Research Article | Open Access

# Fuzzy-Fated Filters of -Algebras

**Academic Editor:**Jinde Cao

#### Abstract

Fuzzy set theory of fated filters in -algebras is considered. A characterization of a fuzzy-fated filter is established, and conditions for a fuzzy filter to be a fuzzy-fated filter are provided. The notion of an -fuzzy-fated filter is introduced. Characterizations of an -fuzzy-fated filter are provided. Implication-based fuzzy-fated filters are discussed.

#### 1. Introduction

One important task of artificial intelligence is to make the computers simulate beings in dealing with certainty and uncertainty in information. Logic appears in a βsacredβ (resp., a βprofaneβ) form which is dominant in proof theory (resp., model theory). The role of logic in mathematics and computer science is twofoldβas a tool for applications in both areasβand a technique for laying the foundations. Nonclassical logic including many-valued logic and fuzzy logic, takes the advantage of classical logic to handle information with various facets of uncertainty (see [1] for generalized theory of uncertainty), such as fuzziness randomness and Nonclassical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information. Among all kinds of uncertainties, incomparability is an important one which can be encountered in our life. The concept of -algebras was first introduced by Wang in [2] by providing an algebraic proof of the completeness theorem of a formal deductive system [3]. Obviously, -algebras are different from the BL-algebras. Jun and Lianzhen [4] studied (fated) filters of -algebras. Lianzhen and Kaitai [5] discussed the fuzzy set theory of filters in -algebras. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [6]. Murali [7] proposed a definition of a fuzzy point belonging to fuzzy subset under a natural equivalence on fuzzy subset. The idea of quasicoincidence of a fuzzy point with a fuzzy set, which is mentioned in [8], played a vital role to generate some different types of fuzzy subsets. It is worth pointing out that Bhakat and Das [9, 10] initiated the concepts of -fuzzy subgroups by using the βbelongs toβ relation and βquasicoincident withβ relation between a fuzzy point and a fuzzy subgroup and introduced the concept of an -fuzzy subgroup. In particular, an -fuzzy subgroup is an important and useful generalization of Rosenfeld's fuzzy subgroup. As a generalization of the notion of fuzzy filters in -algebras, Ma et al. [11] dealt with the notion of -fuzzy filters in -algebras.

In this paper, we deal with the fuzzy set theory of fated filters in -algebras. We provide conditions for a fuzzy filter to be a fuzzy-fated filter. We also introduce the notion of -fuzzy-fated filters and investigate related properties. We establish a relation between an -fuzzy filter and an -fuzzy-fated filter and provide conditions for an -fuzzy filter to be an -fuzzy-fated filter. We deal with characterizations of an -fuzzy-fated filter. Finally, we discuss the implication-based fuzzy-fated filters of an -algebra.

#### 2. Preliminaries

Let be a bounded distributive lattice with order-reversing involution and a binary operation , then is called an *-algebra* (see [2]) if it satisfies the following axioms: (R1),
(R2),
(R3),
(R4),
(R5),
(R6).

Let be an -algebra. For any , we define and . It is proved that and are commutative, associative, and , and is a residuated lattice.

For any elements , and of an -algebra , we have the following properties (see [12]): (a1) if and only if ,(a2), (a3), (a4), (a5) implies ,(a6) implies ,(a7), (a8), (a9) and ,(a10) and ,(a11), (a12), (a13) if and only if ,(a14) implies ,(a15), (a16).

A nonempty subset of an -algebra is called a *filter* of if it satisfies the following two conditions: (b1),
(b2).

It can be easily verified that a nonempty subset of an -algebra is a filter of if and only if it satisfies the following conditions: (b3), (b4).

*Definition 2.1. *A fuzzy subset of an -algebra is called a *fuzzy filter* of if it satisfies: (c1),
(c2) is order preserving, that is, .

Denote by the set of all filters of , and by the set of all fuzzy filters of .

Theorem 2.2. *A fuzzy subset of an -algebra is a fuzzy filter of if and only if it satisfies the following: *(c3)*,
*(c4)*.
*

For any fuzzy subset of and , the set

is called a *level subset* of . It is well known that a fuzzy subset of is a fuzzy filter of if and only if the nonempty level subset , of is a filter of .

A fuzzy subset of a set of the form
is said to be a * fuzzy point* with support and value and is denoted by .

#### 3. Fuzzy-Fated Filters

In what follows, is an -algebra unless otherwise specified. In [4], the notion of a fated filter of is introduced as follows.

A nonempty subset of is called a *fated filter* of (see [4]) if it satisfies (b1) and

Denote by the set of all fated filters of . Note that is a complete lattice under the set inclusion with the largest element and the least element . Now, we consider the fuzzy form of a fated filter of .

Lemma 3.1 (see [4]). *A filter of is fated if and only if the following assertion is valid:
*

Lemma 3.2 (see [4]). *A filter of is fated if and only if the following assertion is valid:
*

*Definition 3.3. *A fuzzy subset of is called a *fuzzy-fated filter* of if it satisfies the following assertion:

Denote by the set of all fuzzy-fated filters of .

*Example 3.4. *Let be a set with Hasse diagram and Cayley tables which are given in Table 1, then is an -algebra (see [5]), where and . Define a fuzzy subset of by
Then
which is a fated filter of . Therefore, is a fuzzy-fated filter of .

We provide a characterization of a fuzzy-fated filter.

Theorem 3.5. *For a fuzzy subset of if and only if it satisfies the following conditions: *(1)*,
*(2)*.
*

*Proof. *Suppose that is a fuzzy-fated filter of . For any , let , then , that is, , and so is a fated filter of . Thus, , and hence for all . For any , let
Then and . Since is a fated filter of , it follows from (3.1) that , so that
for all .

Conversely, let satisfy two conditions (1) and (2), and let be such that , then there exists such that . Using (1), we have , and so . Let be such that and , then and . It follows from (2) that
so that . Hence, is a fated filter of , and therefore is a fuzzy-fated filter of .

Theorem 3.6. *For any fuzzy subset of , one has .*

*Proof. *Let be a fuzzy-fated filter of . Replacing and by and , respectively, in Theorem 3.5(2) and using (R2), we have
for all . Using Theorem 2.2, is a fuzzy filter of .

The following example shows that the converse of Theorem 3.6 may not be true.

*Example 3.7. *Let be a set with Hasse diagram and Cayley tables which are given in Table 2, then is an -algebra (see [5]), where and . Define a fuzzy subset of by
Then is a fuzzy filter of , but it is not a fuzzy-fated filter of since

Theorem 3.8. *For any fuzzy filter of , the following assertions are equivalent: *(1)* is a fuzzy-fated filter of ,*(2)* satisfies the following inequality:*

*Proof. *Assume that is a fuzzy-fated filter of . Putting in Theorem 3.5(2) and using (R2) and Theorem 3.5(1), we have
for all .

Conversely, suppose that satisfies (3.13), it follows from (c4) that
for all , so from Theorem 3.5 that is a fuzzy-fated filter of .

Theorem 3.9. *Let be a fuzzy filter of , then if and only if it satisfies
**
for all .*

*Proof. *Assume that . If for all , then . Suppose that
for some , then there exists such that
It follows that and , so from Lemma 3.1 that , that is, . This is a contradiction, and so for all .

Conversely, let be a fuzzy filter of that satisfies (3.16). Let be such that , then by Theorem 3.5 in [5]. Assume that and for all , then and . Using (3.16), we have
and so . Therefore, , and thus is a fuzzy-fated filter of .

*Remark 3.10. *Based on Theorem 3.9 and [5, Definition 4.1], we know that the notion of a fuzzy-fated filter is equivalent to the notion of a fuzzy implicative filter.

#### 4. Fuzzy-Fated Filters Based on Fuzzy Points

For a fuzzy point and a fuzzy subset of , Pu and Liu [8] introduced the symbol , where . We say that (i)*belong to *, denoted by if ,(ii) is *quasicoincident with *, denoted by , if ,(iii) if or ,(iv) if does not hold for .

*Definition 4.1 (see [11]). * A fuzzy subset of is said to be an *-fuzzy filter* of if it satisfies (1),
(2),
for all and .

Theorem 4.2 (see [11]). *A fuzzy subset of is an -fuzzy filter of if and only if the following conditions are valid: *(1)*,
*(2)*.
*

*Definition 4.3. *A fuzzy subset of is said to be an *-fuzzy-fated filter* of if it satisfies (1),
(2),
for all and .

If a fuzzy subset of satisfies (c3) and Definition 4.3(2), then we say that is a *strong **-fuzzy-fated filter* of .

*Example 4.4. *Consider an -algebra which appeared in Example 3.4. Define a fuzzy subset of by
It is routine to verify that is a strong -fuzzy-fated filter of . A fuzzy subset of defined by
is an -fuzzy-fated filter of , but it is not a strong -fuzzy-fated filter of .

Obviously, every strong -fuzzy-fated filter is an -fuzzy-fated filter, but not converse as seen in Example 4.4.

We provided characterizations of an -fuzzy-fated filter.

Theorem 4.5. *A fuzzy subset of is an -fuzzy-fated filter of if and only if it satisfies the following inequalities: *(1)*,
*(2)*.
*

*Proof. *Let be an -fuzzy-fated filter of . Assume that there exists such that , then for some , and so . It follows from Definition 4.3(1) that , that is, or , so that or . This is a contradiction. Hence, for all . Suppose that there exist such that
then for some . Thus and . Using Definition 4.3(2), we have , which implies that or . This is a contradiction, and therefore
for all . Conversely, let be a fuzzy subset of that satisfies two conditions (1) and (2). Let and be such that , then , which implies from (1) that . If , then , that is, . If , then and so , that is, . Hence, . Let and be such that and , then and . It follows from (2) that
If , then , which shows that . If , then , and thus , that is, . Hence, . Consequently, is an -fuzzy-fated filter of .

Corollary 4.6. *Every strong -fuzzy-fated filter of satisfies the following inequalities: *(1)*,
*(2)*.
*

Theorem 4.7. *A fuzzy subset of is an -fuzzy-fated filter of if and only if it satisfies the following assertion:
*

*Proof. *Assume that is an -fuzzy-fated filter of . Let be such that , then there exists , and so . Using Theorem 4.5(1), we get
that is, . Assume that for all and , then and . It follows from Theorem 4.5 (2) that
so that . Therefore, is a fated filter of .

Conversely, let be a fuzzy subset of satisfying the assertion (4.6). Assume that for some . Putting , we have and so . Hence, is a fated filter of by (4.6), which implies that . Thus, , which is a contradiction. Therefore, for all . Suppose that
for some . Taking , we get and . It follows from (3.1) that , that is, . This is a contradiction. Hence,
for all . Using Theorem 4.5, we conclude that is an -fuzzy-fated filter of .

Proposition 4.8. *Every -fuzzy-fated filter of satisfies the following inequalities: *(1)*,
*(2)*,
** for all .*

*Proof. *(1) Suppose that there exist such that
Taking implies that , and . Since by Theorem 4.7, it follows from Lemma 3.1 that , that is, . This is a contradiction, and therefore satisfies (1).

(2) If is an -fuzzy-fated filter of , then for all by Theorem 4.7. Hence, for all . Suppose that
for some and then , which implies from Lemma 3.2 that , that is, . This is a contradiction. Hence, for all .

Theorem 4.9. *If is a fated filter of , then a fuzzy subset of defined by
**
where and is an -fuzzy-fated filter of .*

*Proof. *Note that
which is a fated filter of . It follows from Theorem 4.7 that is an -fuzzy-fated filter of .

Theorem 4.10. *Let be an -fuzzy-fated filter of . If , then is a fuzzy-fated filter of .*

*Proof. *It is straightforward.

For any fuzzy subset of and any , we consider two subsets:
It is clear that .

Theorem 4.11. *If is an -fuzzy-fated filter of , then
*

*Proof. *Assume that for all , then there exists , and so . Using Theorem 4.5(1), we have
which implies that . Assume that and for all , then and , that is, and . Using Theorem 4.5(2), we get
Thus, if , then
If , then . It follows that so that . Therefore, is a fated filter of .

Corollary 4.12. *If is a strong -fuzzy-fated filter of , then
*

The converse of Corollary 4.12 is not true as shown by the following example.

*Example 4.13. *Consider the -fuzzy-fated filter of which is given in Example 4.4, then
is a fated filter of . But is not a strong -fuzzy-fated filter of .

Theorem 4.14. *For a fuzzy subset of , the following assertions are equivalent: *(1)* is an -fuzzy-fated filter of ,*(2)*.
*

*Proof. *Assume that is an -fuzzy-fated filter of , and let be such that , then there exists , and so or . If , then . It follows from Theorem 4.5(1) that
so that . If , then , that is, . Thus,
and so . Let be such that and , then or , and or . We can consider four cases:
For the first case, Theorem 4.5(2) implies that
so that or , that is, . Hence, . Case (4.25) implies that
Thus, . Similarly, for the case (4.26). The final case implies that
so that . Consequently, is a fuzzy-fated filter of .

Conversely, let be a fuzzy subset of such that is a fated filter of whenever it is nonempty for all . If there exists such that , then for some . It follows that but . Also, and so . Hence, , which is a contradiction. Therefore, for all . Suppose that
for some . Taking implies that , and . Since , it follows that . But (4.31) induces and , that is, . This is a contradiction, and thus for all . Using Theorem 4.5, we conclude that is an -fuzzy-fated filter of .

#### 5. Implication-Based Fuzzy-Fated Filters

Fuzzy logic is an extension of set theoretic multivalued logic in which the truth values are linguistic variables or terms of the linguistic variable truth. Some operators, for example in fuzzy logic, are also defined by using truth tables, and the extension principle can be applied to derive definitions of the operators. In fuzzy logic, the truth value of fuzzy proposition is denoted by . For a universe of discourse, we display the fuzzy logical and corresponding set-theoretical notations used in this paper The truth valuation rules given in (5.3) are those in the Εukasiewicz system of continuous-valued logic. Of course, various implication operators have been defined. We show only a selection of them in the following. (a)Gaines-Rescher implication operator (): (b)GΓΆdel implication operator (): (c)The contraposition of GΓΆdel implication operator ():

Ying [13] introduced the concept of fuzzifying topology. We can expand his/her idea to -algebras, and we define a fuzzifying fated filter as follows.

*Definition 5.1. *A fuzzy subset of is called a *fuzzifying fated filter* of if it satisfies the following conditions: (1)for all , we have
(2)for all , we get

Obviously, conditions (5.9) and (5.10) are equivalent to Theorem 3.5(1) and Theorem 3.5(2), respectively. Therefore, a fuzzifying fated filter is an ordinary fuzzy-fated filter.

In [14], the concept of -tautology is introduced, that is,

*Definition 5.2. *Let be a fuzzy subset of and , then is called a *-implication-based fuzzy-fated filter* of if it satisfies the following conditions: (1)for all , we have
(2)for all , we get

Let be an implication operator. Clearly, is a -implication-based fuzzy-fated filter of if and only if it satisfies (1),
(2).

Theorem 5.3. *For any fuzzy subset of , one has the following: *(1)*if , then is a 0.5-implication-based fuzzy-fated filter of if and only if is a fuzzy-fated filter of ,*(2)*if , then is a 0.5-implication-based fuzzy-fated filter of if and only if is an -fuzzy-fated filter of ,*(3)*if , then is a 0.5-implication-based fuzzy-fated filter of if and only if satisfies the following conditions:(3.1) ,
(3.2)*

*for all .*

*Proof. *(1) It is Straightforward.

(2) Assume that is a 0.5-implication-based fuzzy-fated filter of , then(i),(ii). From (i), we have or , and so for all . The second case implies that
or . It follows that
for all . Using Theorem 4.5, we know that is an -fuzzy-fated filter of .

Conversely, suppose that is an -fuzzy-fated filter of . From Theorem 4.5(1), if , then . Otherwise, . From Theorem 4.5(2), if
then , and so
If , then , and thus
Consequently, is a 0.5-implication-based fuzzy-fated filter of .

(3) Suppose that satisfies (3.1) and (3.2). In (3.1), if , then , and hence . If , then
If in (5.19), then . Hence,
If in (5.19), then which implies that
In (3.2), if , then
and so . Therefore,
If
then
Thus, if in (5.25), then and
Therefore,
whenever ,
whenever . Now, if
in (5.25), then , and so

Consequently, is a 0.5-implication-based fuzzy-fated filter of .

Conversely, assume that is a 0.5-implication-based fuzzy-fated filter of , then (iii),
(iv),
for all . The case (iii) implies that , that is, , or and so . It follows that
From (iv), we have
that is, , or
Hence,
for all . This completes the proof.

#### 6. Conclusion

Using the βbelongs toβ relation () and quasicoincidence with relation () between fuzzy points and fuzzy sets, we introduced the notion of an -fuzzy-fated filter, this is a generalization of a fuzzy implicative filter. In fuzzy logic, one can see that various implication operators have been defined. We used Gaines-Rescher implication operator (), GΓΆdel implication operator (), and the contraposition of GΓΆdel implication operator () to study -implication-based fuzzy-fated filters.

There are also other situations concerning the relations between this kind of results, another type of structures (e.g., -fuzzy-fated filter), and (fuzzy) soft and rough set theory. How to deal with these situations will be one of our future topics. We will also try to study the intuitionistic fuzzy version of several type of filters in -algebras related to the intuitionistic βbelongs toβ relation () and intuitionistic quasicoincidence with relation () between intuitionistic fuzzy points and intuitionistic fuzzy sets.

#### Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable comments and suggestions for improving this paper.

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#### Copyright

Copyright © 2011 Jeong Soon Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.