AlgebraVolume 2013 (2013), Article ID 368962, 8 pageshttp://dx.doi.org/10.1155/2013/368962

Research Article

Applications of Soft Sets in BE-Algebras

Young Bae Jun¹ and Sun Shin Ahn²

¹Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, Republic of Korea²Department of Mathematics Education, Dongguk University, Seoul 100-715, Republic of Korea

Received 11 December 2012; Accepted 14 March 2013

Academic Editor: Antonio M. Cegarra

Copyright © 2013 Young Bae Jun and Sun Shin Ahn. 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 intersectional soft subalgebras of a BE-algebra is introduced, and related properties are investigated. Characterization of an intersectional soft subalgebra is discussed. The problem of classifying intersectional soft subalgebras by their inclusive subalgebras will be 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. Ma et al. studied -tyle (interval-valued) fuzzy ideals in BCI-algebras and soft -algebras (see [3–5]). As a generalization of a BCK-algebra, H. S. Kim and Y. H. Kim [6] introduced the notion of a BE-algebra and investigated several properties. In [7], Ahn and So introduced the notion of ideals in BE-algebras. They gave several descriptions of ideals in BE-algebras. Song et al. [8] considered the fuzzification of ideals in BE-algebras. They introduced the notion of fuzzy ideals in -algebras and investigated related properties. They gave characterizations of a fuzzy ideal in BE-algebras.

Various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [9]. In response to this situation, Zadeh [10] 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 [11]. 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 [12]. Maji et al. [13] and Molodtsov [12] 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 [12] 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. [13] described the application of soft set theory to a decision making problem. Maji et al. [14] also studied several operations on the theory of soft sets. Chen et al. [15] 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. Çağman et al. [16] 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 [17] considered the application of soft rough approximations in multicriteria group decision making problems. Aktaş and Çağman [18] 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. After that, many algebraic properties of soft sets are studied (see [19–29]).

In this paper, we introduce the notion of int-soft subalgebras of a BE-algebra and investigate their properties. We consider characterization of an int-soft subalgebra, and solve the problem of classifying int-soft subalgebras by their inclusive subalgebras.

2. Preliminaries

Let be the class of all algebras of type . By a BE-algebra we mean a system in which the following axioms hold (see [6]):

A relation “” on a BE-algebra is defined by

A BE-algebra is said to be transitive (see [7]) if it satisfies

A BE-algebra is said to be self distributive (see [6]) if it satisfies

Note that every self distributive BE-algebra is transitive, but the converse is not true in general (see [7]).

A mapping of BE-algebras is called a homomorphism if for all .

A soft set theory is introduced by Molodtsov [12], and Çağman and Enginoğlu [30] provided new definitions and various results on soft set theory.

In what follows, let be an initial universe set, and let be a set of parameters. Let denote the power set of and .

Definition 1 (see [12, 13]). A soft set over is defined to be the set of ordered pairs where such that if .

The function is called an approximate function of the soft set .

In what follows, denote by the set of all soft sets over by Çağman and Enginoğlu [30].

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

Definition 2 (see [31, 32]). Assume that has a binary operation . For any nonempty subset of , a soft set over is said to be intersectional over 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 Subalgebras

In what follows, we take as a set of parameters, which is a BE-algebra under the operation “” unless otherwise specified.

Definition 3. A soft set over is called an intersectional soft subalgebra (briefly, int-soft subalgebra) over if it satisfies

Example 4. Let be the set of parameters where is a -algebra with the following Cayley table: Let be a soft set over defined as follows: where , and are subsets of with . It is easy to check that is an int-soft subalgebra over .

Example 5. Let be the set of parameters, and let be the initial universe set, where is a -algebra [7] with the following Cayley table: Let be a class of subsets of which is a poset under the following Hasse diagram:                                                      Let be a soft set over defined as follows: It is easy to check that is an int-soft subalgebra over .

Theorem 6. A soft set over is an int-soft subalgebra over if and only if is a subalgebra of for all .

The subalgebra in Theorem 6 is called the inclusive subalgebra of .

Proof. Assume that is an int-soft subalgebra over . Let and . Then, and . It follows from (11) that that is, . Thus, is a subalgebra of .Conversely, suppose that is a subalgebra of for all . Let be such that and . Take . Then, , and so by assumption. Hence, Therefore, is an int-soft subalgebra over .

Lemma 7. Every int-soft subalgebra over satisfies the following inclusion:

Proof. Using (1) and (11), we have for all .

Proposition 8. For any int-soft subalgebra over , if a fixed element satisfies , then

Proof. Assume that a fixed element satisfies . Then, for all .

Proposition 9. Let be an int-soft subalgebra over . If a fixed element satisfies the following condition: then .

Proof. Taking in (22) implies that by (3). It follows from Lemma 7 that .

For any -algebras and , let be a function and , and let be soft sets over .

(1) The soft set where , is called the soft preimage of under .

(2) The soft set where is called the soft image of under .

Proposition 10. For any -algebras and , let be a function. Then,

Proof. Note that for all . Hence, for all , and therefore (26) is valid.

Theorem 11. Let be a homomorphism of -algebras and . If is an int-soft subalgebra over , then the soft preimage of under is also an int-soft subalgebra over .

Proof. For any , we have Hence, is also an int-soft subalgebra over .

Theorem 12. Let be a homomorphism of -algebras and . If is an int-soft subalgebra over and is injective, then the soft image of under is also an int-soft subalgebra over .

Proof. Let . If at least one of and is empty, then the inclusion is clear. Assume that and . Since is injective, we have Therefore, is an int-soft subalgebra over .

Theorem 13. Let and define a soft set over by where is any subset of and is a subset of satisfying . If is an int-soft subalgebra over , then so is .

Proof. If is an int-soft subalgebra over , then is a subalgebra of for all by Theorem 6. Let . If , then . Hence, If or , then or . Thus, Therefore, is an int-soft subalgebra over .

Theorem 14. If and are int-soft subalgebras over , then the soft intersection of and is an int-soft subalgebra over .

Proof. Let . Then, Hence, is an int-soft subalgebra over .

The following example shows that the soft union of int-soft subalgebras over may not be an int-soft subalgebra over .

Example 15. Let be the set of parameters where is a -algebra [7] with the following Cayley table: Let and be soft sets over defined, respectively, as follows: where , and are subsets of with . It is easy to check that and are int-soft subalgebras over . But is not an int-soft subalgebra over , since .

Theorem 16. Let be an int-soft subalgebra over . 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. Straightforward.

The converse of Theorem 16 is not true in general as seen in the following example.

Example 17. Let be the set of parameters, and let be the initial universe set where is a -algebra as in Example 4. Consider a soft set over which is given by Then, is an int-soft subalgebra over . The -inclusive sets of are described as follows: If we take and , then and there is no such that . But .

Theorem 18. Let be an int-soft subalgebra over . Let and be subsets of such that and are 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 inclusion and there is no such that , it follows that , that is, . Therefore, the -inclusive set of is equal to the -inclusive set of .

Theorem 19. Let be a soft set over in which is totally ordered by set inclusion. For each subset of , if the -inclusive set of is a subalgebra of , then is an int-soft subalgebra over .

Proof. Let be such that and . Then, either or . We may assume that without loss of generality. Then, , and . Since is a subalgebra of , it follows that so that Therefore, is an int-soft subalgebra over .

We have the following question.

Question. Let be an int-soft subalgebra over . Does any subalgebra can be represented as a -inclusive set of ?

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

Example 20. Let be the set of parameters, and let be the initial universe set where is a -algebra as in Example 15. Consider a soft set over which is given by Then, is an int-soft subalgebra over . The -inclusive sets of are described as follows: The subalgebra cannot be a -inclusive set , since there is no such that .

However, we have the following theorem.

Theorem 21. Every subalgebra of a -algebra can be represented as a -inclusive set of an int-soft subalgebra.

Proof. Let be a subalgebra of a -algebra . For a subset of , define a soft set over by Obviously, . We now prove that is an int-soft subalgebra over . Let . If , then because is a subalgebra of . Hence, , and so . If and , then and which imply that Similarly, if and , then . Obviously, if and , then . Therefore, is an int-soft subalgebra over .

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

Theorem 22. Let be an int-soft subalgebra over and let 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, .

Remark 23. As a consequence of Theorem 22, if is a finite -algebra, then the -inclusive sets of an int-soft subalgebra over form a chain. But for all . Therefore, , where , is the smallest inclusive subalgebra but not always as seen in the following example, and so we have the chain where .

Example 24. Let be a subalgebra of a -algebra such that . Let be the int-soft subalgebra over which is given in the proof of Theorem 21. Then, . Further, the -inclusive sets of are and . Thus, we have but .

Corollary 25. Let be a finite -algebra, and let be an int-soft subalgebra over . If , then the family of -inclusive sets , constitutes all the -inclusive sets of .

Proof. Let and . If , where , then by Theorem 22. If , where is the least element (under the set inclusion) of , then . Assume that , where is the greatest element (under the set inclusion) of . If there is such that and , then . It is a contradiction. It follows from Theorem 22 that . Thus, for any , the inclusive subalgebra is one of .

The following example shows that two int-soft subalgebras over may have an identical family of -inclusive sets but the int-soft subalgebras over may not be equal.

Example 26. Let be the set of parameters, and let be the initial universe set where is a -algebra as in Example 15. Consider a soft set over which is given by where are subsets of . It is easy to verify that is an int-soft subalgebra over . The -inclusive sets of are and . Now let , and be subsets of such that and for and . Define a soft set over as follows: Then, is an int-soft subalgebra over , and the -inclusive sets of are and . Hence, the two int-soft subalgebras and over have an identical family of -inclusive sets. However, it is clear that is not equal to .

Lemma 27. Let be a finite -algebra, and let be an int-soft subalgebra over . If and are elements of such that , then .

Proof. Straightforward.

Theorem 28. Let be a finite -algebra, and let and be two int-soft subalgebras over having the identical family of -inclusive sets. If and , where then we have (1) , (2) , (3) .

Proof. Corollary 25 implies that the only -inclusive sets of and are the two families and . Since and have the same family of -inclusive sets, we have which proves (1).(2) Using (1) and Remark 23, we have two chains of -inclusive sets: Clearly, we have Since two families of -inclusive sets are identical, it is clear that . By hypothesis, for some . Assume that . Then, for some , and for some . Thus, by (49) and (50), we have and . This is a contradiction, and so . By mathematical induction on , we finally obtain .(3) Let be such that and , where and . It is sufficient to show that . Now, implies that . This gives from (50) that . Since , it follows from (2) that and so . Hence, by (49). Using (2), we have . Thus, , and so . This completes the proof.

Theorem 29. Let be a -algebra. Given any chain of subalgebras there exists an int-soft subalgebra over whose -inclusive sets are exactly the subalgebras of this chain.

Proof. Consider a class of subsets of such that Define a soft set by and . We will prove that is an intersectional soft -algebra over . Note that if , then . If , then either or for . Thus, if , then . If , then . Let . We distinguish two cases as follows:Case  1. Let . Then, . Since is a subalgebra, we have , and so either or . In any case we know that Case  2. For , let and . Then, and . It follows that Hence, we conclude that is an intersectional soft -algebra over . From the definition of , we have . Thus, the -inclusive sets of are given by the chain of subalgebras Now, . Finally, we prove that for . Clearly . If , then , and so for . Hence, , and thus for some . Since , we have , and so for . This completes the proof.

Theorem 29 is illustrated as an example.

Example 30. Let be the initial universe set, and let be the set of parameters where is a -algebra as in Example 5. Consider subalgebras , and . Then, . Define a soft set over by Then, is an int-soft subalgebra over with , , , and .

Theorem 31. Let be a soft set over , and let be a subset of . Define a soft set over by If is an int-soft subalgebra over , then so is .

Proof. Let . If , then , and so If or , then or . Hence, Therefore, is an int-soft subalgebra over .

4. Conclusion

Using the notion of int-soft sets, we have introduced the concept of int-soft subalgebras in -algebras and investigated related properties. We have considered characterization of an int-soft subalgebra and solved the problem of classifying int-soft subalgebras by their inclusive subalgebras. We have shown that(1)every soft image of an int-soft subalgebra is also an int-soft subalgebra;(2)the soft intersection of int-soft subalgebras is an int-soft subalgebra.

We have made a new int-soft subalgebra from the old one. Work is ongoing. Some important issues for future work are as follows:(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,(3)to study the soft set application in ideal and filter theory of -algebras.

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.

References

Y. Imai and K. Iséki, “On axiom systems of propositional calculi. XIV,” Proceedings of the Japan Academy, vol. 42, pp. 19–22, 1966.

K. Iséki, “An algebra related with a propositional calculus,” Proceedings of the Japan Academy, vol. 42, pp. 26–29, 1966.

X. Ma, J. Zhan, and Y. B. Jun, “New types of fuzzy ideals of BCI-algebras,” Neural Computing and Applications, vol. 21, pp. S19–S27, 2012.

X. Ma, J. Zhan, and Y. B. Jun, “Soft -algebras based on fuzzy sets,” Journal of Multiple-Valued Logic and Soft Computing, vol. 19, pp. 547–563, 2012.

X. Ma, J. Zhan, B. Davvaz, and Y. B. Jun, “Some kinds of ()-interval-valued fuzzy ideals of BCI-algebras,” Information Sciences, vol. 178, no. 19, pp. 3738–3754, 2008.

H. S. Kim and Y. H. Kim, “On BE-algebras,” Scientiae Mathematicae Japonicae, vol. 66, no. 1, pp. 113–116, 2007.

S. S. Ahn and K. S. So, “On ideals and upper sets in -algebras,” Scientiae Mathematicae Japonicae, vol. 68, no. 2, pp. 279–285, 2008.

S. Z. Song, Y. B. Jun, and K. J. Lee, “Fuzzy ideals in BE-algebras,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 33, no. 1, pp. 147–153, 2010.

L. A. Zadeh, “From circuit theory to system theory,” Proceedings of the Institute of Radio Engineers, vol. 50, pp. 856–865, 1962.

L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.

L. A. Zadeh, “Toward a generalized theory of uncertainty (GTU)—an outline,” Information Sciences, vol. 172, no. 1-2, pp. 1–40, 2005.

D. Molodtsov, “Soft set theory—first results,” Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999.

P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets in a decision making problem,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1077–1083, 2002.

P. K. Maji, R. Biswas, and A. R. Roy, “Soft set theory,” Computers & Mathematics with Applications, vol. 45, no. 4-5, pp. 555–562, 2003.

D. Chen, E. C. C. Tsang, D. S. Yeung, and X. Wang, “The parameterization reduction of soft sets and its applications,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 757–763, 2005.

N. Çağman, F. Çitak, and S. Enginoglu, “FP-soft set theory and its applications,” Annals of Fuzzy Mathematics and Informatics, vol. 2, no. 2, pp. 219–226, 2011.

F. Feng, “Soft rough sets applied to multicriteria group decision making,” Annals of Fuzzy Mathematics and Informatics, vol. 2, no. 1, pp. 69–80, 2011.

H. Aktaş and N. Çağman, “Soft sets and soft groups,” Information Sciences, vol. 177, no. 13, pp. 2726–2735, 2007.

U. Acar, F. Koyuncu, and B. Tanay, “Soft sets and soft rings,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3458–3463, 2010.

A. O. Atagün and A. Sezgin, “Soft substructures of rings, fields and modules,” Computers & Mathematics with Applications, vol. 61, no. 3, pp. 592–601, 2011.

F. Feng, Y. B. Jun, and X. Zhao, “Soft semirings,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2621–2628, 2008.

Y. B. Jun, “Soft BCK/BCI-algebras,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1408–1413, 2008.

Y. B. Jun, H. S. Kim, and J. Neggers, “Pseudo -algebras,” Information Sciences, vol. 179, no. 11, pp. 1751–1759, 2009.

Y. B. Jun, K. J. Lee, and C. H. Park, “Soft set theory applied to ideals in -algebras,” Computers & Mathematics with Applications, vol. 57, no. 3, pp. 367–378, 2009.

Y. B. Jun, K. J. Lee, and J. Zhan, “Soft -ideals of soft BCI-algebras,” Computers & Mathematics with Applications, vol. 58, no. 10, pp. 2060–2068, 2009.

Y. B. Jun, K. J. Lee, and A. Khan, “Soft ordered semigroups,” Mathematical Logic Quarterly, vol. 56, no. 1, pp. 42–50, 2010.

Y. B. Jun and C. H. Park, “Applications of soft sets in ideal theory of BCK/BCI-algebras,” Information Sciences, vol. 178, no. 11, pp. 2466–2475, 2008.

C. H. Park, Y. B. Jun, and M. A. Öztürk, “Soft WS-algebras,” Communications of the Korean Mathematical Society, vol. 23, no. 3, pp. 313–324, 2008.

J. Zhan and Y. B. Jun, “Soft -algebras based on fuzzy sets,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 2037–2046, 2010.

N. Çağman and S. Enginoğlu, “Soft set theory and uni-int decision making,” European Journal of Operational Research, vol. 207, no. 2, pp. 848–855, 2010.

Y. B. Jun, M. S. Kang, and K. J. Lee, “Intersectional soft sets and applications to BCK/BCI-algebras,” Communications of the Korean Mathematical Society, vol. 28, pp. 11–12, 2013.

Y. B. Jun, K. J. Lee, and E. H. Roh, “Intersectional soft BCK/BCI-ideals,” Annals of Fuzzy Mathematics and Informatics, vol. 4, no. 1, pp. 1–7, 2012.