Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 983507 | 7 pages | https://doi.org/10.1155/2013/983507

Implicative Int-Soft Filters of -Algebras

Academic Editor: Cengiz Çinar
Received01 Jun 2013
Accepted24 Oct 2013
Published03 Dec 2013

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:
983507.fig.001
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:
983507.fig.002

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.

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Copyright © 2013 Young Bae Jun 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.

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