The Scientific World Journal

Volume 2014 (2014), Article ID 536709, 6 pages

http://dx.doi.org/10.1155/2014/536709

Research Article

Soft Translations and Soft Extensions of BCI/BCK-Algebras

Nazra Sultana,¹ Nazia Rani,1,2 Muhammad Irfan Ali,³ and Azhar Hussain¹

¹Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan

²Department of Mathematics, University of Sargodha, Women Campus, Faisalabad, Pakistan

³Islamabad Model College for Girls F-6/2, Islamabad 44000, Pakistan

Received 12 June 2014; Accepted 8 August 2014; Published 14 September 2014

Academic Editor: Feng Feng

Copyright © 2014 Nazra Sultana 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 concept of soft translations of soft subalgebras and soft ideals over BCI/BCK-algebras is introduced and some related properties are studied. Notions of Soft extensions of soft subalgebras and soft ideals over BCI/BCK-algebras are also initiated. Relationships between soft translations and soft extensions are explored.

1. Introduction

Recently soft set theory has emerged as a new mathematical tool to deal with uncertainty. Due to its applications in various fields of study researchers and practitioners are showing keen interest in it. As enough number of parameters is available here, so it is free from the difficulties associated with other contemporary theories dealing with uncertainty. Prior to soft set theory, probability theory, fuzzy set theory, rough set theory, and interval mathematics were common mathematical tools for dealing with uncertainties, but all these theories have their own difficulties. These difficulties may be due to lack of parametrization tools [1, 2]. To overcome these difficulties, Molodtsov [2] introduced the concept of soft sets. A detailed overview of these difficulties can be seen in [1, 2]. As a new mathematical tool for dealing with uncertainties, Molodtsov has pointed out several directions for the applications of soft sets. Theoretical development of soft sets is due to contributions from many researchers. However in this regard initial work is done by Maji et al. in [1]. Later Ali et al. [3] introduced several new operations in soft set theory.

At present, work on the soft set theory is progressing rapidly. Maji et al. [4] described the application of soft set theory in decision making problems. Aktaş and Çağman studied the concept of soft groups and derived their basic properties [5]. Chen et al. [6] proposed parametrization reduction of soft sets, and then Kong et al. [7] presented the normal parametrization reduction of soft sets. Feng and his colleagues studied roughness in soft sets [8, 9]. Relationship between soft sets, fuzzy sets, and rough sets is investigated in [10]. Park et al. [11] worked on notions of soft WS-algebras, soft subalgebras, and soft deductive system. Jun and Park [12] presented the notions of soft ideals, idealistic soft, and idealistic soft BCI/BCK-algebras. Further applications of soft sets can be seen in [13–25].

The study of BCI/BCK-algebras was initiated by Imai and Iseki [26] as the generalization of concept of set theoretic difference and propositional calculus. For the general development of BCI/BCK-algebras, the ideal theory and its fuzzification play an important role. Jun et al. [27–30] studied fuzzy trends of several notions in BCI/BCK-algebras. Application of soft sets in BCI/BCK is given in [12, 31].

Translations play a vital role in reducing the complexity of a problem. In geometry it is a common practice to translate a system to some new position to study its properties. In linear algebra translations help find solution to many practical problems. In this paper idea of translations is being extended to soft BCI/BCK algebras.

This paper is arranged as follows: in Section 2, some basic notions about BCI/BCK-algebra and soft sets are given. These notions are required in the later sections. Concept of translation is introduced in Section 3 and some properties of it are discussed here. Section 4 is devoted for the study of soft ideal translation in BCI/BCK-algera. In Section 5, concept of ideal extension is introduced and some of its properties are studied.

2. Preliminaries

First of all some basic concepts about BCI/BCK-algebra are given. For a comprehensive study on BCI/BCK-algebras [32] is a very nice monograph by Meng and Jun. Then some notions about soft sets are presented here as well.

An algebra is called a BCI-algebra if it satisfies the following conditions:

(1),(2),(3),(4).

If a BCI-algebra satisfies the following identity:

(5),

then is called a BCK-algebra. Any BCK-algebra satisfies the following axioms:

(i),(ii),(iii),(iv).

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

A subset of a BCI/BCK-algebra is called an ideal of , denoted by , if it satisfies:

(1),(2).

Now we recall some basic notions in soft set theory. Let be a universe and be a set of parameters. Let denote the power set of and let , be nonempty subsets of .

Definition 1 (see [2]). A pair is called a soft set over , where is a mapping given by .

Definition 2 (see [3]). Let be a universe, let be the set of parameters, and let .

(a) is called a relative null soft set (with respect to the parameters set ), denoted by , if , for all .(b) is called a relative whole soft set (with respect to the parameters set ), denoted by , if , for all .

Definition 3 (see [3]). The complement of a soft set is denoted by and is defined by , where is a mapping given by , . Clearly, .

Definition 4 (see [8]). A soft set over is called a full soft set if .

3. Soft Translations of Soft Subalgebras

Here notion of translations in soft BCI/BCK-algebra is initiated. Concept of soft extensions is introduced here also.

Let be set valued map defined as

where . Then also denotes a soft set over a BCI/BCK algebra . From here onward a soft set will be denoted by symbols like , unless stated otherwise.

A soft set over a BCI/BCK-algebra is called a soft subalgebra of if it satisfies

In what follows denote a BCI/BCK-algebra, and for any soft set over , we denote unless otherwise specified.

That is .

It is easy to see that for all . If is a full soft set then is an empty set. Therefore throughout this paper only those soft set are considered which are not full.

Definition 5. Let be a soft set over and let . A mapping is called a soft -translation of if, for all ,

Lemma 6. Let and be a soft set over , then implies , for all .

Proof. Since , and . Let then this implies or but because . So that is , for all .

Proposition 7. Let be a soft subalgebra of and . Then the soft -translation of is a soft subalgebra of .

Proof. Let . Then

Hence is a soft subalgebra of .

Proposition 8. Let be a soft set over such that the -translation of is a soft subalgebra of for some . Then is a soft subalgebra of .

Proof. Assume is a soft subalgebra of for some . Let , we have

Now by Lemma 6 we have

for all . Hence is a soft subalgebra of .

From Propositions 7 and 8 we have the following.

Theorem 9. A soft set of is a soft subalgebra of if and only if -translation of is a soft subalgebra of for some .

Definition 10. Let and be two soft sets over . If for all , then we say that is a soft extension of .

Example 11. Consider a BCI/BCK-algebra presented as follows:

Define two soft sets and of as in Table 1.

Here , , , and , which implies that is a soft extension of .

Table 1

Next the concept of soft -extension is being introduced.

Definition 12. Let and be two soft sets over . Then is called a soft -extension of , if the following conditions hold:

(1) is a soft extension of .(2)If is a soft subalgebra of , then is a soft subalgebra of .

As we know for all . As a consequence of Definition 12 and Theorem 9, we have the following.

Theorem 13. Let be a soft subalgebra of and . Then the soft -translation of is a soft -extension of .

The converse of Theorem 13 is not true in general as seen in the following example.

Example 14. Consider a BCI/BCK-algebra given as follows:

Define a soft set of by Table 2.

Then is a soft subalgebra of . For soft set , . Let be a soft set over given by Table 3.

Then is a soft -extension of . But it is not a soft -translation of for any nonempty .

Table 2

Table 3

0
1
2
3

For a soft set of , and with , let

If is a soft subalgebra of , then it is clear that is a subalgebra of for all with . But, if we do not give condition that is a soft subalgebra of , then may not be a subalgebra of as seen in the following example.

Example 15. Let be a BCI/BCK-algebra presented as follows:

Define a soft subset of by Table 4.

Then is not a soft subalgebra of with . Since For and , we obtain which is not a subalgebra of since .

Table 4

0
1
2
3
4

In the following theorem, relationship between -translations and is studied in case of soft subalgebra.

Theorem 16. Let be a soft set over and . Then the soft -translation of is a soft subalgebra of if and only if is a subalgebra of for all with .

Proof. Assume that the soft -translation of is a soft subalgebra of . Then by Theorem 9, is a soft subalgebra of if is a soft subalgebra of . Further let , then and are subalgebras of for all with . Consider

Therefore , which shows that is a subalgebra of , for all , with .

Conversely, suppose that is a subalgebra of for all with . Now assume that there exist such that

Then and but . This shows that and , which is a contradiction and so for all . Hence is a soft subalgebra of .

Theorem 17. Let be a soft subalgebra of and let . If , then the soft -translation of is a soft -extension of the soft -translation of .

Proof. Since , this implies , for all . So -translation is an extension of -translation, and from Theorem 9, and are soft subalgebras of . Hence soft -translation of is a soft -extension of the soft -translation of .

For every soft subalgebra of and , the soft -translation of is a soft subalgebra of . If is a soft -extension of and then there exists such that and , for all . Thus, we have the following theorem.

Theorem 18. Let be a soft subalgebra of and . For every soft -extension of soft -translation of , there exists a such that and are a soft -extension of -translation of .

Proof. For every soft subalgebra of and , the soft -translation of is a soft subalgebra of . If is a soft -extension of and then there exists such that and , for all . Then by Theorem 17, is a soft -extension of -translation of .

Definition 19. A soft -extension of a soft subalgebra of is said to be normalized if there exists such that .

Definition 20. Let be a soft subalgebra of . A soft set of is called a maximal soft -extension of if it satisfies the following conditions:

(1) is a soft -extension of ,(2)there does not exist another soft subalgebra of which is a soft extension of .

Example 21 (see [33]). Let be a set of positive integers and let “” be a binary operation on defined by

, where is the greatest common divisor of and . Then is a BCK-algebra. Let and be soft sets of which are defined by and for all . Clearly, and are soft subalgebras of . By using definition of maximal soft -extension, then it is easy to see that is a maximal soft -extension of .

Proposition 22. If a soft set of is a normalized soft -extension of a soft subalgebra of , then .

Proof. Assume that is a normalized soft -extension of a soft subalgebra of then there exists such that , for some . Consider

This implies .

Theorem 23. Let be a soft subalgebra of . Then every maximal soft -extension of is normalized.

Proof. This follows from the definitions of the maximal and normalized soft -extensions.

4. Soft Translations of Soft Ideals in Soft BCI/BCK-Algebras

Now concept of translation of a soft ideal of a BCI/BCK-algebra is introduced.

Definition 24. A soft subset of a BCI/BCK-algebra is called a soft ideal of , denoted by , if it satisfies:

(1),(2).

Theorem 25. If is a soft subset of , then is a soft ideal of if and only if soft -translation of is a soft ideal of for all .

Proof. Assume that and let . Then and

Conversely, assume that is a soft ideal of for some . Let . Then

and so . Next

which implies that (by Lemma 6). Hence is a soft ideal of .

5. Soft Extensions and Soft Ideal Extensions of Soft Subalgebras

In this section concept of soft ideal extension is being introduced and some of its properties are studied.

Definition 26. Let and be the soft subsets of . Then is called the soft ideal extension of , if the following conditions hold:

(1) is a soft extension of .(2).

For a soft subset of , and with , define .

It is clear that if , then for all with .

Theorem 27. For , let be the soft -translation of . Then the following are equivalent:

(1).(2).

Proof. Consider and let be such that . Since for all , we have

for .

Let be such that and . Then and , that is, and . Since , it follows that

that is, so that . Therefore .

Suppose that for every with . If there exists with such that and then but . This shows that and . This is a contradiction, and so , for all .

Now assume that there exist such that . Then and , but . Hence and , but . This is impossible and therefore , for all . Consequently .

Theorem 28. Let and . If  , then the soft -translation of is a soft ideal extension of the soft -translation of .

Proof. Since

, this implies that . This shows that is a soft extension of .

Now, let is a soft ideal of , then for all , so we have . Consider

That is so is a soft ideal of . Hence is a soft ideal extension of .

6. Conclusion

Soft set theory is a mathematical tool to deal with uncertainties. Translation and extension are very useful concepts in mathematics to reduce the complexity of a problem. These concepts are frequently employed in geometry and algebra. In this papers, we presented some new notions such as soft translations and soft extensions for BCI/BCK-algebras. We also examined some relationships between soft translations and soft extensions. Moreover, soft ideal extensions and translations have been introduced and investigated as well. It is hoped that these results may be helpful in other soft structures as well.

Conflict of Interests

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

Acknowledgment

Authors are grateful to referees and Professor Feng Feng, the Lead Guest Editor of this special issue, for his kind suggestions to improve this paper.

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