Soft Translations and Soft Extensions of BCI/BCK-Algebras

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.


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 Ç agman 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][14][15][16][17][18][19][20][21][22][23][24][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][28][29][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 2 The Scientific World Journal ideal translation in BCI/BCK-algera. In Section 5, concept of ideal extension is introduced and some of its properties are studied.

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.
A subset of a BCI/BCK-algebra is called an ideal of , denoted by ⊲ , if it satisfies: (1) 0 ∈ , (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 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 0 , if ( ) = 0, for all ∈ .
(b) ( , ) is called a relative whole soft set (with respect to the parameters set ), denoted by , if ( ) = , for all ∈ .

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 = ( , * , 0) denote a BCI/BCK-algebra, and for any soft set over , we denote := − ∪{ ( ) | ∈ X} unless otherwise specified.
That is = (⋃ ∈ ( )) = ⋂ ∈ ( ). It is easy to see that ∩ ( ) = 0 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. Proof.

Proposition 7.
Let be a soft subalgebra of and 1 ⊆ . Then the soft 1 -translation 1 of is a soft subalgebra of . Proof. Let , ∈ . Then Hence 1 is a soft subalgebra of . Proposition 8. Let be a soft set over such that the 1translation 1 of is a soft subalgebra of for some 1 ⊆ . Then is a soft subalgebra of .
Proof. Assume 1 is a soft subalgebra of for some 1 ⊆ . Let , ∈ , we have Now by Lemma 6 we have for all , ∈ . Hence is a soft subalgebra of .
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: is a soft extension of .
(2) If is a soft subalgebra of , then is a soft subalgebra of .
As we know 1 ( ) ⊇ ( ) for all ∈ . As a consequence of Definition 12 and Theorem 9, we have the following.

Theorem 13.
Let be a soft subalgebra of and 1 ⊆ T. Then the soft 1 -translation 1 of is a soft -extension of .
The converse of Theorem 13 is not true in general as seen in the following example. Table 1   0  1  2  3  {0}  {0, 1}  {0, 2}  {1, 2}  {0}  {0, 1, 2}  {0, Define a soft set of by Table 2. Then is a soft subalgebra of . For soft set , = {3}. Let be a soft set over given by Table 3. Then is a soft -extension of . But it is not a soft 1translation of for any nonempty 1 ⊆ .
For a soft set of , 1 ⊆ and 2 ∈ ( ) with 2 ⊇ 1 , let If is a soft subalgebra of , then it is clear that 1 ( ; 2 ) is a subalgebra of for all 2 ∈ ( ) with 2 ⊇ 1 . But, if we do not give condition that is a soft subalgebra of , then 1 ( ; 2 ) may not be a subalgebra of as seen in the following example.
Conversely, suppose that 1 ( ; 2 ) is a subalgebra of for all 2 ⊆ ( ) with 2 ⊇ 1 . Now assume that there exist , ∈ such that Then ( ) ⊇ 2 − 1 and ( ) ⊇ 2 − 1 but ( * ) ⊂ 2 − 1 . This shows that , ∈ 1 ( ; 2 ) and * ∉ 1 ( ; 2 ), which is a contradiction and so 1 ( * ) ⊇ 1 ( ) ∩ 1 ( ) for all , ∈ . Hence 1 is a soft subalgebra of . 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: 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 Z + be a set of positive integers and let " * " be a binary operation on Z + defined by * = ( , ) , ∀ , ∈ Z + , where ( , ) is the greatest common divisor of and . Then (Z + ; * , 1) is a BCK-algebra. Let and be soft sets of Z + which are defined by ( ) = {1, 2, 3} and ( ) = Z + for all ∈ Z + . Clearly, and are soft subalgebras of Z + . 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 softextension of a soft subalgebra
of , then (0) = .
Proof. Assume that is a normalized soft -extension of a soft subalgebra of then there exists 0 ∈ such that ( 0 ) = , for some 0 ∈ . Consider This implies (0) = .
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.

Soft Translations of Soft Ideals in Soft BCI/BCK-Algebras
Now concept of translation of a soft ideal of a BCI/BCKalgebra is introduced.

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: is a soft extension of .

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 6 The Scientific World Journal 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.