Abstract
The new concept of the fuzzy filter degree was given by means of the implication operator, which enables to measure a degree to which a fuzzy subset of a BL-algebra is a fuzzy filter. In this paper, we put forward several equivalent characterizations of the fuzzy filter degree by studying its properties and the relationship with level cut sets. Furthermore, we study the fuzzy filter degrees of the intersection and fuzzy direct products of fuzzy subsets and investigate the fuzzy filter degrees of the image and the preimage of a fuzzy subset under a homomorphism.
1. Introduction
As we know, nonclassical logic algebras are one of the focuses in the basic study of artificial intelligence. Among these logic algebras, BL-algebra is one of the most fundamental algebraic structures, which was put forward by Hájek [1] in 1998. MV-algebras, Gödel algebras, and product algebras are the most known classes of BL-algebras.
Filters theory plays an important role in the theoretical study of these algebras. From logical point of view, a kind of filter corresponds to a set of provable formula. In [2], Liu proposed the notion of fuzzy filters in BL-algebras and investigated some of their properties and showed that fuzzy filters are a useful tool to obtain results on classical filters of BL-algebras.
In the application of fuzzy theory, a series of concepts of degree were introduced to depict the level of similarity among objects. When these thoughts are applied in the research of fuzzy logic algebras, the question is how to measure the degree to which a fuzzy subset is a fuzzy logic algebra. In 2010, Shi gave an idea in [3]. Shi proposed the concept of fuzzy subgroup degree in the paper to depict the degree to which a fuzzy subset is a fuzzy subgroup. Using this idea, Wang, Shi, Liao, and others have done a lot of work in recent years and got a lot of good results [4–15].
Inspired by the ideas mentioned above, we present a new concept of fuzzy filter, by which the degree to which a fuzzy subset of lattice L is a fuzzy filter is depicted and some academic results are obtained.
This paper is organized as follows: In Section 2, the basic knowledge required in the paper is proposed. In Section 3, we put forward the concept of fuzzy filter degree and its equivalent characterizations. Furthermore, the fuzzy filter degrees of the intersection and fuzzy direct products of fuzzy subsets are investigated and the fuzzy filter degrees of the image and the preimage of a fuzzy subset under a homomorphism are also discussed.
In what follows, let L denote a BL-algebra.
2. Preliminaries
In this section, we recollect some basic definitions and results which will be used in the following.
Definition 1 (see [1]). A BL-algebra is a structure such that(i) is a bounded lattice(ii) is an abelian monoid, i.e., is commutative and associative and (iii)The following conditions hold for all :(B1) if and only if (residuation)(B2) (divisibility)(B3) (prelinearity)
Definition 2 (see [1]). Let L be a BL-algebra. A subset F of L is called a filter of L if it satisfies(1)(2) and imply
Definition 3 (see [1]). Let A be a fuzzy set in L. A is called a fuzzy filter if it satisfies(1) for all (2) for all x,
Definition 4 (see [1]). A mapping f: from a BL-algebra L into a BL-algebra M is called a BL homomorphism if for any x, :(1)(2)(3)where and are the least elements of L and M, respectively.
If f is a surjection, then f is called an epimorphism. If is an injective, then is called a monomorphism. If is a bijective, then is called an isomorphism.
It is easy to verify that , where 1L and 1M are the largest elements of L and M, respectively.
Definition 5 (see [16]). A mapping I: [0,1] × [0,1] ⟶ [0,1] is called a fuzzy implication operator, if it satisfies(1) implies for all x, y, (2) implies for all x, y, (3), For any x, , let . Obviously, I is an implication, which is called the R-implication generated by “min,” and is usually denoted as x ⟶ y.
The implication operator has the following properties.
Proposition 1 (see [17]). For any a, , the following hold:(1)(2)(3)
Definition 6 (see [18]). Let A be a fuzzy set in L, , then the set is called a level subset of A; the set is called a strong level subset of A.
Definition 7 (see [16]) (extension principle). With the mapping , two mappings can be induced by this mapping, respectively, denoted as f and as follows:where the membership function of and is defined as
Definition 8 (see [19]). Let be a family of BL-algebras. is defined as . The operations “, , , and ⟶” are given as follows:where 1i and 0i are the largest element and the least element of Li, respectively. (, , ) is called the BL direct product.
It is easy to verify that (, , , , ⟶, O, I) is a BL-algebra.
Definition 9 (see [20]). Let Ai be a fuzzy subset of Li (i ∈ I). The fuzzy subset of is defined aswhere is called the fuzzy direct product of .
Definition 10 (see [21]). Let f be a mapping of a BL-algebra L into a BL-algebra M. Then, a fuzzy subset A of L is called f-invariant if for all , and implies .
3. Fuzzy Filters Degree
Definition 11. Let A be a fuzzy subset in L. The fuzzy filter degree of A is defined asFor convenience, we denoteSo, .
Remark 1. In Definition 11, can be used to measure the degree of a fuzzy subset A to be a fuzzy filter of L.
Example 1. Let L = {0, a, b, 1} be a lattice, and the order “” on L be determined by using Figure 1. For all x, y ∈ L, the two binary operations ⟶ and are defined by Table 1 and 2. Then, (, , , , ⟶, 0, 1) is a BL-algebra.
Let A1, A2, and A3 be the fuzzy sets in L given byOne can easily check that A1 is a fuzzy filter of L. Neither A2 nor A3 is a fuzzy filter of L. From Definition 11, we have , , and .
Since I (x, y) is a continuous t-norm on [0,1], it is not difficult to have the following.
Theorem 1. Let A be a fuzzy subset of L. A is a fuzzy filter of L if and only if .
The properties of are discussed below, and their equivalent characterizations are also given.
Lemma 1. Let A be a fuzzy subset of L. if and only if for all x, y ∈ L, and .
Proof. Suppose and then by Definition 11. Thus, and .
Note . There exists for any such that . Therefore, since , we have . So, .
By the arbitrariness of , as well as ; therefore, . Similarly, we can prove .
Conversely, if and for any x, y ∈ L, then and . Hence, we haveThe proof is completed.
Theorem 2. Suppose A be a fuzzy subset of L andThen, .
Proof. We first show that .
Let . By Lemma 1, we get that for any x, y ∈ L, and .
Let b = ∨K, and it is easy to get . There exists for any such that . Since , we have and for any x, y ∈ L. So, and .
By the arbitrariness of ε, we haveMoreover, and c ≥ b by Lemma 1. So, b = c.
Now we prove .
As and for any a ∈ K and x, y ∈ L, we have a ∈ K1 and a ∈ K2. Then, and . Thus, . Therefore, .
Let . There exists , such that for any .
Let . Then, . So, .
Thus, and .
By the arbitrariness of ε, we haveBy Lemma 1, we get that , that is,
Finally, we have .
We know there is a close relationship between the fuzzy filter of a BL-algebra and its level set. Next, we give some characterizations of the fuzzy filter degree of a fuzzy set by means of its level sets.
Lemma 2. Let A be a fuzzy subset of L. If , then nonvoid A[b] is a filter of L for any b ∈ (0, c].
Proof. If for any ; then, for any x, , we have and . Since , the followings are obtained by Lemma 1:So, and . Hence, is a filter of L.
For strong level sets, we can get the similar conclusions such as Lemma 3 by suitable modification to the proof of Lemma 2.
Lemma 3. Let A be a fuzzy subset of L. If , then nonvoid A(b) is a filter of L for any b ∈ (0, c].
Theorem 3. Let A be a fuzzy subset of L. If , then
Proof. Suppose that and . According to Lemma 2, we have c ∈ B; thus, , i.e.,
Next, we prove .
In fact, for any a ∈ B and x, y ∈ L, we have . Otherwise, there exists x0, y0 ∈ L, such that .
Let . Then, , , and . That is, and .
Since a ∈ B and, is a filter of L. This together with implies that , i.e., . This is contrary to
Similarly, for any a ∈ B, x ∈ L, we have . By Lemma 1, it follows that . Thus, .
From the above, we have .
With Lemma 1 and Lemma 3, proceeding similar as in the proof of Theorem 3, we have the following theorem.
Theorem 4. Let A be a fuzzy subset of L. Then, .
Next, we discuss the fuzzy filter degree under fuzzy subset operations.
Firstly, the relation of the fuzzy filter degree between the intersection of a family of fuzzy subsets and each fuzzy subset is given.
Theorem 5. Let {Ai}i∈I be a family of fuzzy subsets of L. Then,
Proof. By Proposition 1 and Definition 5, we haveFollowing in a similar manner, we can getFrom Definition 11, it follows thatSecondly, the fuzzy filter degree of fuzzy subsets under the fuzzy direct product is discussed.
Theorem 6. Let Ai be a fuzzy subset of ; then, .
Proof. Suppose x, . Then, and , where xi, yi ∈ Li.Following in a similar manner, we can getSo, by Definition 11,Finally, we give the relation between the fuzzy ideal degrees of the image and the preimage of a fuzzy subset under a homomorphism.
Theorem 7. Let f: L1⟶ L2 be a BL homomorphism, and A and B be the fuzzy subsets of L1 and L2, respectively. The following assertions hold:(1)If f is a surjective and f-invariant, then (2)(3)If f is a surjective, then .
Proof. (1)By Theorem 2 and Definition 7, we can have that Since f is an epimorphism, there exists x0, y0 ∈ L1 such that and . Thus, . Now that f is an f-invariant, which together with , we have It follows that(2)By Definition 11 and Definition 7, we can have that(3)If f is an epimorphism, then from the proof of (2), we have
4. Conclusions
In order to measure the degree to which a fuzzy subset is a fuzzy filter, we put forward the concept of the fuzzy filter degree of L. Through the research on the fuzzy filter degree of L, we have a deeper understanding about the fuzzy filter, further enriching the theory of fuzzy filter of BL-algebras. Using this idea, we can also measure other fuzzy algebraic structures and it also provides theoretical basis for fuzzy pattern recognition and other applications.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
Thanks are due to the support by the National Science Foundations of China (nos: 61673193, 61170121, and 11401259) and the Natural Science Foundations of Jiangsu Province (no: BK20151117).