#### Abstract

We further develop the theory of vague soft groups by establishing the concept of normalistic vague soft groups and normalistic vague soft group homomorphism as a continuation to the notion of vague soft groups and vague soft homomorphism. The properties and structural characteristics of these concepts as well as the structures that are preserved under the normalistic vague soft group homomorphism are studied and discussed.

#### 1. Introduction

Soft set theory introduced by Molodtsov in 1999 (see [1]) is a general mathematical tool that is commonly used to deal with imprecision, uncertainties, and vagueness that are pervasive in a lot of complicated problems affecting various areas in the real world. Since its inception, research on soft set theory as well as its generalizations and related theories such as soft algebra and soft topology has been developing at an exponential rate. Presently, research pertaining to other areas of generalizations of soft set theory such as fuzzy soft algebraic theory and vague soft algebraic theory is being carried out and is progressing at a rapid rate.

The study of soft algebra was initiated by Aktaş and Çaǧman in 2007 (see [2]) through the introduction of the notion of soft groups. A soft group is a parameterized family of subgroups which includes the algebraic structures of soft sets. Sezgin and Atagün on the other hand (see [3]) introduced the notion of normalistic soft groups and normalistic soft group homomorphism as an extension to the notion of soft groups introduced by [2]. All this led to the study of fuzzy soft algebra by Aygunoglu and Aygün (see [4]) who introduced the notion of fuzzy soft groups. In [4], the authors applied Rosenfeld’s well-known concept of a fuzzy subgroup of a group (see [5]) to fuzzy soft set theory to introduce the concept of a fuzzy soft group of a group which extends the notion of soft groups to include the theory of fuzzy sets and fuzzy algebra.

The concept of vague soft sets which is a combination of the notion of soft sets and vague sets was introduced by Xu et al. (see [6]), as an extension to the notion of soft sets and fuzzy soft sets. Research in the area of vague soft algebra was initiated by Varol et al. (see [7]) who developed the theory of vague soft groups and defined the concepts of vague soft groups, normal vague soft groups, and vague soft homomorphism. Therefore it is now natural to further develop and investigate the concepts introduced in [7] and expand the theory by introducing related concepts as well as generalizations of the existing concepts.

In this paper, we contribute to the further development of the theory of vague soft groups. We define the notion of normalistic vague soft groups which is an extended albeit more comprehensive alternative to the concept of normal vague soft groups introduced in [7]. Varol et al. in [7] defined the concept of normal vague soft groups as an Abelian vague soft group; that is, is a vague soft group that satisfies the commutative law given by and . However, this definition is incomplete as there exist two other statements which describe the normality of a vague soft set that is also equivalent to the commutative law used in [7]. As such, in this paper we incorporate all these three equivalent statements into a single definition with the aim of establishing a more complete, comprehensive, and accurate representation of this concept compared to the notion of normal vague soft groups initiated in [7]. We choose to name this concept as normalistic vague soft groups in order to distinguish it from the existing concept of normal vague soft groups proposed in [7]. Subsequently, we use this concept of normalistic vague soft groups to define the notion of normalistic vague soft group homomorphism as a natural extension to the concept of vague soft homomorphism proposed in [7]. Furthermore, some of the properties and structural characteristics of the concept of normalistic vague soft groups are studied and illustrated with an example. Lastly, we prove that there exists a one-to-one correspondence between normalistic vague soft groups and some of the corresponding concepts in soft group theory and classical group theory.

#### 2. Preliminaries

In this section, some important concepts and definitions on soft set theory, vague soft set theory, and hyperstructure theory will be presented.

*Definition 1 (see [1]). *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 -elements of the soft set or as the -approximate elements of the soft set.

*Definition 2 (see [8]). *For a soft set , the set is called the* support* of the soft set . Thus a nonnull soft set is indeed a soft set with an empty support and a soft set is said to be nonnull if .

*Definition 3 (see [2]). *Let be a group and let be a soft set over . Then defined by is said to be a* soft group* over if and only if , for each .

*Definition 4 (see [2]). *Let and be two soft groups over . Then is a* soft subgroup* of , written as , if (i),(ii) for all .

*Definition 5 (see [3]). *Let be a group and let be a nonnull soft set over . Then is called a* normalistic soft group* over if is a normal subgroup of for all .

*Definition 6 (see [9]). *Let be a space of points (objects) with a generic element of denoted by . A* vague set * in is characterized by a truth membership function and a false membership function . The value is a lower bound on the grade of membership of derived from the evidence for and is a lower bound on the negation of derived from the evidence against . The values and both associate a real number in the interval with each point in , where . This approach bounds the grade of membership of to a subinterval of . Hence a vague set is a form of fuzzy set.

*Definition 7 (see [6]). *A pair is called a* vague soft set* over where is a mapping given by and is the power set of vague sets on . In other words, a vague soft set over is a parameterized family of vague sets of the universe . Every set for all , from this family, may be considered as the set of -approximate elements of the vague soft set . Hence the vague soft set can be viewed as consisting of a collection of approximations of the following form:for all and for all .

*Definition 8 (see [10]). *Let be a vague soft set over . The* support* of denoted by is defined asfor all .

It is to be noted that a* null vague soft set* is a vague soft set where both the truth and false membership functions are equal to zero. Therefore, a vague soft set is said to be nonnull if .

*Definition 9 (see [10]). *The* extended intersection* of two vague soft sets and over a universe , which is denoted as , is a vague soft set , where and, for every and ,where and are the upper and lower bounds of , respectively. This relationship can be written as for every .

*Definition 10 (see [10]). *The* restricted intersection* of two vague soft sets and over a universe , which is denoted as , is a vague soft set , where and, for every and ,where and are the upper and lower bounds of , respectively. This relationship can be written as .

*Definition 11 (see [7]). *Let be a vague soft set over . Then for every , where , the *-cut* or the* vague soft cut* of is a subset of which is defined as follows:for every .

*Definition 12 (see [10]). *Let be a vague soft set over . Then for every , the *-cut* of , denoted as , is a subset of which is defined as follows:for every .

*Definition 13 (see [10]). *Let be a vague soft set over and let be a nonnull subset of . Then is called a* vague soft characteristic set* of in and the lower bound and the upper bound of are defined as follows:where is a subset of , , , and .

#### 3. Vague Soft Groups

In this section, the concept of vague soft groups and some important results pertaining to this concept introduced in [7] are presented. These definitions and results will be extended to the concept of normalistic vague soft groups in the next section.

*Definition 14 (see [7]). *Let be a group and let be a vague soft set over . Then is called a* vague soft group* over if and only if, for every and , the following conditions are satisfied:(i) and ; that is, ,(ii) and ; that is, .In other words, for every is a* vague subgroup* in Rosenfeld’s sense.

Proposition 15 (see [7]). *Let be a vague soft group over and let be the identity element of . Then for every and ,*(i)* and ; that is, ,*(ii)* and ; that is, .*

Proposition 16 (see [7]). *Let be a vague soft set. Then is vague soft group if and only if, for every and ,*

#### 4. Normalistic Vague Soft Groups

In this section, we propose the concept of normalistic vague soft groups and study some of the fundamental properties and structural characteristics of this concept.

Theorem 17 (see [7]). *Let be a vague soft set over . Then the following statements are equivalent for each and for every :*(i)* and ; that is, or*(ii)* and ; that is, or*(iii)* and ; that is, .**If condition (iii) is satisfied, then is said to be an Abelian vague soft set over .*

*Definition 18. *Let be a group and let be a nonnull vague soft group over . Then is called a* normalistic vague soft group* over if, for every and for every , either one of the following conditions is satisfied:(i) and ; that is, or(ii) and ; that is, or(iii) and ; that is, .In other words, for every and , is a* normal vague subgroup* of in Rosenfeld’s sense. If satisfies condition (iii), then is called an* Abelian vague soft group*.

From Definition 18, it is obvious that all normalistic vague soft groups are vague soft groups but the converse is not necessarily true.

Proposition 19. *Let be a nonnull vague soft set over and . If is a normalistic vague soft group over , then is a normalistic vague soft group over if it is nonnull.*

*Proof. *Let be a normalistic vague soft group over . Then is a normal vague subgroup of for every . Since , is a vague soft subgroup of . Therefore is a normal vague subgroup of for every . This implies that is a normal vague subgroup of for every , because is a vague soft subgroup of . Then is a normalistic vague soft group over .

Proposition 20. *Let be a normalistic vague soft group over and let be a nonnull vague soft set over which is as defined below:**for every and while is the identity element of group . Then is a normalistic vague soft group over .*

*Proof. * is a normal vague subgroup of for each . Now let and . Then and and also and . This means that and . Thus the following is obtained:and .

Similarly, it can be proven that and . Therefore is vague subgroup of . Next we prove normality: Similarly, we obtain . Hence it has been proven that is a normal vague subgroup of . As such, is a normalistic vague soft group over .

Theorem 21. *Let be a nonnull subset of , let be a normalistic vague soft group over , let be a vague soft characteristic set over , and let be a vague soft set over as defined in Proposition 20. If is a normalistic vague soft set over , then is a normal subgroup of .*

*Proof. *Suppose that is a nonnull subset of and is a normalistic vague soft group over . Then is a normal vague subgroup of for every . Now let and . Thus we obtain and . Since is a normal vague subgroup of , we obtain the following:Similarly, it can be proven that too. This means that and therefore . Hence is a subgroup of . To prove normality, let and . Then we obtainIn the same manner, it can be proven that . Thus and this implies that . As such, it is proven that is a normal subgroup of .

*Definition 22. *Let be a normalistic vague soft group over . Then(i) is called a* trivial normalistic vague soft group* over if for all and the truth and false membership function of are defined as follows: for every and .(ii) is called an* absolute normalistic vague soft group* over if for all and the truth and false membership function of are defined as follows:

*Example 23. *Let be a group with the operations of group as given in the Cayley table in Table 1.

(i) Then let and let be a nonnull vague soft group over , where is a set-valued function that is as defined below: for all . Then and .

The truth membership function and false membership function of are as given below:for all and . Since for all is a trivial normal vague subgroup of . As such, is a trivial normalistic vague soft group over .

(ii) Now let and let be a nonnull vague soft group over , where is a set-valued function that is as defined below: for all . Thus , , and The truth membership function and false membership function of are as given below:for every and . Hence for all and therefore is an absolute normal vague subgroup of . As such, is an absolute normalistic vague soft group over .

Theorem 24. *Let and be normalistic vague soft groups over . Then consider the following.*(i)*If and are trivial normalistic vague soft groups over , then is a trivial normalistic vague soft group over .*(ii)*If and are absolute normalistic vague soft groups over , then is an absolute normalistic vague soft group over .*(iii)*If is a trivial normalistic vague soft group over and is an absolute normalistic vague soft group over , then is a trivial normalistic vague soft group over .*

*Proof. *The proofs are straightforward and are therefore omitted.

Theorem 25. *Let be a group epimorphism and let be a normalistic vague soft group over . Then is a normalistic vague soft group over .*

*Proof. *Since is a normalistic vague soft group over , it must be nonnull and therefore it follows that must be nonnull too. Now let and assume that, for every , there exists such that and . Thus we obtain the following:As such, is a normalistic vague soft group over .

Theorem 26. *Let be a group isomorphism and let be a normalistic vague soft group over . Then is a normalistic vague soft group over .*

*Proof. *The proof is similar to that of Theorem 25 and is therefore omitted.

Theorem 27. *Let and be normalistic vague soft groups over and , respectively, and let be a group epimorphism. Then consider the following.*(i)*If for all , then is a trivial normalistic vague soft group over .*(ii)*If is an absolute normalistic vague soft group over , then is an absolute normalistic vague soft group over .*(iii)*If is injective and for every , then is an absolute normalistic vague soft group over .*(iv)*If is injective and is a trivial normalistic vague soft group over , then is a trivial normalistic vague soft group over .*

*Proof. *(i) Let be a group epimorphism from to . Then the kernel of is as given below: where is the identity element of . Let for every . Then for every and , it follows that . Then the truth and false membership function of are as given below:for every and . As such, is a trivial normalistic vague soft group over .

(ii) The proof is similar to the proof of part (i) and is therefore omitted.

(iii) Let be an injective group epimorphism and, for every , . Thus for every because is injective. Therefore by Definition 22(ii), is an absolute normalistic vague soft group over . As such, we obtainfor every because is injective. Thus for every , . This means that is an absolute normalistic vague soft group over with truth and false membership function as given below:for every . Hence is an absolute normalistic vague soft group over .

(iv) The proof is similar to the proof of part (iii) and is therefore omitted.

#### 5. The Homomorphism of Normalistic Vague Soft Groups

In [7], the notion of vague soft functions was introduced whereas the concepts of the image and preimage of a vague soft set under a vague soft function were introduced in [10]. Here we extend these concepts to include normalistic vague soft groups and subsequently introduce the notion of normalistic vague soft group homomorphism. Lastly, we prove that this homomorphism preserves normalistic vague soft groups.

*Definition 28 (see [7]). *Let and be two functions, where and are the set of parameters for the classical sets and , respectively. Let and be vague soft groups over and , respectively. Then the ordered pair is called a* vague soft function* from to , denoted as .

*Definition 29 (see [10]). *Let and be two vague soft groups over and , respectively. Let be a vague soft function.(i)The* image* of under the vague soft function , denoted as , is a vague soft set over , which is defined as where for every , , and .(ii)The* preimage* of under the vague soft function , denoted as , is a vague soft set over , which is defined as where for every , , and .If and are injective (surjective), then the vague soft function is said to be injective (surjective).

Next we introduce the notion of normalistic vague soft group homomorphism as an extension to the notion of vague soft homomorphism introduced in [7].

*Definition 30. *Let and be normalistic vague soft groups over and , respectively. Then the vague soft function is called as follows.

(a) It is called* normalistic vague soft group homomorphism* if the following conditions are satisfied:(i) is a homomorphism from to ,(ii) is a function from into ,(iii) for every .Then is said to be* normalistic vague soft homomorphic* to and this is denoted as .

(b) It is called* normalistic vague soft group isomorphism* if the following conditions are satisfied:(i) is an isomorphism from to ,(ii) is a bijective mapping from to ,(iii) for every .Then is said to be* normalistic vague soft isomorphic* to and this is denoted as .

Theorem 31. *Let and be normalistic vague soft groups over and , respectively, and let be a normalistic vague soft group homomorphism. Then consider the following:*(i)* is a normalistic vague soft group over .*(ii)* is a normalistic vague soft group over .*

*Proof. *The proofs follow from Definitions 18 and 30 and are therefore omitted.

Theorem 32. *Let , , and be groups, let , , and be normalistic vague soft groups over , , and , respectively, and let and be normalistic vague soft group homomorphisms. Then is a normalistic vague soft group homomorphism from to .*

*Proof. *Suppose that represents a normalistic vague soft homomorphism from to . Then is a vague soft function such that is a group homomorphism from into and is a function from to . Also satisfies the following condition:Now suppose that represents a normalistic vague soft homomorphism from to . Then is a vague soft function such that represents a group homomorphism from to , is a function from into , and the following condition is satisfied: Therefore we have which is a group homomorphism from into and the composition represents a function from into . Therefore it follows that is a vague soft function from to . Furthermore for all we have Hence is a normalistic vague soft group homomorphism from to .

Theorem 33. *Let and be groups and let and be vague soft sets over and , respectively. If is a normalistic vague soft group over and , then is a normalistic vague soft group over .*

*Proof. *Since , there exists a vague soft group isomorphism from to . This means that there exists an isomorphism from to and a bijective mapping from to which satisfies for every . Now suppose that is a normalistic vague soft group over . Then is a normal vague subgroup of for all . Thus is a normal vague subgroup of for all . Since is a bijective mapping, for all , there exists an such that . Hence we have for all . As such, is a normal vague subgroup of for all .

Corollary 34. *Let and be normalistic vague soft groups over and , respectively, and . If is a normal vague subgroup of , then is a normal vague subgroup of and .*

#### 6. Conclusion

In this paper, we continue to further develop the initial theory of vague soft groups. We successfully introduced the novel concept of normalistic vague soft group homomorphisms as an extension to the notion of vague soft homomorphism. We also further developed and studied the concept of normalistic vague soft groups through the introduction of the notion of trivial normalistic vague soft groups and absolute normalistic vague soft groups as well as studying the behavior of normalistic vague soft groups under the normalistic vague soft group homomorphism. Lastly, it is proven that the homomorphic image and preimage of a normalistic vague soft group are preserved under the normalistic vague soft group homomorphism.

#### Conflict of Interests

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

#### Acknowledgments

The authors would like to gratefully acknowledge the financial assistance received from the Ministry of Education, Malaysia, and UCSI University, Malaysia, under Grant no. FRGS/1/2014/ST06/UCSI/03/1.