Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 950897, 7 pages

http://dx.doi.org/10.1155/2013/950897

Research Article

Intersection-Soft Filters in -Algebras

Young Bae Jun,¹ Sun Shin Ahn,² and Kyoung Ja Lee³

¹Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea

²Department of Mathematics Education, Dongguk University, Seoul 100-715, Republic of Korea

³Department of Mathematics Education, Hannam University, Daejeon 306-791, Republic of Korea

Received 26 January 2013; Revised 1 March 2013; Accepted 6 March 2013

Academic Editor: Yong Zhou

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.

Abstract.

The notions of (strong) intersection-soft filters in -algebras are introduced, and related properties are investigated. Characterizations of a (strong) intersection-soft filter are established, and a new intersection-soft filter from old one is constructed. A condition for an intersection-soft filter to be strong is given, and an extension property of a strong intersection-soft filter is established.

1. Introduction

To solve complicated problem in economics, engineering, and 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] in order to get an algebraic proof of the completeness theorem of a formal deductive system [6]. The filter theory in -algebras is discussed in [7].

In this paper, we apply the notion of intersection-soft sets to the filter theory in -algebras. We introduced the concept of (strong) intersection-soft filters in -algebras and investigate related properties. We establish characterizations of a (strong) intersection-soft filter and make a new intersection-soft filter from old one. We provide a condition for an intersection-soft filter to be strong and construct an extension property of a strong intersection-soft filter.

2. Preliminaries

2.1. Basic Results on -Algebras

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, associative, and , and is a residuated lattice. In the following, let denote where appears times for .

We refer the reader to the book [8] 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

(i), (ii).

Lemma 4 (see [7]). Let be a nonempty subset of . Then is a filter of if and only if it satisfies

(1), (2).

2.2. Basic Results on Soft Set Theory

Soft set theory was introduced by Molodtsov [1] and Çağman and Enginoğlu [9].

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., ) denotes the power set of (resp., ).

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

Definition 5 (see [1, 9]). A soft set of over (a soft set of for short) is any function , or, equivalently, any set

for .

Definition 6 (see [9]). Let and be soft sets of . We say that is a soft subset of , denoted by , if for all .

3. Intersection-Soft Filters

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

Definition 7. 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 .

Example 8. 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 .

We provide characterizations of an int-soft filter.

Theorem 9. Let . Then is an int-soft filter of if and only if the following assertions are valid:

(1), (2).

Proof. Assume that is an int-soft filter of . For any , let . Then . Since is a filter of , we have and so . For any , let . Then and . Since is a filter of , it follows that . Hence .

Conversely, suppose that satisfies two conditions (1) and (2). Let such that . Then there exists , and so . It follows from (1) that . Thus . Let such that and . Then and . It follows from (2) that

that is, . Thus is a filter of , and hence is an int-soft filter of .

Proposition 10. Let be an int-soft filter of . Then the following properties are valid.

(1) is order preserving, that is,

(2). (3). (4). (5). (6). (7). (8). (9). (10). (11).

Proof. Let such that . Then , and so

by and of Theorem 9.

Let such that . Then

by and of Theorem 9.

Since for all , it follows from that . Using (11) and , we have . It follows from Theorem 9 that . Therefore . Since and for all , we have and by . Hence for all .

It follows from .

Note that for all . Using , we have

for all .

Combining (15), (1), and (3), we have the desired result.

It follows from and .

Since for all , it follows from that

Since for all , we have

by and . Hence . Similarly,   ∩    =   for all .

Using , we have

for all . It follows from that .

Note that

for all . Using and , we have

for all .

Note that for all . It follows from and that

for all .

Theorem 11. Let . Then is an int-soft filter of if and only if the following assertions are valid:

(1) is order preserving, (2).

Proof. The necessity follows from and of Proposition 10.

Conversely, suppose that satisfies two conditions and . Let . Since , we have by . Note that . It follows from and that

Therefore is an int-soft filter of .

Proposition 12. Let such that . If is an int-soft filter of , then for all .

Proof. Suppose that there exists such that . Then , and so and where . Since is a filter of , we have . This shows that , and it is a contradiction. Hence for all .

Theorem 13. Let . Then is an int-soft filter of if and only if the following assertion is valid:

Proof. Assume that is an int-soft filter of , and let . Since , it follows from Proposition 10 and Theorem 9 that

Conversely, suppose that satisfies the inclusion (32). Let . Then

by (32). Therefore is an int-soft filter of by Theorem 9.

Theorem 14. Let . Then is an int-soft filter of if and only if the following assertion is valid:

Proof. Suppose that is an int-soft filter of . Let such that . Then by Proposition 10, and so

by Theorem 9 (2).

Conversely, assume that satisfies the condition (35). Let . Since , we have by (35). Note that . It follows from (35) that . Therefore is an int-soft filter of by Theorem 9.

Proposition 15. Every int-soft filter of satisfies.

Proof. Let . Since , we have

by Proposition 10  . It follows from (2) and Theorem 9 that

This completes the proof.

The following example shows that the converse of Proposition 15 may not be true in general.

Example 16. 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 satisfies the condition (37), but is not an int-soft filter of since .

Proposition 17. For an int-soft filter of , the following are equivalent:

Proof. Assume that (42) is valid, and let . Using (R4), (5), and (14), we have

It follows from Proposition 10 (1), (42), and (R4) that

Conversely, suppose that (43) holds. If we use instead of in (43), then

which proves (42).

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

Theorem 18. Let . For a subset of , define a soft set of by

If is an int-soft filter of , then so is .

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

If or , then or . Thus

Therefore is an int-soft filter of .

Theorem 19. Any filter of can be realized as an inclusive filter of some int-soft filter of .

Proof. Let be a filter of . For a nonempty subset of , let be a soft set of defined by

Obviously for all . For any , if and , then . Hence . If or , then or . Thus . Therefore is an int-soft filter of and clearly . This completes the proof.

Definition 20. An int-soft filter of is said to be strong if the following assertion is valid:

Example 21. The int-soft filter in Example 8 is strong.

Theorem 22. Let . Then is a strong int-soft filter of if and only if the following assertions are valid:

(1), (2).

Proof. Suppose that is a strong int-soft filter of . Obviously, is valid. For every , we have

by Theorem 9 (2) and (51).

Conversely, assume that satisfies two conditions (1) and (2). If we take in (2), then for all . Hence is an int-soft filter of . Now if we put in (2), then

by (R2) and (1). Therefore is a strong int-soft filter of .

Example 23. Let . For any , we define

Then is an -algebra (see [5]). Let be given as follows:

where and are subsets of with . Then is an int-soft filter of . But

and so is not a strong int-soft filter of by Theorem 22.

We provide a condition for an int-soft filter to be strong.

Theorem 24. Let be an -algebra satisfying the following inequality:

Then every int-soft filter of is strong.

Proof. Let be an int-soft filter of . Using (5), (6), and (57), we have

for all . It follows from Proposition 10 (1) that

for all . Therefore is a strong int-soft filter of .

We consider an extension property of a strong int-soft filter.

Theorem 25. Let and be two int-soft filters of such that and . If is strong, then so is .

Proof. Assume that is a strong int-soft filter of . For any , let . Since is a strong int-soft filter of , we have

by (51) and assumption, and so

Since is an int-soft filter of , it follows that

Using (R4) and (14), we have

It follows from (62) and Theorem 14 that

Therefore is a strong int-soft filter of .

4. Conclusion

Using the notion of int-soft sets, we have introduced the concept of (strong) int-soft filters in -algebras and investigated related properties. We have established characterizations of a (strong) int-soft filter and made a new int-soft filter from old one. We have provided a condition for an int-soft filter to be strong and constructed an extension property of a strong 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 (soft) algebraic structures; and (3) to study the notions of implicative int-soft filters and 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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