#### Abstract

Notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice are introduced and some related properties are investigated. The characterizations of (subpositive implicative, Boolean) vague filters is obtained. We prove that the set of all vague filters of a residuated lattice forms a complete lattice and we find its distributive sublattices. The relation among subpositive implicative vague filters and Boolean vague filters are obtained and it is proved that subpositive implicative vague filters are equivalent to Boolean vague filters.

#### 1. Introduction

In the classical set, there are only two possibilities for any elements: in or not in the set. Hence the values of elements in a set are only one of and . Therefore, this theory cannot handle the data with ambiguity and uncertainty.

Zadeh introduced fuzzy set theory in 1965 [1] to handle such ambiguity and uncertainty by generalizing the notion of membership in a set. In a fuzzy set each element is associated with a point-value selected from the unit interval , which is termed the grade of membership in the set. This membership degree contains the evidences for both supporting and opposing .

A number of generalizations of Zadeh’s fuzzy set theory are intuitionistic fuzzy theory, L-fuzzy theory, and vague theory. Gau and Buehrer proposed the concept of vague set in 1993 [2], by replacing the value of an element in a set with a subinterval of . Namely, a true membership function and a false-membership function are used to describe the boundaries of membership degree. These two boundaries form a subinterval of . The vague set theory improves description of the objective real world, becoming a promising tool to deal with inexact, uncertain, or vague knowledge. Many researchers have applied this theory to many situations, such as fuzzy control, decision-making, knowledge discovery, and fault diagnosis. Recently in [3], Jun and Park introduced the notion of vague ideal in pseudo MV-algebras and Broumand Saeid [4] introduced the notion of vague BCK/BCI-algebras.

The concept of residuated lattices was introduced by Ward and Dilworth [5] as a generalization of the structure of the set of ideals of a ring. These algebras are a common structure among algebras associated with logical systems (see [6–9]). The residuated lattices have interesting algebraic and logical properties. The main example of residuated lattices related to logic is and BL-algebras. A basic logic algebra (BL-algebra for short) is an important class of logical algebras introduced by Hajek [10] in order to provide an algebraic proof of the completeness of “Basic Logic” (BL for short). Continuous t-norm based fuzzy logics have been widely applied to fuzzy mathematics, artificial intelligence, and other areas. The filter theory plays an important role in studying these logical algebras and many authors discussed the notion of filters of these logical algebras (see [11, 12]). From a logical point of view, a filter corresponds to a set of provable formulas.

In this paper, the concept of vague sets is applied to residuated lattices. In Section 2, some basic definitions and results are explained. In Section 3, we introduce the notion of vague filters of a residuated lattice and investigate some related properties. Also, we obtain the characterizations of vague filters. In Section 4, the smallest vague filters containing a given vague set are established and we obtain the distributive sublattice of the set of all vague filters of a residuated lattice. In Section 5, notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice are introduced and some related properties are obtained.

#### 2. Preliminaries

We recall some definitions and theorems which will be needed in this paper.

*Definition 1 (see [6]). *A* residuated lattice* is an algebraic structure such that(1) is a bounded lattice with the least element 0 and the greatest element 1,(2) is a monoid; that is, is associative and ,(3) if and only if if and only if , for all .

In the rest of this paper, we denote the residuated lattice by , , and .

Proposition 2 (see [6, 9]). *Let be a residuated lattice. Then we have the following properties: for all ,*(1)*,*(2)*,*(3)*,*(4)* if and only if if and only if ,*(5)*if , then , ,*(6)*if , then , ,*(7)*if , then , ,*(8)*, ,*(9)*, ,*(10)*, .*

*Definition 3 (see [6, 9]). *Let be a nonempty subset of a residuated lattice . is called a* filter* if(F1) imply ,(F2) and imply .

Theorem 4 (see [6, 9]). *A nonempty subset of a residuated lattice is a filter if and only if*(F3)* and imply , or*(F3′)* and imply .*

*Definition 5 (see [11]). *Let be a nonempty subset of a residuated lattice . is called a* subpositive implicative filter* of if(SPIF1),(SPIF2) and imply for any ,(SPIF3) and imply for any .

Proposition 6 (see [11]). *Let be a filter of . is a subpositive implicative filter if and only if*(SPIF4)* imply for any ,*(SPIF5)* imply for any .*

*Definition 7 (see [1]). *Let be a nonempty set. A* fuzzy set* in is a mapping .

Proposition 8 (see [12]). *Let be a residuated lattice. A fuzzy set is called a fuzzy filter in if it satisfies *

*(fF1)*

*imply ,*

*(fF2)*

*,*

*for all .*

*Definition 9 (see [13]). *A* vague set * in the universe of discourse is characterized by two membership functions given by(i)a true membership function ,(ii)a false membership function ,where is a lower bound on the grade of membership of derived from the “evidence for ,” is a lower bound on the negation of derived from the “evidence against ,” and .

Thus the grade of membership of in the vague set is bounded by a subinterval of . This indicates that if the actual grade of membership of is , then . The vague set is written as , where the interval is called the* vague value* of in , denoted by .

*Definition 10 (see [13]). *A vague set of a set is called(1)the* zero vague set* of if and for all ,(2)the* unit vague set* of if and for all ,(3)the *-vague set* of if and for all , where .

Let denote the family of all closed subintervals of . Now we define the refined minimum (briefly, imin) and an order on elements and of as

Similarly we can define , , and . Then the concept of and could be extended to define and of infinite number of elements of . It is a known fact that is a lattice with universal bounds zero vague set and unit vague set. For we now define -cut and -cut of a vague set.

*Definition 11 (see [13]). *Let be a vague set of a universe with the true-membership function and false-membership function . The -cut of the vague set is a crisp subset of the set given by
Clearly . The -cuts are also called vague-cuts of the vague set .

*Definition 12 (see [13]). *The -cut of the vague set is a crisp subset of the set given by . Equivalently, we can define the -cut as .

#### 3. Vague Filters of Residuated Lattices

In this section, we define the notion of vague filters of residuated lattices and obtain some related results.

*Definition 13. *A vague set of a residuated lattice is called a* vague filter* of if the following conditions hold:(VF1)if , then , for all ,(VF2), for all .

That is,(1) implies and , for all ,(2) and , for all .

In the following proposition, we will show that the vague filters of a residuated lattice exist.

Proposition 14. *Unit vague set and -vague set of a residuated lattice are vague filters of .*

*Proof. *Let be a -vague set of . It is clear that satisfies (VF1). For we have
Thus , for all . Hence it is a vague filter of . The proof of the other cases is similar.

Lemma 15. *Let be a vague set of a residuated lattice such that it satisfies (VF1). Then the following are equivalent:*(VF4)*,*(VF5)*,**
for all .*

*Proof. *Suppose that satisfies (VF4). We have . By (VF1), . We get that
Conversely, if satisfies (VF5), then the inequality analogously entails (VF4).

Theorem 16. *Let be a vague set of a residuated lattice . Then is a vague filter if and only if it satisfies (VF4) and *(VF6)*, for all .*

*Proof. *Let be a vague filter of . Since , then , for all by (VF1). We have . By (VF1) and (VF2) we get
Hence (VF4) hold.

Conversely, let satisfy (VF4) and (VF6). Since , then ;
Hence is order-preserving and (VF1) holds. By Proposition 2 part (10), . By (VF4), we have
By Lemma 15,
Hence (VF2) holds.

Theorem 17. *Let be a vague set of a residuated lattice . Then is a vague filter if and only if it satisfies (VF5) and (VF6).*

*Definition 18. *A vague set of a residuated lattice is called a* vague lattice filter* of if it satisfies (VF1) and(VLF2), for all .

Proposition 19. *Let be a vague filter of a residuated lattice ; then is a vague lattice filter.*

In the following example, we will show that the converse of Proposition 19 may not be true in general.

*Example 20. *Let such that , , where and are incomparable. Define the operations , , and as follows:
Then is a residuated lattice. Let be the vague set defined as follows:
It is routine to verify that is a vague lattice filter, but it is not a vague filter of .

Let be the vague set defined as follows:
Then is a vague filter of .

In the following theorems, we will study the relationship between filters and vague filters of a residuated lattice.

Theorem 21. *Let be a vague set of a residuated lattice . Then is a vague filter of if and only if, for all , the set is either empty or a filter of , where .*

*Proof. *Suppose that is a vague filter of and . Let . We will show that is a filter of .(1)Suppose that and . By (VF1), . Therefor .(2)Suppose that . Then . By (VF2), we have . Hence and then is a filter of .

Conversely, suppose that, for all , the sets is either empty or a filter of . Let , , , and . Put and . Then and . Hence ; that is, . So and, by assumption, is a filter of . We obtain that ; that is,
Hence (VF2) holds.

Suppose that , , and . Hence . By assumption is a filter. We get that ; that is, . Hence is a vague filter of .

The filters like are also called* vague-cuts filters* of .

Corollary 22. *Let be a vague filter of a residuated lattice . Then for all , the set is either empty or a filter of .*

Corollary 23. *Any filter of a residuated lattice is a vague-cut filter of some vague filter of .*

*Proof. *For all , define
Hence we have , if and , otherwise, where . It is clear that . Let . If , then . We have . If one of or does not belong to , then one of and is equal to . Therefore . Suppose that . If , then . Hence . If , then . We have . Hence is a vague filter of .

In the following proposition, we will show that vague filters of a residuated lattice are a generalization of fuzzy filters.

Proposition 24. *Let be a vague set of a residuated lattice . Then is a vague filter of if and only if the fuzzy sets and are fuzzy filters in .*

#### 4. Lattice of Vague Filters

Let and be vague sets of a residuated lattice . If , for all , then we say containing and denote it by .

The unite vague set of a residuated lattice is a vague filter of containing any vague filter of . For a vague set, we can define the intersection of two vague sets and of a residuated lattice by , for all . It is easy to prove that the intersection of any family of vague filters of is a vague filter of . Hence we can define a vague filter generated by a vague set as follows.

*Definition 25. *Let be a vague set of a residuated lattice . A vague filter is called a* vague filter generated* by , if it satisfies(i),(ii) that implies , for all vague filters of .

The vague filter generated by is denoted by .

*Example 26. *Consider the vague lattice filter in Example 20 which is not a vague filter of . Then the vague filter generated by is

Theorem 27. *Let be a vague set of a residuated lattice . Then the vague value of in is
**
for all .*

*Proof. *First, we will prove that is a vague set of :
Now, we will show that is a vague filter of .(VF1)Suppose that is arbitrary. For every , we can take such that , , and
By Proposition 2 part (4), we have and then
Since is arbitrary, we obtain that .

Similarly, we can show that . Hence , for all .(VF2)Suppose that where . Let such that . Then . And we get that
Since , where is arbitrary, then
Therefore is a vague filter of by Theorem 16. It is clear that . Now, let be a vague filter of such that . Then we have
for all . Hence and then is a vague filter generated by .

We denote the set of all vague filters of a residuated lattice by .

Corollary 28. * is a complete lattice.*

*Proof. *Clearly, if is a family of vague filters of a residuated lattice , then the infimum of this family is and the supremum is . It is easy to prove that is a complete lattice.

Let and be vague sets of a residuated lattice . Then is defined as follows: for all .

Theorem 29. *Let and be vague sets of a residuated lattice . Then is a vague set of .*

*Proof. *Since and are vague sets of , we have and for all . Thus
Hence and then is a vague set of .

Theorem 30. *Let and be vague filters of a residuated lattice . Then is a vague filter of .*

*Proof. *By Theorem 29, is a vague set of . We will prove that is a vague filter of .(VF2)Let and . We get that . Since is arbitrary, we have
(VF1)Let . Then
By Definition 13, is a vague filter of .

Theorem 31. *Let and be vague filters of a residuated lattice such that . Then is the least upper bound of and .*

*Proof. *By Definition 13, we have for all . Let . Since , we get that
Hence . Similarly, we can show that . Suppose that is the vague filter of containing and . Then
Hence ; that is, is the least upper bound of and .

Lemma 32. *Let be a vague filter of a residuated lattice and . Then *(1)*, for all ;*(2)*, for .*

*Proof. *(1) Since , then by (VF2). Hence . Also, we have . We obtain that

(2) It follows from Proposition 2 part (2) and Definition 13.

Let VFD() be the set of all vague filters and of such that .

Theorem 33. * is a distributive lattice.*

*Proof. *Let be such that . We will show that . By Lemma 32,

#### 5. Subpositive Implicative Vague Filters

*Definition 34. *Let be a vague subset of a residuated lattice . is called a* subpositive implicative vague filter* of if(VF6), for all ,(SVF1) for any ,(SVF2) for any .

Proposition 35. *Let be a vague filter of a residuated lattice . Then is a subpositive implicative vague filter of L if and only if it satisfies*(SVF3)* for any ,*(SVF4)* for any .*

Proposition 36. *Let be a subpositive implicative vague filter of a residuated lattice . Then is a vague filter of .*

*Proof. *We have . By (VS1) and Theorem 17, is a vague filter of a residuated lattice .

The relation between subpositive implicative vague filter and its level subset is as follows.

Theorem 37. *Let be a vague set of a residuated lattice . Then is a subpositive implicative vague filter of if and only if for all , the set is either empty or a subpositive implicative filter of , where .*

*Proof. *Suppose that is a subpositive implicative filter vague set of and . Let . Since is a vague filter, then is a filter of by Theorem 21. Now, let . Then . By (SVF3), ; that is, . Similarly, we can show that satisfies (SVF4). Hence is a subpositive implicative filter of .

Conversely, let and . Thus . We get that . So . Thus ; that is, . Similarly, we can prove that satisfies (SVF4).

In the following theorems, we obtain some characterizations of subpositive implicative vague filter.

Theorem 38. *Let be a vague filter of a residuated lattice . Then is a subpositive implicative vague filter of if and only if it satisfies*(SVF5)* for any ,*(SVF6)* for any .*

*Proof. *Let be a subpositive implicative vague filter of . By Proposition 2 part (3), we have . By Proposition 2 part (7), . We get that . Hence . Thus it satisfies (SVF5). Similarly, we can prove (SVF6). Conversely, let satisfy (SVF5) and (SVF6); . Similarly, we can prove that (SVF5) holds. Hence is a vague subpositive implicative filter.

Theorem 39. *Let be a vague filter of a residuated lattice . Then is a subpositive implicative vague filter of if and only if it satisfies*(SVF7)* for any ;*(SVF8)* for any .*

*Proof. *Let (SVF7) and (SVF8) hold. We will show that satisfies (SVF5) and (SVF6). We have . Also, we have . Since is a vague filter, . Hence satisfies (SVF5). Analogously, we can prove that satisfies (SVF6). Hence is a subpositive implicative vague filter of by Theorem 38. Conversely, let be a subpositive implicative vague filter of . We have
We get that , by (SVF5). Similarly, we can prove that (SVF8) holds.

Theorem 40. *Let be a vague filter of a residuated lattice . Then is a subpositive implicative vague filter of if and only if it satisfies*(SVF9)* for any ,*(SVF10)* for any .*

*Proof. *Let be a subpositive implicative vague filter of . Then satisfies (SVF5) and (SVF6):
Similarly, we can prove that (SVF10) holds. Conversely, let satisfy (SVF9) and (SVF10). We will show that satisfies (SVF5) and (SVF6); . Hence satisfies (SVF5). Similarly, we can prove (SVF7) holds. Therefore is a subpositive implicative vague filter of L by Theorem 38.

Theorem 41. * for any ,*(SVF12)* for any .*

*Proof. *Let be a subpositive implicative vague filter of . Then (SVF11) and (SVF12) follow by (SVF9) and (SVF10). Conversely, since , then . Hence by Proposition 2 part (6). Therefore . Similarly, we can prove that (SVF10) holds. Hence is a subpositive implicative vague filter of by Theorem 40.

The extension theorem of subpositive implicative vague filters is obtained from the following theorem.

Theorem 42. *Let and be two vague filters of a residuated lattice which satisfies and . If is a subpositive implicative vague filter of L, so is.*

*Proof. *Suppose that . Then
We have . Therefore . By (SVF10), . Hence . Since , then . We get that . Similarly, we can show that satisfies (SVF11). Hence is a vague subpositive implicative filter of , by Theorem 40.

*Definition 43. *Let be a vague filter of a residuated lattice . is called a* vague Boolean filter* of if(BVF1), for all ,(BVF2), for all .

Similar to Theorem 40, we have the following theorem with respect to Boolean vague filters.

Theorem 44. *Let be a vague set of a residuated lattice . Then is a Boolean vague filter of if and only if for all ; the set is either empty or a Boolean filter of , where .*

Theorem 45. *Let be a vague filter of a residuated lattice . Then is a subpositive implicative vague filter of if and if only if is a Boolean vague filter.*

*Proof. *Let be a subpositive implicative vague filter. We have
Hence . Similarly, we can prove that satisfies (BVF2). Hence is a Boolean vague filter.

Conversely, let . We have . Similarly, we can show that satisfies (SVF12). Hence is a subpositive implicative vague filter by Theorem 41.

#### 6. Conclusion

In this paper, we introduced the notions of vague filters, subpositive implicative vague filters, and Boolean vague filters of a residuated lattice and investigated some related properties. We studied the relationship between filters and vague filters of a residuated lattice and showed that vague filters of a residuated lattice is a generalization of fuzzy filters. We proved that the set of all vague filters of a residuated lattice forms a complete lattice and obtained some characterizations of subpositive implicative vague filter. The extension theorem of subpositive implicative vague filters was obtained. Finally, it was proved that subpositive implicative vague filters are equivalent to Boolean vague filters.

#### Conflict of Interests

There is no conflict of interests regarding the publication of this paper.