#### Abstract

The notion of implicative int-soft filters is introduced, and related properties are investigated. A relation between an int-soft filter and an implicative int-soft filter is discussed, and conditions for an int-soft filter to be an implicative int-soft filter are provided. Characterizations of an implicative int-soft filter are considered, and a new implicative int-soft filter from an old one is displayed. The extension property of an implicative int-soft filter is established.

#### 1. Introduction

To solve complicated problems in economics, engineering, and the environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. Uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [1]. Maji et al. [2] and Molodtsov [1] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory.

To overcome these difficulties, Molodtsov [1] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [2] described the application of soft set theory to a decision making problem. Maji et al. [3] also studied several operations on the theory of soft sets. Chen et al. [4] presented a new definition of soft set parametrization reduction and compared this definition to the related concept of attributes reduction in rough set theory.

-algebras, which are different from -algebras, have been introduced by Wang [5]. The filter theory in -algebras is discussed in [7]. In [8], Jun et al. applied the notion of intersection-soft sets to the filter theory in -algebras. They introduced the concept of strong int-soft filters in -algebras and investigated related properties. They established characterizations of a strong int-soft filter and provided a condition for an int-soft filter to be strong. They also constructed an extension property of a strong int-soft filter.

In this paper, we introduce a new notion which is called an implicative int-soft filter and investigate related properties. We discuss a relation between an int-soft filter and an implicative int-soft filter. We provide conditions for an int-soft filter to be an implicative int-soft filter. We consider characterizations of an implicative int-soft filter and construct a new implicative int-soft filter from an old one. We establish the extension property of an implicative int-soft filter.

#### 2. Preliminaries

*Definition 1 (see [5]). *Let be a bounded distributive lattice with order-reversing involution and a binary operation . Then is called an *-algebra* if it satisfies the following axioms: (R1) , (R2) , (R3) , (R4) , (R5) , (R6) .

Let be an -algebra. For any , we define and . It is proven that and are commutative and associative and , and is a residuated lattice. In the following, let denote , where appears times for .

We refer the reader to the book [9] for further information regarding -algebras.

Lemma 2 (see [7]). *Let be an -algebra. Then the following properties hold:
*

*Definition 3 (see [7]). *A nonempty subset of is called a *filter* of if it satisfies the following:(i),
(ii).

*Definition 4 (see [10]). *A subset of is called an *implicative filter* of if it satisfies the following:(i),
(ii).

Note that every implicative filter is a filter. The following is a characterization of filters.

Soft set theory was introduced by Molodtsov [1] and Çağman and Enginğlu [11].

In what follows, let be an initial universe set and let be a set of parameters. We say that the pair is a* soft universe.* Let (resp., ) denote the power set of (resp., ).

By analogy with fuzzy set theory, the notion of soft set is defined as follows.

*Definition 5 (see [1, 11]). *A soft set of over (a soft set of for short) is
or, equivalently, any set
for .

#### 3. Implicative Int-Soft Filters

In what follows, denote by the set of all soft sets of over , where is an -algebra unless otherwise specified.

*Definition 6 (see [8]). *A soft set is called an *int-soft filter* of if it satisfies
where which is called the *-inclusive set* of .

If is an int-soft filter of , every -inclusive set is called an *inclusive filter* of .

Lemma 7 (see [8]). *Let . Then is an int-soft filter of if and only if the following assertions are valid:
*

*Definition 8. *Let . Then is called an* implicative int-soft filter* of if and only if it satisfies (19) and

*Example 9. *Let be a set with the order , and the following Cayley tables:

Then is an -algebra (see [10]), where and . Let be given as follows:
where and are subsets of with . Then is an implicative int-soft filter of .

We provide a relation between an int-soft filter and an implicative int-soft filter.

Theorem 10. *Every implicative int-soft filter is an int-soft filter.*

*Proof. *Let be an implicative int-soft filter of . If we take in (21), then
for all . Therefore, is an int-soft filter of .

The following example shows that the converse of Theorem 10 is not true in general.

*Example 11. *Let be a set with the order and the following Cayley tables:

Then is an -algebra (see [10]), where and . Let be given as follows: where and are subsets of with . Then is an int-soft filter of (see [8]). But it is not an implicative int-soft filter of since

We provide conditions for an int-soft filter to be an implicative int-soft filter.

Theorem 12. *An int-soft filter of is implicative if and only if it satisfies
**
for all .*

*Proof. *Let be an int-soft filter of that satisfies condition (26). Using (R1), (R4), and (26), we have
for all . Thus, is an implicative int-soft filter of .

Conversely, suppose that is an implicative int-soft filter of . Then
for all by (R1), (R4), and (21).

Lemma 13 (see [8]). *Every int-soft filter is order preserving; that is,
*

Proposition 14. *Every implicative int-soft filter of satisfies the following assertions:
*

*Proof. *If we put in (26), then
for all . Since for all , it follows from (29) and (R4) that
for all . Consequently, we get for all . Equation (33) follows from (32) and (20).

If we put and in (21), then
for all . Since for all , (29) implies that

Combining (40) and (41), we have for all .

Using (20) and (34), we have
for all . The proof of (36) is by induction on . For , if we use (34), then

for all . Suppose (36) holds for ; that is, for all . It follows from (34) that

for all . Therefore, (36) is valid. Note that
and for all . It follows from (21) and (R1) that

for all . Similarly for all . It follows from (31) that
for all .

Theorem 15. *Let be an int-soft filter of . If satisfies condition (33), then is an implicative int-soft filter of .*

*Proof. *Let be an int-soft filter of which satisfies condition (33). If we take and in (33) and use (30), then
for all . It follows from Theorem 12 that is an implicative int-soft filter of .

Corollary 16. *Let be an int-soft filter of . If satisfies condition (32), then is an implicative int-soft filter of .*

Theorem 17. *If an int-soft filter of satisfies condition (36), then it is implicative.*

*Proof. *Since for all , it follows from (29) that
for all . Now condition (36) implies that

for all . Combining (49) and (50) induces for all . Therefore, is an implicative int-soft filter of by Corollary 16.

Theorem 18. *If an int-soft filter of satisfies condition (37), then it is implicative.*

*Proof. *Using (19), (20), (37), and (R4), we have
for all . Since for all , it follows from (29) that for all . Therefore,

for all , and so is an implicative int-soft filter of by Corollary 16.

Theorem 19. *Let satisfy condition (19) and
**
Then is an implicative int-soft filter of .*

*Proof. *Using (R4) and (6), we have
for all . It follows from (R2), (R4), (19), (53), and (54) that

for all . Thus, is an implicative int-soft filter of .

Corollary 20. *Every int-soft filter satisfying condition (53) is an implicative int-soft filter.*

*Proof. *Let be an int-soft filter of that satisfies condition (53). Since satisfies two conditions (19) and (54) (see [8]), we know that is an implicative int-soft filter of .

Theorem 21. *A soft set of is an implicative int-soft filter of if and only if the nonempty -inclusive set is an implicative filter of for all .*

The implicative filters in Theorem 21 are called *inclusive implicative filters* of .

*Proof. *Assume that is an implicative int-soft filter of . Let be such that . Then is an int-soft filter of (see Theorem 10), and so is a filter of . Let be such that and . Then and . It follows from (21) that
and so that . Therefore, is an implicative filter of for all .

Conversely, suppose that the nonempty -inclusive set is an implicative filter of for all . Then is a filter of , and so is an int-soft filter of . For every , let . Then and , which imply that . Hence,
Thus, is an implicative int-soft filter of .

Theorem 22 (extension property). *Let and be two int-soft filters of such that and for all . If is implicative, then so is .*

*Proof. *Assume that is implicative. Then by (37). It follows from the hypothesis that
for all . Therefore, is an implicative int-soft filter of by Theorem 18.

We finally make a new implicative int-soft filter from an old one.

Theorem 23. *For any , let be a soft set of over defined by
**
where and are subsets of with . If is an implicative int-soft filter of , then so is .*

*Proof. *Assume that is an implicative int-soft filter of . Then is an implicative filter of for all with . Hence, , and so
for all . Let . If and , then . Hence,

If or , then or . Thus,

Therefore, is an implicative int-soft filter of .

Theorem 24. *For any implicative filter of , there exists an implicative int-soft filter of such that its inclusive implicative filter is .*

*Proof. *Let be a soft set of over in which is given by
where is a nonempty subset of . Since , we have for all . For every , if and , then . Hence,

If or , then or . Thus,

Therefore, is an implicative int-soft filter of . Obviously, .

#### 4. Conclusion

In [8], Jun et al. have applied the notion of intersection-soft sets to the filter theory in -algebras. They have introduced the concept of strong int-soft filters in -algebras and investigated related properties. They have established characterizations of a strong int-soft filter and provided a condition for an int-soft filter to be strong. They also have constructed an extension property of a strong int-soft filter.

In this paper, we have introduced a new notion which is called an implicative int-soft filter and investigated related properties. We have discussed a relation between an int-soft filter and an implicative int-soft filter. We have provided conditions for an int-soft filter to be an implicative int-soft filter. We have considered characterizations of an implicative int-soft filter and constructed a new implicative int-soft filter from an old one. We also have established the extension property of an implicative int-soft filter.

Work is ongoing. Some important issues for future work are (1) to develop strategies for obtaining more valuable results, (2) to apply these notions and results for studying related notions in other algebraic structures with applications in soft set theory, and (3) to study the notions of the Boolean int-soft filters.

#### Acknowledgments

The authors wish to thank the anonymous reviewer(s) for their valuable suggestions. This work (RPP-2012-021) was supported by the fund of Research Promotion Program, Gyeongsang National University, 2012.