In this paper, we develop fuzzy filter theory on a BE-algebra and apply the concept of interval-valued hesitant fuzzy filter to BE-algebras. Moreover, some types of interval-valued hesitant fuzzy filters such as interval-valued hesitant fuzzy implicative filter and interval-valued hesitant fuzzy fantastic filter are introduced and some of properties are investigated.
The concept of fuzzy set was introduced by Zadeh [20]. 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 multi sets. Intuitionistic fuzzy sets (IFSs, for short) were introduced in 1983 by Atanassov [2]. Interval-valued fuzzy sets (IVFSs, for short), apparently first studied by Sambuc [15] who called them φ-flou functions, serve to capture a feature of uncertainty w.r.t. the assignment of membership degrees. Also, another generalization of this theory was proposed by Torra and Narukawa [17] and Torra [16] as hesitant fuzzy set (HFS), which permitting a set of several possible membership values. Then some researchers who have defined divers concepts, extensions, aggregation operators, decision making and measures to handle with hesitant information [1, 19]. Z. Pei et al. investigated some properties of operators and algebraic structure of hesitant fuzzy sets with confidence levels and based on closed interval-valued hesitant fuzzy sets, an equivalence relation on hesitant fuzzy sets was defined, then lattices and distributive lattices on hesitant fuzzy sets were constructed [9].
H.S. Kim and Y.H. Kim introduced the notion of a BE-algebra as a generalization of a dual BCK-algebra [7]. A. Borumand Saeid et al. defined some types of filters in BE-algebras and showed the relationship between them [4]. Recently, A. Rezaei et al. introduced the notion of hesitant fuzzy (implicative) filters and got some useful properties [13].
In this paper, we introduce the notion of interval-valued hesitant fuzzy (implicative, fantastic) filters and get some useful properties. In fact, we show that in self distributive BE-algebras two concepts of interval-valued hesitant fuzzy implicative filter and interval-valued hesitant fuzzy filter are equivalent. Translation of interval-valued hesitant fuzzy filters is studied.
Preliminaries
In this section, we cite the fundamental definitions that will be used in the sequel:
Definition 2.1. [7] By a BE-algebra we shall mean an algebra of type (2, 0) satisfying the following axioms:
x ∗ x = 1,
x ∗ 1 =1,
1 ∗ x = x,
x ∗ (y ∗ z) = y ∗ (x ∗ z), for all x, y, z ∈ X .
From now on is a BE-algebra, unless otherwise is stated. We introduce a relation “≤” on by x ≤ y if and only if x ∗ y = 1. A BE-algebra is said to be a self distributive if x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z), for all x, y, z ∈ X.
Proposition 2.2.[7] In a BE-algebra , the following hold:
x ∗ (y ∗ x) =1,
y ∗ ((y ∗ x) ∗ x) =1, for all x, y ∈ X .
Proposition 2.3.[11] Let be a self distributive BE-algebra. The following hold:
if x ≤ y, then z ∗ x ≤ z ∗ y,
y ∗ z ≤ (x ∗ y) ∗ (x ∗ z), for all x, y, z ∈ X .
From now on, is a BE-algebra unless otherwise specified.
A subset F of X is called a filter of if it satisfies: (F1) 1 ∈ F, (F2) x ∈ F and x ∗ y ∈ F imply y ∈ F. Define
which is called an upper set of x and y. It is easy to see that 1, x, y ∈ A (x, y), for any x, y ∈ X . Every upper set A (x, y) need not be a filter of in general.
Definition 2.4. [4] A non-empty subset F of X is called an implicative filter of if satisfies the following conditions:
1 ∈ F,
x ∗ (y ∗ z) ∈ F and x ∗ y ∈ F imply that x ∗ z ∈ F, for all x, y, z ∈ X .
Definition 2.5. [12] A fuzzy set μ of X is called a fuzzy filter of if satisfies the following conditions:
μ (1) ≥ μ (x),
μ (y) ≥ min {μ (x ∗ y), μ (x)}, for allx, y ∈ X.
Definition 2.6. [10] A fuzzy set μ of X is called a fuzzy implicative filter of if satisfies the following conditions:
μ (1) ≥ μ (x),
μ (x ∗ z) ≥ min {μ (x ∗ (y ∗ z)), μ (x ∗ y)}, for all x, y, z ∈ X.
Definition 2.7. [16] Let X be a reference set. A hesitant fuzzy set (HFS, for short) A in is represented mathematical as:
where ρ ([0, 1]) is the power set of [0, 1].
So, we can define a set of fuzzy sets an HFS by union of their membership functions.
Definition 2.8. [18] Let and λ ∈ (0, 1]. The operations, complement, union and intersection are defined as follows:
hc = {1 - γ : γ ∈ h},
h1 ⊔ h2 = {max(γ1, γ2) : γ1 ∈ h1, γ2 ∈ h2},
h1 ⊓ h2 = {min(γ1, γ2) : γ1 ∈ h1, γ2 ∈ h2},
h1 ⊕ h2 = {γ1 + γ2 - γ1γ2 : γ1 ∈ h1, γ2 ∈ h2},
h1 ⊗ h2 = {γ1γ2 : γ1 ∈ h1, γ2 ∈ h2},
λh = {1 - (1 - γ) λ : γ ∈ h}.
Definition 2.9. [13] Hesitant fuzzy set A of is called a hesitant fuzzy filter if satisfies the following conditions:
hA (x) ⊑ hA (1),
hA (x) ⊓ hA (x ∗ y) ⊑ hA (y), for allx, y ∈ X .
Denote the set of all hesitant fuzzy filters of by HFF ().
Theorem 2.11.[13] Algebra of type (2, 2, 1, 0, 0) is a complete lattice, but it is not a Boolean algebra.
By an interval we mean an interval [a-, a+] where a-, a+ are real numbers and 0 ≤ a- ≤ a+ ≤ 1 .The set of all intervals is denoted by D [0, 1]. The interval [a, a] is identified with the real number a.
For intervals , , where i ∈ I, I is an index set, we define , , , .
Furthermore, we have
if and only if and ,
if and only if and ,
, where 0 ≤ k ≤ 1.
Therefore it can be shown that (D [0, 1], ≤, ∧, ∨) is a complete lattice, as its least element and as its greatest element.
By an interval-valued fuzzy set F on , we mean that set
where and are two fuzzy set of such that , for any x ∈ X. Putting , we see that , where .
For any , the set is called the interval-valued level subset of [5].
Definition 2.12. [6] Let X be a reference set. An interval-valued hesitant fuzzy set (IVHFS) on is where denotes all possible interval-valued membership degrees of the element x ∈ X to the set . For convenience, we call an interval-valued hesitant fuzzy element (IVHFE), which reads Here is an interval number. An IVHFE is the basic unit of an IVHFS, and it can be considered as a special case of the IVHFS. The relationship between IVHFE and IVHFS is similar to that between interval-valued fuzzy number and interval-valued fuzzy set.
It can easily seen that both hesitant fuzzy sets and interval-valued fuzzy sets are all particular cases of interval-valued hesitant fuzzy sets.
Definition 2.13. [6] Let X be a reference set and be an interval-valued hesitant fuzzy set on . For any x, y ∈ X,
if and only if for any interval , there exists an interval such that
Interval-valued hesitant fuzzy filters
In this section, we will apply the interval-valued fuzzy set to generalize the notion of hesitant fuzzy filters in BE-algebras.
Definition 3.1. Interval-valued hesitant fuzzy set of is called an interval-valued hesitant fuzzy filter if satisfies the following conditions:
, for all x, y ∈ X .
Denote the set of all interval-valued hesitant fuzzy filters of by IVHFF ().
Note. It can be seen that both hesitant fuzzy filters and interval-valued fuzzy filters are all particular cases of interval-valued hesitant fuzzy filters.
Example 3.2. Let X : = {1, a, b, c} be a set 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 is a BE-algebra. For anyx ∈ X, we define as follows:
, and .
Then is an interval-valued hesitant fuzzy filter of .
Proposition 3.3.Let and x, y, z, a, b, ai ∈ X for i = 1, …, n. Then
if x ≤ y, then
,
,
,
,
if , then is an order reversing (i.e., if x ≤ y, then ),
if z ∈ A (x, y), then ,
if , then , where
Proof (i). Let x ≤ y. Then x ∗ y = 1 and so
(ii). Since x ≤ y ∗ x, by using (i) we have .
(iii). By using (ii) we have
(iv). It follows from Definition 3.1 that
(v). From (iv) we have
(vi). Let x ≤ y, that is, x ∗ y = 1. Then
(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 , for all a1, …, ak, ak+1, x ∈ X . Then
Since is a hesitant fuzzy filter of , we have
Proposition 3.4.Let be a self distributive BE-algebra and . If x ≤ y, then
Proof. Let . Since is self distributive and x ≤ y, we have z ∗ x ≤ y ∗ z by Proposition 2.3(i). It follows from Proposition 3.3(i) that □
Proposition 3.5.Let be an indexed family of interval-valued hesitant fuzzy filter of . Then , is also an interval-valued hesitant fuzzy filter of .
Theorem 3.6.Let be self distributive BE-algebraand . Then if and only if satisfies following two conditions:
,
, for all x, y, z ∈ X .
Proof. Assume that . Since is self distributive BE-algebra, we have x ∗ y ≤ (y ∗ z) ∗ (x ∗ z), it follows from Proposition 3.3(i) that
Also, since , we have
Therefore,
Conversely, if put x = 1 in (ii), by (BE3), we have
Therefore, . □
Theorem 3.7.Let . Then if and only if satisfies the following conditions:
,
, for all x, y, z ∈ X .
Proof. Assume that . The proof is similar to the proof of Proposition 3.3(ii) and (v).
Conversely, let . If we put y = x in (i), by (BE1), we have
Also, by using (ii), (BE3) and (BE1), we have
Therefore, . □
Theorem 3.8.Let . Then if and only if the set is a filter of , for all with
Proof. Let and be such that Then and It follows from (IVHFF1) that and so . On the other hand, we have
Hence Therefore,
Conversely, assume that for all with Let x ∈ X. Put . Then . Now, since is a filter of , then . Hence
Also, for any x, y ∈ X, let and . Hence and , which imply that and so Then
Therefore, . □
Corollary 3.9.Let and a ∈ X be such that . Then the set is a filter of .
Definition 4.1. Interval-valued hesitant fuzzy set of is called an interval-valued hesitant fuzzy implicative filter if satisfies the following conditions:
, for all x, y, z ∈ X .
Denote the set of all interval-valued hesitant fuzzy implicative filters on by IVHFIF().
Taking x = 1 in (IVHFIF2), we can see that every interval-valued hesitant fuzzy implicative filter is an interval-valued hesitant fuzzy filter.
Theorem 4.2.Let be a self distributive BE-algebra. Then every interval-valued hesitant fuzzy filter is an interval-valued hesitant fuzzy implicative filter.
Proof. Let . Obvious that , for all x ∈ X . By using self distributivity and (IVHFF2), we have
Therefore, . □
In the following example shows that the condition, self distributivity of Theorem 4.2, is necessary.
Example 4.3. [10] 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 is a BE-algebra, but it is not self distributive, because
For any x ∈ X, we define to a set as follows: , . Then is an interval-valued hesitant fuzzy filter, but it is not an interval-valued hesitant fuzzy implicative filter because,
Theorem 4.4.Let and λ ∈ [0, 1]. Then
,
,
,
,
.
Theorem 4.5.Algebra of type (2, 2, 2, 0, 0) is a complete lattice.
Theorem 4.6.Let be a self distributive BE-algebra and . Then the following conditions are equivalent:
,
,
for all x, y, z ∈ X .
Proof. (i) ⇒ (ii). Let . By (IVHFIF1) and (BE1) we have
(ii) ⇒ (iii). Let be a hesitant fuzzy filter of satisfying the condition (ii). By using (IVHFIF2) and (ii) we have
(iii) ⇒ (i). Since
Hence by Proposition 3.3 (i). Thus
Therefore, . □
Let f : X → Y be a homomorphism of BE-algebras and , where . Define a mapping such that , for all x ∈ X . Then is well defined and , in which
Theorem 4.7.Let f : X → Y be an onto homomorphism of BE-algebras and . Then (resp., ) if and only if (resp., ).
Proof. Assume that . For any x ∈ X, we have
Hence (IVHFF1) is valid. Now, let x, y ∈ X
Therefore, .
Conversely, assume that . Lety ∈ Y. Since f is onto, there exists x ∈ X such that f (x) = y. Hence
Now, let x, y ∈ Y. Then there exist a, b ∈ X such that f (a) = x and f (b) = y. Hence we have
Therefore, . □
Interval-valued hesitant fuzzy fantastic filters
Definition 5.1. Interval-valued hesitant fuzzy set of is called an interval valued hesitant fuzzy fantastic filter if satisfies the following conditions:
, for all x, y, z ∈ X .
Denote the set of all hesitant fuzzy fantastic filters on by . It can be considered every interval-valued hesitant fuzzy fantastic filter as a hesitant fantastic filter.
Note. If , for all x ∈ X, then is a fuzzy fantastic filter.
Theorem 5.2.Let . Then if and only if
Proof. Let be an interval-valued hesitant fuzzy fantastic filter of . Using (BE3) we have x ∗ y = 1 ∗ (x ∗ y). Hence and so by Definition 5.1, we have
Conversely, since is an interval valued hesitant fuzzy filter and using hypothesis, we have
Therefore, . □
Theorem 5.3.Let and λ ∈ [0, 1]. Then
,
,
,
,
.
Theorem 5.4.Algebra of type (2, 2, 2, 0, 0) is a complete lattice.
Let f : X → Y be a homomorphism of BE-algebras and . Define a mapping such that , for all x ∈ X . Then is well defined and , where .
Theorem 5.5.Let f : X → Y be an onto homomorphism of BE-algebras and . Then if and only if .
Proof. Assume that . For any x ∈ X, we have
Hence (IVHFFF1) is valid. Now, let x, y, z ∈ X.
Therefore, .
Conversely, assume that . Let y ∈ Y. Since f is onto, there exist x ∈ X such that f (x) = y. Then
Now, let x, y, z ∈ Y. Then there exist a, b, c ∈ X such that f (a) = x, f (b) = y and f (c) = z. Hence we have
Therefore, . □
Let . Denote
For any β ∈ [0, ⊤], we define , for all x ∈ X, where . Obviously, is a mapping from X to [0, 1], that is, . is well-defined. Assume that β ∈ [0, ⊤] and x1 = x2. Then and so . Hence is well-defined.
Theorem 5.6.Let β ∈ [0, ⊤]. If , then .
Proof. Let x, y, z ∈ X . Then
Also, Therefore, . □
Theorem 5.7.If there exists β ∈ [0, ⊤] such that , then .
Proof. Assume that , for some β ∈ [0, ⊤] . Let x, y ∈ X . Since , we can see that .
Now, by canceling β we have . Also, by a similar argument we have . Therefore, . □
Let and γ ∈ [0, 1]. Define
where
and
We note that if γ : =1, then .
Theorem 5.8.Let γ ∈ [0, 1]. If , then .
Proof. Let x, y, z ∈ X . Then
Also,
Therefore, . □
Theorem 5.9.Let γ ∈ (0, 1]. If , then .
Proof. See the proof of Theorem 5.8, by canceling γ, since γ ≠ 0. □
We note that if γ = 0, then , for all . So Theorem 5.9 is not valid in this case.
Corollary 5.10.Let γ ∈ (0, 1]. Then if and only if .
Conclusions and future works
From a logical point of view, various filters, correspond to various sets of provable formulas, plays an important role in several algebraic structures. In this paper, we applied the theory of hesitant fuzzy sets to BE-algebras and developed the interval-valued hesitant fuzzy (implicative, fantastic) filters in BE-algebras. The relation between interval-valued hesitant fuzzy (implicative, fantastic) filters and interval-valued hesitant fuzzy filters is discussed and some properties is studied. This result can be used in other algebraic structures. In our future research, we will define different types of interval-valued hesitant fuzzy filters and discuss on relation between them and construct quotient algebras via these filters. Congruence relations on a BE-algebra with respect to these notions will be studied.
Compliance with ethical standards
The author declare that there is no conflict of interest. This article does not contain any studies with human participants or animals performed by the author.
Footnotes
Acknowledgments
The author would like to express their thanks to the reviewers for their comments and suggestions which improved the paper.
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