Journal of Applied Mathematics

Volume 2012, Article ID 178159, 15 pages

http://dx.doi.org/10.1155/2012/178159

## Double-Framed Soft Sets with Applications in BCK/BCI-Algebras

^{1}Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea^{2}Department of Mathematics Education, Dongguk University, Seoul 100-715, Republic of Korea

Received 26 August 2012; Revised 23 October 2012; Accepted 23 October 2012

Academic Editor: Morteza Rafei

Copyright © 2012 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

In order to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties, the notion of double-framed soft sets is introduced, and applications in BCK/BCI-algebras are discussed. The notions of double-framed soft algebras in BCK/BCI-algebras are introduced, and related properties are investigated. Characterizations of double-framed soft algebras are considered. Product and int-uni structure of double-framed soft algebras are discussed, and several examples are provided.

#### 1. Introduction

The real world is inherently uncertain, imprecise, and vague. Various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [1]. In response to this situation, Zadeh [2] 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 [3]. To solve a 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 [4]. Maji et al. [5] and Molodtsov [4] 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 [4] 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. Worldwide, there has been a rapid growth in interest in soft set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on soft sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. Maji et al. [5] described the application of soft set theory to a decision-making problem. Maji et al. [6] also studied several operations on the theory of soft sets. Aktaş and Çağman [7] 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. Jun and Park [8] studied applications of soft sets in ideal theory of BCK/BCI-algebras. Jun et al. [9, 10] introduced the notion of intersectional soft sets and considered its applications to BCK/BCI-algebras. Also, Jun [11] discussed the union soft sets with applications in BCK/BCI-algebras. We refer the reader to the papers [12–24] for further information regarding algebraic structures/properties of soft set theory.

The aim of this paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. To do this, we introduce the notion of double-framed soft sets and apply it to BCK/BCI-algebras. We discuss double-framed soft algebras in BCK/BCI-algebras and investigate related properties. We consider characterizations of double-framed soft algebras and deal with the product and int-uni structure of double-framed soft algebras. We provide several examples.

#### 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 BCI-algebra if it satisfies the following conditions: (i), (ii), (iii), (iv). If a BCI-algebra satisfies the following identity: (v), then is called a BCK-algebra. Any BCK/BCI-algebra satisfies the following conditions: where if and only if .

A nonempty subset of a BCK/BCI-algebra is called a *subalgebra* of if for all .

We refer the reader to the books [25, 26] for further information regarding BCK/BCI-algebras.

Molodtsov [4] defined the soft set in the following way: let be an initial universe set, and let be a set of parameters. Let denote the power set of and .

*Definition 2.1 (see [4]). *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 [4].

#### 3. Double-Framed Soft Algebras

In what follows, we take as a set of parameters, which is a BCK/BCI-algebra and as subalgebras of unless otherwise specified.

*Definition 3.1. *A double-framed pair is called a *double-framed soft set* over , where and are mappings from to .

*Definition 3.2. *A double-framed soft set over is called a double-framed soft algebra over if it satisfies

*Example 3.3. *Suppose that there are five houses in the initial universe set given by
Let a set of parameters be a set of status of houses in which stands for the parameter “beautiful,” stands for the parameter “cheap,” stands for the parameter “in good location,” stands for the parameter “in green surroundings,”with the following binary operation:
Then is a BCK-algebra. For a subalgebra of , consider a double-framed soft set over as follows:
It is routine to verify that is a double-framed soft algebra over .

*Example 3.4. *Consider the initial universe set and a set of parameters which are given in Example 3.3. Define a binary operation on by the following Cayley table:
Consider a double-framed soft set over as follows:
Then is a double-framed soft algebra over .

*Example 3.5. *There are six women patients in the initial universe set given by
Let a set of parameters be a set of status of patients in which stands for the parameter “chest pain,” stands for the parameter “headache,” stands for the parameter “mental depression,” stands for the parameter “migraine,” stands for the parameter “neurosis,” stands for the parameter “omodynia,” stands for the parameter “period pains,” stands for the parameter “toothache,” with the following binary operation:
Then is a BCI-algebra. Consider a double-framed soft set over as follows:
Then we have
and/or
Hence, is not a double-framed soft algebra over .

Lemma 3.6. *Every double-framed soft algebra over satisfies the following condition:
*

*Proof. *It is straightforward.

Proposition 3.7. *For a double-framed soft algebra over , the following are equivalent: *(1),
(2).

*Proof. *Assume that (2) is valid. Taking and using (2.1), we have and . It follows from Lemma 3.6 that and .

Conversely, suppose that and for all . Using (3.1), we have
for all . This completes the proof.

Proposition 3.8. *In a BCI-algebra , every double-framed soft algebra over satisfies the following condition:
*

*Proof. *Using Lemma 3.6, we have
for all .

Let and be double-framed soft sets over a common universe . Then is called a *double-framed soft subset* of , denoted by , if (i),
(ii).

Theorem 3.9. *Let be a double-framed soft subset of a double-framed soft set . If is a double-framed soft algebra over , then so is .*

*Proof. *Let . Then , and so
Hence, is a double-framed soft algebra over .

The converse of Theorem 3.9 is not true as seen in the following example.

*Example 3.10. *Suppose that there are six houses in the initial universe set given by
Let a set of parameters be a set of status of houses in which stands for the parameter “beautiful,” stands for the parameter “cheap,” stands for the parameter “in good location,” stands for the parameter “in green surroundings,” stands for the parameter “luxury,” with the following binary operation:
Then is a BCI-algebra. For a subalgebra , define a double-framed soft set over as follows:
It is routine to verify that is a double-framed soft algebra over . Take , and define a double-framed soft set over as follows:
Then is not a double-framed soft algebra over since
and/or

For two double-framed soft sets and over , the *double-framed soft int-uni set* of and is defined to be a double-framed soft set where
It is denoted by .

Theorem 3.11. *The double-framed soft int-uni set of two double-framed soft algebras and over is a double-framed soft algebra over .*

*Proof. *For any , we have
Therefore, is a double-framed soft algebra over .

For a double-framed soft set over and two subsets and of , the *-inclusive set* and the *-exclusive set* of , denoted by and , respectively, are defined as follows:
respectively. The set
is called a *double-framed including set* of . It is clear that

Theorem 3.12. *For a double-framed soft set over , the following are equivalent: *(1)* is a double-framed soft algebra over ,*(2)*for every subsets and of with and , the -inclusive set and the -exclusive set of are subalgebras of .*

*Proof. *Assume that is a double-framed soft algebra over . Let be such that and for every subsets and of with and . It follows from (3.1) that
Hence, and , and therefore, and are subalgebras of .

Conversely suppose that (2) is valid. Let be such that , , and . Taking and implies that and . Hence, and , which imply that
Therefore, is a double-framed soft algebra over .

Corollary 3.13. *If is a double-framed soft algebra over , then the double-framed including set of is a subalgebra .*

For any double-framed soft set over , let be a double-framed soft set over defined by where , and are subsets of with and .

Theorem 3.14. *If is a double-framed soft algebra over , then so is .*

*Proof. *Assume that is a double-framed soft algebra over . Then and are subalgebras of for every subsets and of with and . Let . If , then . Thus,
If or , then or . Hence,
Now, if , then . Thus,
If or , then or . Hence,
Therefore, is a double-framed soft algebra over .

The following example shows that the converse of Theorem 3.14 is not true in general.

*Example 3.15. *Suppose that there are ten houses in the initial universe set given by
Let a set of parameters be a set of status of houses in which stands for the parameter “beautiful,” stands for the parameter “cheap,” stands for the parameter “in good location,” stands for the parameter “in green surroundings,” with the following binary operation:
Then is a BCI-algebra. Consider a double-framed soft set over as follows:
Note that for and . Let be a double-framed soft set over defined by
that is,
It is routine to verify that is a double-framed soft algebra over . But is not a double-framed soft algebra over since and/or .

Let and be double-framed soft sets over . The *-product* of and is defined to be a double-framed soft set over in which

Theorem 3.16. *For any BCK/BCI-algebras and as sets of parameters, let and be double-framed soft algebras over . Then the -product of and is also a double-framed soft algebra over .*

*Proof. *Note that is a BCK/BCI-algebra. For any , we have
Hence, is a double-framed soft algebra over .

Theorem 3.16 is illustrated from the following example.

*Example 3.17. *Consider the initial universe set given by
which consists of six women patients, and take five parameters as follows: stands for the parameter “chest pain,” stands for the parameter “headache,” stands for the parameter “mental depression,” stands for the parameter “migraine,” stands for the parameter “neurosis.” Let and with the following Cayley tables:
Then
is a BCK-algebra with the following Cayley table:
Let and be double-framed soft sets over given by
respectively. Then and are double-framed soft algebras over . Moreover, the -product of and is a double-framed soft algebra over .

#### 4. Conclusion

We have introduced the notion of double-framed soft algebras in a BCK/BCI-algebras and provided several examples. We have considered the characterization of double-framed soft algebra. We have shown the following.(1)Every double-framed soft subset of a double-framed soft algebra is also a double-framed soft algebra. (2)The double-framed soft int-uni set of two double-framed soft algebras is a double-framed soft algebra. (3)The , -product of double-framed soft algebras and is also a double-framed soft algebra.

Given a double-framed soft algebra, we have made a new double-framed soft algebra. We have provided an example which shows that the new constructed double-framed soft set is a double-framed soft algebra, but the given double-framed soft set is not a double-framed soft algebra.

Future research will focus on applying the idea/result in this paper to the ideal theory of BCK/BCI-algebras and related algebraic structures.

#### Acknowledgment

The authors wish to thank the anonymous reviewers for their valuable suggestions.

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