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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 853907, 9 pages
http://dx.doi.org/10.1155/2013/853907
Research Article

Positive Implicative Ideals of BCK-Algebras Based on Intersectional Soft Sets

1Department of Mathematics Education, Chinju National University of Education, Jinju 660-756, Republic of Korea
2Department of Mathematics Education (and RINS), Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 14 September 2012; Revised 17 March 2013; Accepted 8 May 2013

Academic Editor: Mina Abd-El-Malek

Copyright © 2013 Eun Hwan Roh and Young Bae Jun. 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 aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion of int-soft positive implicative ideals is introduced, and related properties are investigated. Relations between an int-soft ideal and an int-soft positive implicative ideal are established. Characterizations of an int-soft positive implicative ideal are obtained. Extension property for an int-soft positive implicative ideal is constructed. The -product and -product of int-soft positive implicative ideals are considered, and the soft intersection (resp., union) of int-soft positive implicative ideals is discussed.

1. Introduction

Various problems in many fields involve data containing uncertainties which dealt with wide range of existing theories such as the theory of probability, (intuitionistic) fuzzy set theory, vague sets, theory of interval mathematics, and rough set theory. All of these theories have their own difficulties which are pointed out in [1]. To overcome these difficulties, Molodtsov [1] introduced the soft set theory as a new mathematical tool for dealing with uncertainties that is free from the difficulties. Molodtsov successfully applied the soft set theory in several directions, such as smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, and theory of measurement (see [14]). Soft set theory is applied to algebraic structures, and many algebraic properties of soft sets are studied (see [516]). Jun et al. introduced the notion of int-soft sets and applied the notion of soft set theory to BCK/BCI-algebras (see [14, 17, 18]). In the paper [17], the notion of int-soft BCK/BCI-algebras is discussed, and in the paper [18], the notion of int-soft ideals of BCK/BCI-algebras is introduced, and a few results are considered.

As a continuation of the papers [17, 18], in this paper, we first investigate more properties of int-soft ideals in BCK-algebras. We introduce the new notion, the so-called int-soft positive implicative ideals in BCK-algebras, and investigate related properties. We consider relations between an int-soft ideal and an int-soft positive implicative ideal and establish characterizations of an int-soft positive implicative ideal. We construct extension property for an int-soft positive implicative ideal. We deal with the -product and -product of int-soft positive implicative ideals and discuss the soft intersection (resp., union) of int-soft positive implicative ideals.

2. Preliminaries

A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki and was extensively investigated by several researchers.

An algebra of type is called a -algebra if it satisfies the following conditions: (I), (II),(III),(IV). If a -algebra satisfies the following identity:(V), then is called a -algebra. Any BCK/BCI-algebra satisfies the following axioms:(a1),(a2), (a3),(a4),where if and only if .

A nonempty subset of a BCK/BCI-algebra is called a subalgebra of if for all . A subset of a BCK/BCI-algebra is called an ideal of if it satisfies

A subset of a BCK-algebra is called a positive implicative ideal (briefly, PI-ideal) of if it satisfies (1) and

Note that every PI-ideal is an ideal, but the converse is not true (see [19]).

We refer the reader to the books [19, 20] for further information regarding BCK/BCI-algebras.

Molodtsov [1] defined the soft set in the following way. Let be an initial universe set and a set of parameters. Let denote the power set of and .

A pair is called a soft set over , where is a mapping given by

In other words, a soft set over is a parameterized family of subsets of the universe . For , may be considered as the set of -approximate elements of the soft set . Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [1]. We refer the reader to the papers [1, 2123] for further information regarding soft sets.

Let and be soft sets over . The -product of and is defined to be a soft set over which is defined by The -product of and is defined to be a soft set over which is defined by

Assume that has a binary operation . For any nonempty subset of , a soft set over is called an int-soft set over (see [17, 18]) if it satisfies

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

3. Intersectional Soft Positive Implicative Ideals

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

Definition 1 (see [18]). A soft set over is called an int-soft algebra over if it satisfies

Definition 2 (see [24]). A soft set over is called an int-soft ideal over if it satisfies

Let be a soft set over and a fixed element of . If is an int-soft ideal over , then by (10). We have the following question.

Question. Let be a soft set over . If satisfies the condition (10), then is the -inclusive set an ideal of ?

The following example provides a negative answer to this question; that is, there exists an element such that is not an ideal of .

Example 3. Let be a BCK-algebra with the following Cayley table: Let ,,,, and be subsets of such that . Define a soft set over by Then satisfies the condition (10), but it is not an int-soft ideal over since and is not an ideal of . Note that is an ideal of .

We give conditions for the -inclusive set to be an ideal.

Theorem 4. Let be a soft set over and a fixed element of . If is an int-soft ideal over , then the -inclusive set is an ideal of .

Proof. Recall that . Let be such that and . Then and . Since is an int-soft ideal over , it follows from (11) that which shows that . Therefore the -inclusive set is an ideal of .

Theorem 5. Let be a soft set over and . Then the following hold.(1)If the -inclusive set is an ideal of , then satisfies the following condition: (2)If satisfies (10) and (16), then the -inclusive set is an ideal of .

Proof. (1) Assume that the -inclusive set is an ideal of . Let be such that . Then and . It follows from (2) that ; that is, . Hence (16) is valid.
(2) Suppose that satisfies (10) and (16). Obviously by (10). Let be such that and . Then and , which imply that . It follows from (16) that ; that is, . Therefore is an ideal of .

Definition 6. A soft set over is called an int-soft PI-ideal over if it satisfies (10) and

Example 7. Let be a BCK-algebra with the following Cayley table: Let ,, and be subsets of such that . Define a soft set over by Then is an int-soft PI-ideal over .

Example 8. Let be a BCK-algebra with the following Cayley table: Let be a class of subsets of which is a poset with the following Hasse diagram (see Figure 1).
Let be a soft set over defined by Then is an int-soft PI-ideal over .

853907.fig.001
Figure 1

We discuss relations between int-soft ideal and int-soft PI-ideal.

Theorem 9. Every int-soft PI-ideal is an int-soft ideal.

Proof. Let be an int-soft PI-ideal over . If we take in (17) and use (a1), then we have (11). Hence is an int-soft ideal over .

The converse of Theorem 9 is not true as seen in the following example.

Example 10. Let be a BCK-algebra with the following Cayley table: Let ,, and be subsets of such that . Define a soft set over by Then is an int-soft ideal over . But it is not an int-soft PI-ideal over since

We provide conditions for an int-soft ideal to be an int-soft PI-ideal.

Lemma 11 (see [18]). Every int-soft ideal over satisfies

Theorem 12. For a soft set over , the following are equivalent.(1) is an int-soft PI-ideal over .(2) is an int-soft ideal over satisfying the condition (3) is an int-soft ideal over satisfying the condition

Proof. Assume that is an int-soft PI-ideal over . Then is an int-soft ideal over by Theorem 9. If we take in (17), then Hence (2) is valid. Now, let be an int-soft ideal over satisfying (26). Note that for all . Using Lemma 11, we have It follows from (a3) and (26) that Therefore (3) holds. Finally, let be an int-soft ideal over satisfying (27). Then for all by using (11) and (27). Therefore is an int-soft PI-ideal over .

Theorem 13. A soft set over is an int-soft PI-ideal over if and only if satisfies (10) and

Proof. Let be an int-soft PI-ideal over . Then is an int-soft ideal over by Theorem 9. Hence the condition (10) holds. Using (11), (a1), (a3), (III), and (27), we have for all , which proves (33).
Conversely, assume that a soft set over satisfies two conditions (10) and (33). Then for all . Hence is an int-soft ideal over . If we take in (33), then for all by (a1) and (10). It follows from Theorem 12 that is an int-soft PI-ideal over .

Lemma 14. A soft set over is an int-soft ideal over if and only if it satisfies

Proof. Necessity follows from [18, Proposition 3.7]. Assume that satisfies (37). Since for all , we have for all . Note that , that is, , for all . It follows from (37) that for all . Hence is an int-soft ideal over .

The following could be easily proved by induction.

Corollary 15. A soft set over is an int-soft ideal over if and only if it satisfies, for all , where .

Theorem 16. A soft set over is an int-soft PI-ideal over if and only if it satisfies

Proof. Suppose is an int-soft PI-ideal over . Then is an int-soft ideal over by Theorem 9. Let be such that . It follows from (26) and Lemma 14 that .
Conversely, assume that satisfies the condition (39). For any , let , which is equivalent to . Thus by (a1) and (39). It follows from Lemma 14 that is an int-soft ideal over . Since for all , we have by (39) and (10). Therefore is an int-soft PI-ideal over by Theorem 12.

Theorem 17. A soft set over is an int-soft PI-ideal over if and only if it satisfies, for all , and  ,

Proof. Suppose is an int-soft PI-ideal over . Then is an int-soft ideal over by Theorem 9. Let be such that . It follows from (27) and Lemma 14 that which proves (42).
Conversely, assume that satisfies the condition (42). Let for all . Then by (III), (a1), and (42). It follows from Theorem 16 that is an int-soft PI-ideal over .

The above two theorems have more general forms.

Theorem 18. A soft set over is an int-soft PI-ideal over if and only if it satisfies, for all ,

Proof. Suppose is an int-soft PI-ideal over . Then is an int-soft ideal over . Let be such that . By (26) and Corollary 15, we have which proves (45).
Conversely, assume that satisfies the condition (45). Let be such that . Then by (45). It follows from Theorem 16 that is an int-soft PI-ideal over .

Theorem 19. A soft set over is an int-soft PI-ideal over if and only if it satisfies, for all ,

Proof. It is similar to Theorem 18.

Lemma 20 (see [18]). Every int-soft ideal is an int-soft algebra.

Theorem 21. A soft set over is an int-soft PI-ideal over if and only if it satisfies

Proof. Assume that is an int-soft PI-ideal over . Then is an int-soft ideal over by Theorem 9, and so it is an int-soft algebra by Lemma 20. Let be such that . Then there exists such that . Using (9) and (III), we have and so . Let be such that and . Then and . It follows from (17) that Hence . Therefore is a PI-ideal of .
Conversely, suppose that is a PI-ideal of for all with . Then is a subalgebra of . Let be such that and . If we take , then , and so . Hence If we put in (51) and use (III), then for all . Let be such that and . Taking implies that and . It follows from (3) that . Thus Therefore is an int-soft PI-ideal over .

The PI-ideals in Theorem 21 are called inclusive PI-ideals of .

Corollary 22. If is an int-soft PI-ideal over , then is a PI-ideal of for all with .

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

Proof. Assume that is an int-soft PI-ideal over . Then is a PI-ideal of for all with . Hence , and so for all . Let . If and , then by (3). Thus If or , then or . Hence Therefore is an int-soft PI-ideal over .

Theorem 24. Every PI-ideal of can be realized as an inclusive PI-ideal of an int-soft PI-ideals over .

Proof. Let be a PI-ideal of . Define a soft set over as follows: where is a nonempty subset of . Clearly, for all . For every , if and , then . Thus If or , then or . It follows that Therefore is an int-soft PI-ideal over , and obviously . This completes the proof.

Note that an int-soft ideal might not be an int-soft PI-ideal (see Example 10). But we have the following extension property for int-soft PI-ideals.

Theorem 25 (extension property for int-soft PI-ideals). Let and be int-soft ideals over such that and for all . If is an int-soft PI-ideal over , then so is .

Proof. Assume that is an int-soft PI-ideal over . For any , we have by (a3), (27), and (III). It follows from (10) and (11) that for all . Therefore, by Theorem 12, is an int-soft PI-ideal over .

Theorem 26. Let and be -algebras. If and are int-soft PI-ideals over , then so is their -product.

Proof. Note that is a BCK-algebra. If and are int-soft PI-ideals over , then and are int-soft ideals over by Theorem 9. For any , we have Let . Then Therefore is an int-soft ideal over . Also, we have It follows from Theorem 12 that is an int-soft PI-ideal over .

The following example shows that the -product of int-soft PI-ideals is not an int-soft PI-ideal.

Example 27. Consider two BCK-algebras and with the following Cayley tables: Then , is a BCK-algebra with the following Cayley table: Let be the set of alphabets. Let and be soft sets over given by respectively. Then and are int-soft PI-ideals over . But the -product is not an int-soft PI-ideal over . Note that and so is not an int-soft ideal over . Thus is not an int-soft PI-ideal over .

Let and be soft sets over . The soft intersection of and is defined to be a soft set over which is defined by The soft union of and is defined to be a soft set over which is defined by

Theorem 28. If and are int-soft PI-ideals over , then so is their soft intersection .

Proof. Assume that and are int-soft PI-ideals over . For any , we have Therefore is an int-soft PI-ideal over .

The following example shows that the soft union of int-soft PI-ideals is not an int-soft PI-ideal over .

Example 29. Let be a BCK-algebra with the following Cayley table: Let be the set of alphabets. Let and be soft sets over given by respectively. Then and are int-soft PI-ideals over , and Note that Hence is not an int-soft ideal over , and therefore is not an int-soft PI-ideal over .

4. Conclusion

With the aim of providing a soft algebraic tool in considering many problems that contain uncertainties, we have introduced the notion of int-soft positive implicative ideals and investigated related properties. We have considered relations between an int-soft ideal and an int-soft positive implicative ideal and established characterizations of an int-soft positive implicative ideal. We have constructed extension property for an int-soft positive implicative ideal. We have dealt with the -product and -product of int-soft positive implicative ideals and discussed the soft intersection (resp., union) of int-soft positive implicative ideals. Our future research will be focused on studying the application of this structure to other algebraic structures.

Acknowledgments

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

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