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: