 = {0.2,0.3} and t3 = {0.5}. Define A as
Then A is a hesitant fuzzy implicative filter.
Theorem 12.
Let X be a self distributive BE-algebra. Then every hesitant fuzzy filter is a hesitant fuzzy implicative filter.
Proof.
Let A ∈ HFF(X). Obvious that hA(x) ⊑ hA(1), for all x ∈ X. By using self distributivity and (HFF2), we have
Therefore, A ∈ HFIF(X).
In the following example shows that the condition self distributivity of Theorem 12, is necessary.
Example 7. (Rao [9])
Let X = {1,a,b,c,d} with the following table:
| * | 1 | a | b | c | d |
| 1 | 1 | a | b | c | d |
| a | 1 | 1 | b | c | d |
| b | 1 | a | 1 | b | a |
| c | 1 | a | 1 | 1 | a |
| d | 1 | 1 | 1 | b | 1 |
Then (X; *, 1) is a BE-algebra but it is not self distributive because,
Let t1 = {0.8,0.7} and t2 = {0.4}. Define A as hA(1) = hA(a) = t1 and hA(b) = hA(c) = hA(d) = t2. Then A is a hesitant fuzzy filter, but it is not a hesitant implicative filter because,
Theorem 13.
Let F be a (implicative) filter of X. Then there exists a hesitant (implicative) fuzzy filter hA of X such that iA(hA;γ) = F, for some γ ∈ ρ([0,1]).
Proof.
Define hesitant fuzzy set hA as follows
- Case 1.
If x * y,x ∈ F, then y ∈ F. Hence
- Case 2.
If x * y ∈ F and x ∉ F, Then hA(x * y) = F and hA(x) = ∅. Hence
- Case 3.
If x *y ∉ F and x ∈ F, Then hA(x * y) = ∅ and hA(x) = F. Hence
- Case 4.
If x * y ∉ F and x ∉ F, Then hA(x * y) = ∅ and hA(x) = ∅. Hence
Clearly iA(hA;γ) = F.
Theorem 14.
Let X be a self distributive BE-algebra and A ∈ HFF(X). Then the following conditions are equivalent:
- (i)
A ∈ HFIF(X),
- (ii)
hA(y * (y * x)) ⊑ hA(y * x),
- (iii)
hA((z * (y * (y * x))) ⊓ hA(z) ⊑ hA(y * x), for all x,y,z ∈ X.
Proof.
(i) ⇒ (ii). Let A ∈ HFIF(X). By using (HFIF1) and (BE1) we have
(ii) ⇒ (iii). Let A be a hesitant fuzzy filter of X satisfying the condition (ii). By using (HFIF2) and (ii) we have
(iii) ⇒ (i). Since
Therefore, A ∈ HFIF(X).
Let f : X → Y be a homomorphism of BE-algebras and A ∈ HFS(Y). Define a mapping hAf : X → ρ([0,1]) such that hAf(x) = hA(f(x)), for all x ∈ X.
Then hAf(x) is well-defined and Af ∈ HFS(X), in which Af = {x ∈ X : f(x) ∈ A}.
Theorem 15.
Let f : X → Y be an onto homomorphism of BE-algebras and A ∈ HFS(Y). Then A ∈ HFF(Y)(resp. A ∈ HFIF(Y)) if and only if Af ∈ HFF(X)(resp. Af ∈ HFIF(X)).
Proof.
Assume that A ∈ HFF(Y). For any x ∈ X, we have
Hence (HFF1) is valid. Now, let x,y ∈ X
Therefore, Af ∈ HFF(X).
Conversely, Assume that Af ∈ HFF(X). Let y ∈ Y. Since f is onto, there exists x ∈ X such that f(x) = y. Then
Now, let x,y ∈ Y. Then there exists a,b ∈ X such that f(a) = x and f(b) = y. Hence we have
Therefore, A ∈ HFF(Y).
Let A ∈ HFS(X). Denote
Then for any β ∈ [0, ⊤], define hAβ (x) ≔ hA(x) + β = {a + β : a ∈ hA(x)}, for all x ∈ X.
Obviously, hAβ is a mapping from X to [0,1], that is, Aβ ∈ HFS(X). hAβ (x) is well define. Assume that β ∈ [0, ⊤] and x1 = x2. Then hA(x1) = hA(x2) and so hAβ (x1) = hA(x1) + β = hA(x2) + β = hAβ (x2).
Hence hAβ is well define. Let β1,β2 ∈ [0, ⊤] be such that β1 ≤ β2. Then hAβ1 ⊑ hAβ2.
Example 8.
In Example 6, sup{sup{hA(x) : x ∈ X}} = sup{0.9,0.3,0.5} = 0.9 and so ⊤ = 1 − 0.9 = 0.1.
Theorem 16.
Let β ∈ [0, ⊤]. If A ∈ HFF(X) (resp. HFIF(X)), then hAβ ∈ HFF(X) (resp. HFIF(X)), too.
Proof.
Let x,y ∈ X. Then
Also, hAβ (x) = hA(x) + β ⊆ hA(1) + β = hAβ (1). Therefore, hAβ ∈ HFF(X).
Theorem 17.
If there exists β ∈ [0, ⊤] such that hAβ ∈ HFF(X)(resp. HFIF(X)), then hA ∈ HFF(X) (resp. HFIF(X)), too.
Proof.
Assume that hAβ ∈ HFF(X), for some β ∈ [0, ⊤]. Let x, y ∈ X. Since hAβ (x * y) ⊓ hAβ (x) ⊑ hAβ (y), we can see that
Now, by canceling β we have
Also, by a similar way, hA(x) ⊑ hA(1). Therefore, hA ∈ HFF(X).
5. Conclusion
Uncertainty usually appears in many real world problems. Fuzzy sets and its extensions have provided successful results dealing with uncertainty in different problems. We have paid attention to one of them, HFS, that manages hesitant situations that often appear when the membership degree of an element to a set must be established. Additionally, it is known that many operators for hesitant fuzzy sets and their extensions have been introduced to deal with such a type of information in different applications where decision making has been the most remarkable one.
In this paper, we applied the theory of hesitant fuzzy sets to BE-algebras and introduced the notions of hesitant fuzzy (implicative) filters and γ-inclusive sets in BE-algebras and many related properties are introduced.
Acknowledgments
We thank the anonymous referees for the careful reading of the paper and the suggestions on improving its presentation.
References
Cite this article
TY - JOUR AU - Akbar Rezaei AU - Arsham Borumand Saeid PY - 2016 DA - 2016/01/01 TI - Hesitant Fuzzy Filters in BE-algebras JO - International Journal of Computational Intelligence Systems SP - 110 EP - 119 VL - 9 IS - 1 SN - 1875-6883 UR - https://doi.org/10.1080/18756891.2016.1144157 DO - 10.1080/18756891.2016.1144157 ID - Rezaei2016 ER -