Research Article | Open Access

# Classes of Int-Soft Filters in Residuated Lattices

**Academic Editor:**Xiaolong Xin

#### Abstract

The notions of int-soft filters, int-soft *G*-filters, regular int-soft filters, and *MV*-int-soft filters in residuated lattices are introduced, and their relations, properties, and characterizations are investigated. Conditions for an int-soft filter to be an int-soft *G*-filter, a regular int-soft filter, or an *MV*-int-soft filter are provided. The extension property for an int-soft *G*-filter is discussed. Finally, it is shown that the notion of an *MV*-int-soft filter coincides with the notion of a regular int-soft filter in *BL*-algebras.

#### 1. Introduction

In order to deal with fuzzy and uncertain information, nonclassical logic has become a formal and useful tool. As the semantic systems of nonclassical logic systems, various logical algebras have been proposed. Residuated lattices are important algebraic structures which are basic of -algebras, -algebras, -algebras, GÃ¶del algebras, -algebras, lattice implication algebras, and so forth. The (fuzzy) filter theory in the logical algebras has an important role in studying these algebras and completeness of the corresponding nonclassical logics, and it is studied in [1â€“8]. Uncertainty is an attribute of information. As a new mathematical tool for dealing with uncertainties, Molodtsov [9] introduced the concept of soft sets. Since then several authors studied (fuzzy) algebraic structures based on soft set theory in several algebraic structures. Acar et al. [10] introduced initial concepts of soft rings. Ahn et al. [11] introduced the notion of int-soft filters of a -algebra and investigated related properties. They also discussed characterization of an int-soft filter and solved the problem of classifying int-soft filters by their -inclusive filter. Aktas and Ã‡agman [12] defined soft groups and derived their basic properties using Molodtsovâ€™s definition of the soft sets. AtagÃ¼n and Sezgin [13] introduced and studied soft subrings and soft ideals of a ring by using Molodtsovâ€™s definition of the soft sets. Moreover, they introduced soft subfields of a field and soft submodule of a left R-module and investigated some related properties about soft substructures of rings, fields, and modules. Ã‡aÄŸman and EnginoÄŸlu [14] constructed a uni-int decision making method which selects a set of optimum elements from the alternatives. Feng et al. [15] improved and further extended Ã‡aÄŸman and EnginoÄŸluâ€™s uni-int decision making method in virtue of choice value soft sets and k-satisfaction relations. Ã‡aÄŸman and EnginoÄŸlu [16] discussed fuzzy parameterized (FP) soft sets and their related properties and proposed a decision making method based on FP-soft set theory. Feng [17] considered the application of soft rough approximations in multicriteria group decision making problems. Feng et al. [18] initiated the study of soft semirings by using the soft set theory. Jun et al. applied the notion of soft sets by Molodtsov to the theory of -algebras, -algebras, and subtraction algebras (see [19â€“22]). Jun et al. [23] discussed (strong) intersection-soft filters in -algebras. Zhan and jun [24] investigated characterizations of (implicative, positive implicative, and fantastic) filteristic soft -algebras by means of -soft sets and -soft sets. Recently, Feng and Li [25] explored some relationships among five different types of soft subsets and investigated free soft algebras with respect to soft product operations. They pointed out that soft sets have some nonclassical algebraic properties which are distinct from those of crisp sets or fuzzy sets.

In this paper, we introduce the notions of int-soft filters, int-soft -filters, regular int-soft filters, and -int-soft filters in residuated lattices and investigate their relations and properties. We consider characterizations of int-soft filters, int-soft -filters, regular int-soft filters, and -int-soft filters. We provide conditions for an int-soft filter to be an int-soft -filter, a regular int-soft filter, or an -int-soft filter. We establish the extension property for an int-soft -filter. Finally, we show that the notion of an -int-soft filter coincides with the notion of a regular int-soft filter in -algebras.

#### 2. Preliminaries

*Definition 1 (see [1, 26, 27]). *A residuated lattice is an algebra of type such that (1) is a bounded lattice;(2) is a commutative monoid;(3) and form an adjoint pair; that is,

In a residuated lattice , the ordering and negation are defined as follows: and for all .

Proposition 2 (see [1, 2, 6, 7, 26, 27]). *In a residuated lattice , the following properties are valid:
*

*Definition 3 (see [5]). *A nonempty subset of a residuated lattice is called a filter of if it satisfies the conditions

Proposition 4 (see [5]). *A nonempty subset of a residuated lattice is a filter of if and only if it satisfies
*

A soft set theory is introduced by Molodtsov [9], and Ã‡aÄŸman and EnginoÄŸlu [14] provided new definitions and various results on soft set theory.

In what follows, let be an initial universe set and let be a set of parameters. Let denote the power set of and .

*Definition 5 (see [9, 14]). *A soft set over is defined to be the set of ordered pairs
where such that if .

#### 3. Int-Soft Filters

In what follows, we take a residuated lattice as a set of parameters.

*Definition 6. *A soft set over is called an int-soft filter of if it satisfies

Proposition 7. * Every int-soft filter of satisfies
*

*Proof. *Let . Since , we have by (22). Since , it follows from (23) and (22) that
This completes the proof.

Lemma 8. *Let be a soft set over that satisfies two conditions (24) and (25). Then one has
*

*Proof. *Let be such that . Then , and so
Since , (28) is from (27).

Theorem 9. * A soft set over is an int-soft filter of if and only if it satisfies two conditions (24) and (25).*

*Proof. *The necessity is from Proposition 7.

Conversely, let be a soft set over that satisfies (24) and (25). If , then and so
Since for all , it follows from (28) that for all . Therefore is an int-soft filter of .

Theorem 10. * A soft set over is an int-soft filter of if and only if it satisfies condition (27).*

*Proof. *The necessity is from Lemma 8 and Theorem 9.

Conversely, let be a soft set over satisfying (27). Since and for all , it follows from (27) that and for all . Hence is an int-soft filter of by Theorem 9.

Theorem 11. * A soft set over is an int-soft filter of if and only if satisfies condition (24) and
*

*Proof. *Assume that is an int-soft filter of . Then condition (24) is valid. Using (6) and (25), we have
for all .

Conversely, let be a soft set over satisfying (24) and (31). Taking in (31) and using (3), we get
for all . Thus is an int-soft filter of by Theorem 9.

Lemma 12. *Every int-soft filter over satisfies the following condition:
*

*Proof. *If we take and in (25), then
This completes the proof.

Theorem 13. *A soft set over is an int-soft filter of if and only if it satisfies the following conditions:
*

*Proof. *Assume that is an int-soft filter of . Using (3), (24), and (25), we have
for all . Using (31) and (34), we get
for all .

Conversely, let be a soft set over satisfying two conditions (36) and (37). If we take in (36), then for all . Using (37) induces
for all . Therefore is an int-soft filter of by Theorem 9.

Theorem 14. * A soft set over is an int-soft filter of if and only if the set
**
is a filter of for all with .*

*Proof. *Assume that is an int-soft filter of . Let and be such that and . Then and . It follows from (24) and (25) that and and so that and . Hence is a filter of by Proposition 4.

Conversely, suppose that is a filter of for all with . For any , let . Then and is a filter of . Hence and so . For any , let and . If we take , then and which imply that . Thus
Therefore is an int-soft filter of by Theorem 9.

Theorem 15. *For a soft set over , let be a soft set over , where
**
where . If is an int-soft filter of , then so is .*

*Proof. *Suppose that is an int-soft filter of . Then is a filter of for all with by Theorem 14. Thus , and so for all . Let . If and , then . Hence
If or , then or . Thus
Therefore is an int-soft filter of .

Theorem 16. * If is an int-soft filter of , then the set
**
is a filter of for every .*

*Proof. *Since for all , we have . Let be such that and . Then and . Since is an int-soft filter of , it follows from (25) that
so that . Hence is a filter of by Proposition 4.

Theorem 17. * Let and let be a soft set over . Then *(1)*if is a filter of , then satisfies the following condition:
*(2)*if satisfies (24) and (48), then is a filter of .*

*Proof. * Assume that is a filter of . Let be such that
Then and . Using (20), we have and so .

(2) Suppose that satisfies (24) and (48). From (24) it follows that . Let be such that and . Then and , which imply that . Thus by (48), and so . Therefore is a filter of by Proposition 4.

#### 4. Int-Soft -Filters

*Definition 18 (see [28]). *A nonempty subset of is called a -filter of if it is a filter of that satisfies the following condition:

*Definition 19. *A soft set over is called an int-soft -filter of if it is an int-soft filter of that satisfies

Note that condition (51) is equivalent to the following condition:

Lemma 20. * Every int-soft filter of satisfies the following condition:
*

*Proof. *Let . Using (6) and (8), we have
It follows from Theorem 10 that
This completes the proof.

Theorem 21. * Let be a soft set over . Then is an int-soft -filter of if and only if it is an int-soft filter of that satisfies the following condition:
*

*Proof. *Assume that is an int-soft -filter of . Then is an int-soft filter of . Note that , and thus for all . It follows from (22) that . Combining this and (52), we have
for all . Using (53) and (57), we have
for all .

Conversely, let be an int-soft filter of that satisfies condition (56). If we put and in (56) and use (3) and (24), then
for all . Therefore is an int-soft -filter of .

Theorem 22. * Let be a soft set over that satisfies condition (24) and
**
Then is an int-soft -filter of .*

*Proof. *If we take in (60) and use (3), then
Hence is an int-soft filter of by Theorem 9. Let . Since
by (6) and (8), we have by (22). It follows from (22), (24), (25), (8), and (60) that
Therefore is an int-soft -filter of by Theorem 21.

The following example shows that any int-soft -filter may not satisfy condition (60).

*Example 23. *Let (unit interval). For any , define
Then is a residuated lattice. Let be a soft set over defined by
where . Then is an int-soft -filter of . But it does not satisfy condition (60). For example,

Proposition 24. * For an int-soft filter of , condition (60) is equivalent to the following condition:
*

*Proof. *Assume that condition (60) is valid. It follows from (24) and (3) that
for all .

Conversely, suppose that condition (67) is valid. It follows from (6) and (25) that
for all .

Combining Theorem 22 and Proposition 24, we have the following result.

Theorem 25. *Every int-soft filter satisfying condition (67) is an int-soft -filter.*

Proposition 26. * Every int-soft filter of with condition (60) satisfies the following condition:
*

*Proof. *Let be an int-soft filter of that satisfies condition (60) and let . Since , that is, , we have by (7). It follows from (8), (6), and (7) that
Using (22), (24), (3), (6), and (60), we have
Hence condition (70) is valid.

Proposition 27. * Every int-soft -filter of with condition (70) satisfies condition (60).*

*Proof. *Let be an int-soft -filter of that satisfies condition (70). For any , we have
by (6), (25), (22), (8), (52), and (70). Since , it follows from (22) that and so from (25) that
Therefore . Hence condition (60) is valid.

Theorem 28. *Let be an int-soft filter of . Then is an int-soft -filter of if and only if the following condition holds:
*

*Proof. *Suppose that is an int-soft -filter of . Since for all , we have . It follows from (56) and (3) that
and so from (24) that for all .

Conversely, let be an int-soft filter of which satisfies condition (75) and let . Since
by (6) and (8), it follows from (22) that
Hence, we have
by using (25), (75), and (24). Hence is an int-soft -filter of .

Theorem 29. *For an int-soft filter of , the following assertions are equivalent: *(1)* is an int-soft -filter of ;*(2)*. *

* Proof. * . Suppose that is an int-soft -filter of and let . Since , it follows from (22) that . Hence by using (52).

. Assume that holds. Using Lemma 20 and , we have
for all , and so is an int-soft -filter of by Theorem 21.

Proposition 30. *Every int-soft -filter of satisfies the following conditions:
*

*Proof. *Let be an int-soft -filter of . Using (6), (56), (8), and (24), we have
for all . Thus (81) holds. Since for all , it follows from (22) that and so that
for all by using (81).

Proposition 31. *Assume that satisfies the divisibility; that is, , for all . If is an int-soft -filter of satisfying (82), then the following equality is true:
*

*Proof. *Using the divisibility and (6), we have
for all . It follows from (6) and (82) that
for all .

Theorem 32. *Let satisfy the divisibility; that is, , for all . Then every int-soft filter of satisfying condition (85) is an int-soft -filter of .*

*Proof. *Using Lemma 20 and (6) and (85), we have
for all . Therefore is an int-soft -filter of by Theorem 21.

Theorem 33 (extension property). *Let and be int-soft filters of such that ; that is, for all and . If is an int-soft -filter of , then so is .*

*Proof. *Assume that is an int-soft -filter of . Using (6) and (3), we have
for all . Thus
by hypotheses and (57), and so
for all by (24). Since is an int-soft filter of , it follows from (25), (6), and (24) that
for all . Therefore is an int-soft -filter of .

#### 5. Regular and -Int-Soft Filters

Zhu and Xu [29] introduced the notion of a regular filter in a residuated lattice.

*Definition 34 (see [29]). *A filter of is said to be* regular* if it satisfies the following condition:

*Definition 35. *An int-soft filter of is said to be* regular* if it satisfies

*Example 36. * Let (unit interval). For any , define
Then is a residuated lattice (see [29]). Let be a soft set over defined by
Then is a regular int-soft filter of .

Theorem 37. * For an int-soft filter of , the following assertions are equivalent. *(1)* is regular.*(2)*. *(3)*. *

*Proof. *Assume that is a regular int-soft filter of and let . Using (7) and (10), we have
It follows from (8) and (7) that
and so from (24), (94), and (25) that
that is, the second condition holds. Since , we have
by (8) and (7). It follows from (24), (94), and (25) that
Hence the third condition holds. Next, suppose that the second condition is valid. Condition (10) together with the second condition induces
for all , and so . Hence is regular. Finally, assume that the third condition is valid. Since for all , it follows from (3) that , and so by (24). Therefore is regular.

Theorem 38. * A soft set over is a regular int-soft filter of if and only if it satisfies condition (24) and
*

*Proof. *Assume that is a regular int-soft filter of . Clearly condition (24) holds. Using (25) and Theorem 37, we get
for all .

Conversely, suppose that satisfies two conditions (24) and (103). Let . Since and , it follows from (3), (24), and (103) that
Therefore is an int-soft filter of by Theorem 9. If we take in (103) and use (3) and (24), then
Hence is regular by Theorem 37.

By a similar way to the proof of Theorem 38, we have the following characterization of a regular int-soft filter.

Theorem 39. *A soft set over is a regular int-soft filter of if and only if it satisfies condition (24) and
*

Lemma 40 (see [29]). *Let be a filter of . Then the following assertions are equivalent: *(1)* is regular;*(2)*. *

Theorem 41. *A soft set over is a regular int-soft filter of if and only if the set
**
is a regular filter of for all with .*

*Proof. *Assume that is a regular int-soft filter of . Let be such that . Since is an int-soft filter of , the set is a filter of by Theorem 14. Let be such that . Then by Theorem 37, and so . Hence is regular by Lemma 40.

Conversely, suppose that is a regular filter of for all with . Then is a filter of , and thus is an int-soft filter of by Theorem 14. For any , let . Then which implies from Lemma 40 that