Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 602959, 8 pages

http://dx.doi.org/10.1155/2013/602959

Research Article

Int-Soft Filters of -Algebras

Sun Shin Ahn,¹ N. O. Alshehri,² and Young Bae Jun³

¹Department of Mathematics Education, Dongguk University, Seoul 100-715, Republic of Korea

²Department of Mathematics, Faculty of Science for Girls, King Abdulaziz University, Jeddah, Saudi Arabia

³Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 7 August 2013; Accepted 2 November 2013

Academic Editor: Cengiz Çinar

Copyright © 2013 Sun Shin Ahn 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 notion of int-soft filters of a -algebra is introduced, and related properties are investigated. Characterization of an int-soft filter is discussed. The problem of classifying int-soft filters by their -inclusive filter is solved.

1. Introduction

In 1966, Imai and Iséki [1] and Iséki [2] introduced two classes of abstract algebras: BCK-algebras and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. As a generalization of a BCK-algebra, H. S. Kim and Y. H. Kim [3] introduced the notion of a -algebra and investigated several properties. In [4], Ahn and So introduced the notion of ideals in -algebras. They gave several descriptions of ideals in -algebras.

Various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [5]. In response to this situation Zadeh [6] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [7]. To solve complicated problem in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. 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 [8]. Maji et al. [9] and Molodtsov [8] 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 [8] 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. [9] described the application of soft set theory to a decision-making problem. Maji et al. [10] also studied several operations on the theory of soft sets. Chen et al. [11] presented a new definition of soft set parametrization reduction and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. Çaman et al. [12] introduced fuzzy parameterized (FP) soft sets and their related properties. They proposed a decision-making method based on FP-soft set theory and provided an example which shows that the method can be successfully applied to the problems that contain uncertainties. Feng [13] considered the application of soft rough approximations in multicriteria group decision-making problems. Aktaş and Çaman [14] studied the basic concepts of soft set theory and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups.

In this paper, we introduce the notion of int-soft filter of a -algebra and investigate its properties. We consider characterization of an int-soft filter and solve the problem of classifying int-soft subalgebras by their -inclusive filters. We provide conditions for a soft set to be an int-soft filter. We make a new int-soft filter from old one.

2. Preliminaries

Let be the class of all algebras of type . By a -algebra we mean a system in which the following axioms hold (see [3]):

A relation “” on a -algebra is defined by

A -algebra is said to be transitive (see [4]) if it satisfies

A -algebra is said to be self-distributive (see [3]) if it satisfies

Every self-distributive -algebra satisfies the following properties:

Note that every self distributive -algebra is transitive, but the converse is not true in general (see [4]).

Definition 1 (see [3]). Let be a -algebra and let be a nonempty subset of . Then is a filter of if

(F1), (F2).

A soft set theory is introduced by Molodtsov [8]. In what follows, let be an initial universe set and a set of parameters. Let denote the power set of and .

Definition 2 (see [8]). A soft set of over is defined to be the set of ordered pairs:

where such that if .

For a soft set of and a subset of , the -inclusive set of , denoted by , is defined to be the set

For any soft sets and of , we call a soft subset of , denoted by , if for all . The soft union of and , denoted by , is defined to be the soft set of over in which is defined by

The soft intersection of and , denoted by , is defined to be the soft set of over in which is defined by

3. Int-Soft Filters

In what follows, we take a -algebra , as a set of parameters unless otherwise specified.

Definition 3 (see [15]). A soft set of is called an int-soft subalgebra of if it satisfies

Definition 4. A soft set of over is called an int-soft filter of if it satisfies

Example 5. Let be the set of parameters where is a -algebra with the following Cayley table:

Let be a soft set of over defined as follows:

where and are subsets of with . It is easy to check that is an int-soft filter of .

Proposition 6. Every int-soft filter of over satisfies the following properties:

(i),(ii).

Proof. (i) Let be such that . Then . It follows from (17) and (18) that

(ii) Using (18) and (4), we obtain

for all .

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

Theorem 7. If a soft set of over satisfies (17) and Proposition 6(ii), then it is an int-soft filter of .

Proof. Taking in Proposition 6(ii) and using (3), we have

for all . Hence is an int-soft filter of .

Corollary 8. Let be a soft set of over . Then is an int-soft filter of over if and only if it satisfies (17) and Proposition 6(ii).

Lemma 9. Every int-soft filter of over satisfies the following inclusion:

Proof. If we take and in (18), then

by using (4), (1), and (17).

Theorem 10. A soft set of over is an int-soft filter of over if and only if it satisfies the following conditions:

(i), (ii).

Proof. Assume that is an int-soft filter of over . Using (18), (4), (1), (2), and (17), we get

for all . Using Proposition 6(ii) and Lemma 9, we have

for any .

Conversely, let be a soft set of over satisfying conditions (i) and (ii). If we take in (i), then for all . Using (ii), we obtain

for all . Hence is an int-soft filter over .

Proposition 11. Let be a soft set of over . Then is an int-soft filter of over if and only if it satisfies

Proof. Assume that is an int-soft filter of over . Let be such that . By Proposition 6(i) and (18), we have

Conversely, suppose that satisfies (29). By (2), we . Hence for all by (29). Thus (17) is valid. Using (1) and (4), we obtain for all . By (29), we get . Hence (18) holds. Therefore is an int-soft filter of .

As a generalization of Proposition 11, we have the following results.

Theorem 12. If a soft set of over is an int-soft filter of over , then

for all , where

Proof. The proof is by induction on . Let be an int-soft filter of over . By Proposition 6(i) and (29), we know that condition (31) is valid for . Assume that satisfies condition (31) for ; that is,

for all . Suppose that for all . Then

Since is an int-soft filter of , it follows from (18) that

This completes the proof.

Now we consider the converse of Theorem 12.

Theorem 13. Let be a soft set of over satisfying (31). Then is an int-soft filter of over .

Proof. Let be such that . Then , and so by (31). It follows from Proposition 11 that is an int-soft filter of over .

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

The filter in Theorem 14 is called the inclusive filter of .

Proof. Assume that is an int-soft filter over . Let and be such that and . Then and . It follows from (17) and (18) that and for all . Hence and . Thus is a filter of .

Conversely, suppose that is a filter of for all with . For any , let . Then . Since is a filter of , we have and so . For any , let and . Take . Then and which imply that . Hence

Thus is an int-soft filter of over .

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

Theorem 15. Let and define a soft set of over by

where is a nonempty subset of . If is an int-soft filter of , then so is .

Proof. Assume that is an int-soft filter of . Then is a filter of over for all by Theorem 14. Hence , and so for all . Let . If and , then . Hence

If or , then or . Thus

Therefore is an int-soft filter of .

For two elements and of , consider a soft set over where

where and are subsets of with . In the following example, we know that there exists such that is not an int-soft filter of .

Example 16. Consider the -algebra which is given in Example 19. Then is not an int-soft filter of over since

Now we provide a condition for the soft set to be an int-soft filter of over for all .

Theorem 17. If is self distributive, then the soft set is an int-soft filter of over for all .

Proof. Let . Obviously, for all . Let be such that or . Then or . Hence

Assume that and . Then

and so . Therefore is an int-soft filter of over for all .

Theorem 18. If and are int-soft filters of , then the soft intersection of and is an int-soft filter of .

Proof. For any , we have

Let . Then

Hence is an int-soft filter of .

The following example shows that the soft union of int-soft filters of may not be an int-soft filter of .

Example 19. Let be the set of parameters and the initial universe set, where is a -algebra with the following Cayley table (see [4]):

Let and be soft sets of over defined, respectively, as follows:

where , , , and are subsets of with . It is easy to check that and are int-soft filters of over . But is not an int-soft filter of over , since

Theorem 20. Let be an int-soft filter of . Let and be subsets of such that . If the -inclusive set of is equal to the -inclusive set of , then there is no such that .

Proof. The proof is straightforward.

The converse of Theorem 20 is not true in general as seen in the following example.

Example 21. Let be the set of parameters and the initial universe set where is a -algebra as in Example 5. Consider a soft set of over which is given by

Then is an int-soft filter of . The -inclusive sets of are described as follows:

If we take and , then and there is no such that . But .

Theorem 22. Let be an int-soft filter of . Let and be subsets of such that and is totally ordered by set inclusion for all . If there is no such that , then the -inclusive set of is equal to the -inclusive set of .

Proof. Since , we have . If , then . Since is totally ordered by set inclusion and there is no such that , it follows that ; that is, . Therefore the -inclusive set of is equal to the -inclusive set of .

We have the following question.

Question. Given an int-soft filter of , does any filter can be represented as a -inclusive set of ?

The following example shows that the answer to the question above is false.

Example 23. Let be the set of parameters and the initial universe set where is a -algebra as in Example 5. Consider a soft set of over which is given by

Then is an int-soft filter of . The -inclusive sets of are described as follows:

The filter cannot be a -inclusive set , since there is no such that .

However, we have the following theorem.

Theorem 24. Every filter of a -algebra can be represented as a -inclusive set of an int-soft filter.

Proof. Let be a filter of a -algebra . For a subset of , define a soft set over by

Obviously, . We now prove that is an int-soft filter of . Since , we have for all . Let . If and , then because is a filter of . Hence , and so . If or , then or . Hence . Therefore is an int-soft filter of .

Note that if is a finite -algebra, then the number of filters of over is finite whereas the number of -inclusive sets of an int-soft filter of over appears to be infinite. But, since every -inclusive set is indeed a filter of , not all these -inclusive sets are distinct. The next theorem characterizes this aspect.

Theorem 25. Let be an int-soft filter of over and let be such that is a chain for all . Two -inclusive sets and are equal if and only if there is no such that .

Proof. Let and be subsets of such that . Assume that there exists such that . Then is a proper subset of , which contradicts the hypothesis.

Conversely, suppose that there is no such that . Obviously, . If , then . It follows from the assumption that ; that is, . Therefore .

Let be a soft set of . For any and ; consider the set

where in which appears -times. Note that for all and .

Proposition 26. Let be a soft set of over such that condition (17) and for all . For any and , if , then for all .

Proof. Assume that . Then , and so

for all by the exchange property of the operation . Hence for all .

Proposition 27. For any soft set of , let satisfy the following assertion:

Then for all and .

Proof. For any , we have

and so . Similarly, .

Proposition 28. Let be a self distributive -algebra and let be an order-preserving soft set of over with the property (17). If in , then for all and .

Proof. Let be such that . For any , if , then

by (4) and (8), and so . Thus , which completes the proof.

The following example shows that there exists a soft set of and such that is not a filter of .

Example 29. Let be the set of parameters and the initial universe set where is a -algebra as in Example 5. Consider a soft set of over which is given by

where and are subsets of with . Then it is a soft set of over . But is not a filter, since and .

We provide conditions for a set to be a filter.

Theorem 30. Let be a soft set over . If is a self distributive -algebra and is injective, then is a filter of for all and .

Proof. Assume that is a self distributive -algebra and is injective. Obviously, . Let and be such that and . Then which implies that since is injective. Using (7), we have

which implies that . Therefore is a filter of for all and .

Theorem 31. Let be a self distributive -algebra. Let be a soft set of over satisfying condition (17) and

Then is a filter of for all and .

Proof. Let and . Obviously, . Let be such that and . Then , which implies from (61) and (17) that

Hence and therefore is a filter of for all and .

Proposition 32. Let be a soft set of over in which is injective. If is a filter of , then the following assertion is valid:

Proof. Assume that is a filter of over and let and . If , then

and so since is injective. Since is a filter of , it follows from (F2) that . Continuing this process, we obtain and so . Therefore for all and .

Theorem 33. Let be a soft set of . For any subset of , if condition (63) holds, then is a filter of .

Proof. Suppose that condition (63) holds. Note that . Let be such that and . Then

and thus where . Therefore is a filter of .

Theorem 34. Let be a soft set of . If is a filter of , then

Proof. Let be a filter of . By Proposition 32, the inclusion

holds. Let . Since for all , it follows that

This completes the proof.

4. Conclusion

Using the notion of int-soft sets, we have introduced the concept of int-soft filters in -algebras and investigated related properties. We have considered characterization of an int-soft filter and solved the problem of classifying int-soft filters by their inclusive filters. We have provided conditions for a soft set to be an int-soft filter. We have made a new int-soft filter from the old one. We have considered the soft intersection of int-soft filters and have shown that the soft union of int-soft filters is not an int-soft filter by providing a counterexample.

Work is ongoing. Some important issues for future work are

(1)to develop strategies for obtaining more valuable results, (2)to study the soft set application in ideal theory of -algebras, (3)to apply these notions and results for studying related notions in other (soft) algebraic structures, (4)to study generalizations of soft set application in ideal and filter theory of -algebras.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors wish to thank the anonymous reviewer(s) for their valuable suggestions. The third author, Young Bae Jun, is an Executive Research Worker of Educational Research Institute Teachers College in Gyeongsang National University.

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