# On Normalistic Vague Soft Groups and Normalistic Vague Soft Group Homomorphism

Authors: Ganeshsree Selvachandran; Abdul Razak Salleh
Journal: Advances in Fuzzy Systems
 (2015)

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## 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.


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## Body

## 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, (F^,A) is a vague soft group that satisfies the commutative law given by tF^ax·y=tF^a(y·x) and 1-fF^ax·y=1-fF^ay·x. 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(F,A) is called asoft set over U, where F is a mapping given by F:A→P(U). In other words, a soft set over U is a parameterized family of subsets of the universe U. For ε∈A, F(ε) may be considered as the set of ε-elements of the soft set F,A or as the ε-approximate elements of the soft set.Definition 2 (see [8]).
For a soft setF,A, the set SuppF,A=x∈A:Fx≠∅ is called thesupport of the soft set (F,A). Thus a nonnull soft set is indeed a soft set with an empty support and a soft set (F,A) is said to be nonnull if Supp(F,A)≠∅.Definition 3 (see [2]).
LetX be a group and let (F,A) be a soft set over X. Then (F,A) defined by F:A→P(X) is said to be asoft group over X if and only if Fa≤X, for each a∈A.Definition 4 (see [2]).
Let(F,A) and (H,K) be two soft groups over G. Then (H,K) is asoft subgroup of (F,A), written as H,K<~(F,A), if (i)
K⊂A,(ii)
H(x)<F(x) for all x∈K.Definition 5 (see [3]).
LetG be a group and let (F,A) be a nonnull soft set over G. Then (F,A) is called anormalistic soft group over G if F(x) is a normal subgroup of G for all x∈Supp(F,A).Definition 6 (see [9]).
LetX be a space of points (objects) with a generic element of X denoted by x. Avague setV in X is characterized by a truth membership function tV:X→[0,1] and a false membership function fV:X→[0,1]. The value tV(x) is a lower bound on the grade of membership of x derived from the evidence for x and fV(x) is a lower bound on the negation of x derived from the evidence against x. The values tV(x) and fV(x) both associate a real number in the interval [0,1] with each point in X, where tVx+fV(x)≤1. This approach bounds the grade of membership of x to a subinterval [tVx,1-fVx] of 0,1. Hence a vague set is a form of fuzzy set.Definition 7 (see [6]).
A pair(F^,A) is called avague soft set over U where F^ is a mapping given by F^:A→V(U) and V(U) is the power set of vague sets on U. In other words, a vague soft set over U is a parameterized family of vague sets of the universe U. Every set F^(e) for all e∈A, from this family, may be considered as the set of e-approximate elements of the vague soft set (F^,A). Hence the vague soft set (F^,A) can be viewed as consisting of a collection of approximations of the following form:(1)F^,A=F^ei:i=1,2,3,…=tF^eixi,1-fF^eixixi:i=1,2,3,…for all e∈A and for all x∈U.Definition 8 (see [10]).
Let(F^,A) be a vague soft set over X. Thesupport of (F^,A) denoted by Supp(F^,A) is defined as(2)Supp⁡F^,A=a∈A:tF^ax≠0,1-fF^ax≠0i.e.F^a≠∅^for all x∈X.
It is to be noted that anull 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 (F^,A) is said to be nonnull if Supp(F^,A)≠∅.Definition 9 (see [10]).
Theextended intersection of two vague soft sets (F^,A) and (G^,B) over a universe X, which is denoted as F^,A∩~G^,B=H^,C, is a vague soft set H^,C, where C=A∪B and, for every c∈C and x∈X,(3)tH^cx=tF^cxc∈A-B,tG^cxc∈B-A,min⁡⁡tF^cx,tG^cxc∈B∩A,1-fH^cx=1-fF^cxc∈A-B,1-fG^cxc∈B-A,min⁡⁡1-fF^cx,1-fG^cxc∈B∩A,where tH^c and fH^c are the upper and lower bounds of H^c, respectively. This relationship can be written as H^cx=F^cx∩~G^c(x) for every x∈X.Definition 10 (see [10]).
Therestricted intersection of two vague soft sets (F^,A) and (G^,B) over a universe X, which is denoted as F^,A⋒~G^,B=H^,C, is a vague soft set H^,C, where C=A∩B and, for every c∈C and x∈X,(4)tH^cx=min⁡⁡tF^cx,tG^cx,1-fH^cx=min⁡⁡1-fF^cx,1-fG^cx,where tH^c and fH^c are the upper and lower bounds of H^c, respectively. This relationship can be written as H^cx=F^cx∩G^cx.Definition 11 (see [7]).
LetF^,A be a vague soft set over U. Then for every α,β∈0,1, where α≤β, the α,β-cut or thevague soft cut of F^,A is a subset of U which is defined as follows:(5)F^,Aα,β=x∈U:tF^ax≥α,1-fF^ax≥β,i.e.F^ax≥α,βfor every a∈A.Definition 12 (see [10]).
LetF^,A be a vague soft set over U. Then for every α∈0,1, the α-cut of F^,A, denoted as (F^,A)(α,α), is a subset of U which is defined as follows:(6)F^,Aα,α=x∈U:tF^ax≥α,1-fF^ax≥α,i.e.F^ax≥α,αfor every a∈A.Definition 13 (see [10]).
Let(F^,A) be a vague soft set over X and let G be a nonnull subset of X. Then (F^,A)G is called avague soft characteristic set of G in [0,1] and the lower bound and the upper bound of (F^a)G are defined as follows:(7)tF^aGx=1-fF^aGxsifx∈G,wotherwise,where (F^a)G is a subset of (F^,A)G, x∈X, s,w∈[0,1], and s>w.


## 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]).
LetX be a group and let (F^,A) be a vague soft set over X. Then (F^,A) is called avague soft group over X if and only if, for every a∈A and x,y∈X, the following conditions are satisfied:(i)
tF^a(xy)≥min⁡⁡tF^ax,tF^ay and 1-fF^axy≥min⁡⁡1-fF^ax,1-fF^ay; that is, F^axy≥min⁡⁡(F^ax,F^ay),(ii)
tF^a(x-1)≥tF^ax and 1-fF^ax-1≥1-fF^ax; that is, F^ax-1≥F^ax.In other words, for every a∈A,F^a is avague subgroup in Rosenfeld’s sense.Proposition 15 (see [7]).
Let(F^,A) be a vague soft group over X and let e be the identity element of X. Then for every a∈A and x∈X,(i)
tF^ax-1=tF^ax and 1-fF^ax-1=1-fF^ax; that is, F^ax-1=F^ax,(ii)
tF^ae≥tF^ax and 1-fF^ae≥1-fF^ax; that is, F^ae≥F^ax.Proposition 16 (see [7]).
Let(F^,A) be a vague soft set. Then (F^,A) is vague soft group if and only if, for every a∈A and x,y∈X,(8)tF^axy-1≥min⁡⁡tF^ax,tF^ay,1-fF^axy-1≥min⁡⁡1-fF^ax,1-fF^ay,thatis,F^ax·y-1≥min⁡⁡F^ax,F^ay.


## 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(F^,A) be a vague soft set over X. Then the following statements are equivalent for each a∈A and for every x,y∈X:(i)
tF^a(y·x·y-1)≥tF^a(x) and 1-fF^ay·x·y-1≥1-fF^ax; that is, F^ay·x·y-1≥F^a(x) or(ii)
tF^ay·x·y-1=tF^a(x) and 1-fF^ay·x·y-1=1-fF^ax; that is, F^ay·x·y-1=F^a(x) or(iii)
tF^ax·y=tF^a(y·x) and 1-fF^ax·y=1-fF^ay·x; that is, F^ax·y=F^a(y·x).If condition (iii) is satisfied, then (F^,A) is said to be an Abelian vague soft set over X.Definition 18.
LetX be a group and let (F^,A) be a nonnull vague soft group over X. Then (F^,A) is called anormalistic vague soft group over X if, for every a∈Supp(F^,A) and for every x,y∈X, either one of the following conditions is satisfied:(i)
tF^a(y·x·y-1)≥tF^a(x) and 1-fF^ay·x·y-1≥1-fF^ax; that is, F^ay·x·y-1≥F^a(x) or(ii)
tF^ay·x·y-1=tF^a(x) and 1-fF^ay·x·y-1=1-fF^ax; that is, F^ay·x·y-1=F^a(x) or(iii)
tF^ax·y=tF^a(y·x) and 1-fF^ax·y=1-fF^ay·x; that is, F^ax·y=F^a(y·x).In other words, for every a∈Supp(F^,A) and x∈X, F^a is anormal vague subgroup of X in Rosenfeld’s sense. If F^a satisfies condition (iii), then (F^,A) is called anAbelian vague soft group.From Definition18, it is obvious that all normalistic vague soft groups are vague soft groups but the converse is not necessarily true.Proposition 19.
Let(F^,A) be a nonnull vague soft set over X and B⊂A. If (F^,A) is a normalistic vague soft group over X, then (F^,B) is a normalistic vague soft group over X if it is nonnull.Proof.
Let(F^,A) be a normalistic vague soft group over X. Then F^a is a normal vague subgroup of X for every a∈SuppF^,A. Since B⊂A, (F^,B) is a vague soft subgroup of F^,A. Therefore F^b is a normal vague subgroup of F^,A for every b∈SuppF^,B. This implies that F^b is a normal vague subgroup of X for every b∈SuppF^,B, because F^,B is a vague soft subgroup of F^,A. Then F^,B is a normalistic vague soft group over X.Proposition 20.
Let(F^,A) be a normalistic vague soft group over X and let (F^,A)+ be a nonnull vague soft set over X which is as defined below:(9)F^,A+=F^a+=x∈X:tF^a+x=tF^ax+1-tF^ae,1-fF^a+x=1-fF^ax+fF^ae,for every a∈Supp⁡F^,A and x∈X while e is the identity element of group X. Then (F^,A)+ is a normalistic vague soft group over X.Proof.
F
^
a is a normal vague subgroup of X for each a∈SuppF^,A. Now let x,y∈F^,A+ and a∈SuppF^,A+. Then tF^a+x=tF^ax+1-tF^ae and 1-fF^a+x=1-fF^ax+fF^ae and also tF^a+y=tF^ay+1-tF^ae and 1-fF^a+y=1-fF^ay+fF^ae. This means that F^a+x=F^ax+1-F^a(e) and F^a+y=F^ay+1-F^ae. Thus the following is obtained:(10)tF^a+xy=tF^axy+1-tF^ae≥min⁡⁡tF^ax,tF^ay+1-tF^ae=min⁡⁡tF^ax+1-tF^ae,tF^ay+1-tF^ae=min⁡⁡tF^a+x,tF^a+yand tF^a+x-1=tF^ax-1+1-tF^ae≥tF^ax+1-tF^ae=tF^a+x.
Similarly, it can be proven that1-fF^a+xy≥min⁡⁡1-fF^a+x,1-fF^a+y and 1-fF^a+x-1≥1-fF^a+x. Therefore (F^a)+ is vague subgroup of X. Next we prove normality: (11)tF^a+y·x·y-1=tF^ay·x·y-1+1-tF^ae≥tF^ax+1-tF^ae=tF^a+x.Similarly, we obtain 1-fF^a+y·x·y-1≥1-fF^a+x. Hence it has been proven that (F^a)+ is a normal vague subgroup of X. As such, (F^,A)+ is a normalistic vague soft group over X.Theorem 21.
LetG be a nonnull subset of X, let (F^,A) be a normalistic vague soft group over X, let (F^,A)G be a vague soft characteristic set over X, and let (F^,A)+ be a vague soft set over X as defined in Proposition 20. If (F^,A)+ is a normalistic vague soft set over X, then G is a normal subgroup of X.Proof.
Suppose thatG is a nonnull subset of X and (F^,A)+ is a normalistic vague soft group over X. Then F^a+ is a normal vague subgroup of X for every a∈Supp(F^,A)+. Now let x,y∈G and a∈SuppF^,A+. Thus we obtain tF^aG+x=tF^aG+y=s and 1-fF^aG+x=1-fF^aG+y=s. Since (F^a)+ is a normal vague subgroup of X, we obtain the following:(12)tF^aG+xy-1=tF^aGxy-1+1-tF^aGe≥min⁡⁡tF^aGx,tF^aGy-1+1-tF^aGe≥min⁡⁡tF^aGx,tF^aGy+1-tF^aGe=min⁡⁡tF^aGx+1-tF^aGe,tF^aGy+1-tF^aGe=min⁡⁡tF^aG+x,tF^aG+y=min⁡⁡s,s=s.Similarly, it can be proven that 1-fF^aG+xy-1≥s too. This means that (F^a)G+xy-1≥s and therefore xy-1∈G. Hence G is a subgroup of X. To prove normality, let x∈G and y∈X. Then we obtain(13)tF^aG+y·x·y-1=tF^aGy·x·y-1+1-tF^aGe=tF^aGx+1-tF^aGe=tF^aG+x=s.In the same manner, it can be proven that 1-fF^aG+y·x·y-1=s. Thus F^aG+y·x·y-1=s and this implies that y·x·y-1∈G. As such, it is proven that G is a normal subgroup of X.Definition 22.
LetF^,A be a normalistic vague soft group over X. Then(i)
(F^,A) is called atrivial normalistic vague soft group over X if F^(a)={eX} for all a∈Supp(F^,A) and the truth and false membership function of F^,A are defined as follows:(14)tF^ax=1ifa∈Supp⁡F^,A,0otherwise,fF^ax=0ifa∈Supp⁡F^,A,1otherwisefor everya∈A and x∈X.(ii)
(F^,A) is called anabsolute normalistic vague soft group over X if F^(a)=X for all a∈Supp(F^,A) and the truth and false membership function of (F^,A) are defined as follows:(15)tF^ax=1ifa∈Supp⁡F^,A,0otherwise,fF^ax=0ifa∈Supp⁡F^,A,1otherwise.Example 23.
LetX=1,-1,i,-i,× be a group with the operations of group X as given in the Cayley table in Table 1.
(i) Then letA={1,-1} and let (F^,A) be a nonnull vague soft group over X, where F^:A→V(X) is a set-valued function that is as defined below: (16)F^a=x∈X:x=a2for all a∈A. Then F^1=F^-1=1=eX and F^i=F^-i=-1.
The truth membership function and false membership function of(F^,A) are as given below:(17)tF^ax=1ifx=a2=1,0otherwise,fF^ax=0ifx=a2=1,1otherwisefor all a∈A and x∈X. Since F^a=eX=1 for all a∈Supp(F^,A),F^a is a trivial normal vague subgroup of X. As such, (F^,A) is a trivial normalistic vague soft group over X.
(ii) Now letB={i,-i} and let (G^,B) be a nonnull vague soft group over X, where G^:B→V(X) is a set-valued function that is as defined below: (18)G^b=x∈X:x=anfor  anyn∈Nfor all b∈B. Thus G^i=G^-i=1,-1,i,-i=X, G^1=1, and (19)G^-1=1ifnis  an  even  integer,-1ifnis  an  odd  integer.The truth membership function and false membership function of (G^,B) are as given below:(20)tG^bx=1ifx=an=X,0otherwise,fG^bx=0ifx=an=X,1otherwisefor every b∈B and x∈X. Hence G^(b)=X for all b∈Supp(G^,B) and therefore G^(b) is an absolute normal vague subgroup of X. As such, (G^,B) is an absolute normalistic vague soft group over X.Table 1
Cayley table.
·
1
-
1
i
-
i
1
1
-
1
i
-
i
-
1
-
1
1
-
i
i
i
i
-
i
-
1
1
-
i
-
i
i
1
-
1Theorem 24.
Let(F^,A) and (G^,B) be normalistic vague soft groups over X. Then consider the following.(i)
If(F^,A) and (G^,B) are trivial normalistic vague soft groups over X, then (F^,A)∩~(G^,B) is a trivial normalistic vague soft group over X.(ii)
If(F^,A) and (G^,B) are absolute normalistic vague soft groups over X, then (F^,A)∩~(G^,B) is an absolute normalistic vague soft group over X.(iii)
If(F^,A) is a trivial normalistic vague soft group over X and (G^,B) is an absolute normalistic vague soft group over X, then (F^,A)⋒~(G^,B) is a trivial normalistic vague soft group over X.Proof.
The proofs are straightforward and are therefore omitted.Theorem 25.
Letφ:X→Y be a group epimorphism and let (F^,A) be a normalistic vague soft group over X. Then (φF^,Supp⁡(F^,A)) is a normalistic vague soft group over Y.Proof.
Since(F^,A) is a normalistic vague soft group over X, it must be nonnull and therefore it follows that (φF^,Supp(F^,A)) must be nonnull too. Now let a∈A and assume that, for every y1,y2∈Y, there exists x1,x2∈X such that φx1=y1 and φx2=y2. Thus we obtain the following:(21)φF^aφx1·φx2=φF^ay1·y2=φF^ay1·φF^ay2=φF^ay2·φF^ay1=φF^ay2·y1=φF^aφx2·φx1.As such, (φF^,Supp(F^,A)) is a normalistic vague soft group over Y.Theorem 26.
Letφ:X→Y be a group isomorphism and let (G^,B) be a normalistic vague soft group over Y. Then (φ-1G^,Supp⁡(G^,B)) is a normalistic vague soft group over X.Proof.
The proof is similar to that of Theorem25 and is therefore omitted.Theorem 27.
Let(F^,A) and (G^,B) be normalistic vague soft groups over X and Y, respectively, and let φ:X→Y be a group epimorphism. Then consider the following.(i)
IfF^(a)=Ker(φ) for all a∈Supp⁡(F^,A), then (φ(F^),Supp⁡(F^,A)) is a trivial normalistic vague soft group over Y.(ii)
If(F^,A) is an absolute normalistic vague soft group over X, then φF^,Supp⁡F^,A is an absolute normalistic vague soft group over Y.(iii)
Ifφ is injective and G^b=φX for every b∈Supp⁡G^,B, then φ-1G^,Supp⁡G^,B is an absolute normalistic vague soft group over X.(iv)
Ifφ is injective and G^,B is a trivial normalistic vague soft group over Y, then φ-1G^,Supp⁡G^,B is a trivial normalistic vague soft group over X.Proof.
(i) Letφ be a group epimorphism from X to Y. Then the kernel of φ is as given below: (22)Ker⁡⁡φ=x∈X:φx=eY,where eY is the identity element of Y. Let F^(a)=Ker⁡⁡(φ) for every a∈SuppF^,A. Then for every a∈SuppF^,A and x∈X, it follows that φF^a=φF^a=φx=eY. Then the truth and false membership function of φF^,SuppF^,A are as given below:(23)tφF^αx=1ifa∈Supp⁡F^,A,0otherwise,fφF^αx=0ifa∈Supp⁡F^,A,1otherwisefor every a∈A and x∈X. As such, (φ(F^),Supp(F^,A)) is a trivial normalistic vague soft group over Y.
(ii) The proof is similar to the proof of part (i) and is therefore omitted.
(iii) Letφ:X→Y be an injective group epimorphism and, for every b∈SuppG^,B, G^b=φX. Thus G^b=φX=Y for every b∈SuppG^,B because φ is injective. Therefore by Definition 22(ii), (G^,B) is an absolute normalistic vague soft group over Y. As such, we obtain(24)φ-1G^b=φ-1G^b=φ-1φX=φ-1Y=Xfor every b∈Supp(G^,B) because φ is injective. Thus for every b∈SuppG^,B, φ-1G^(b)=X. This means that (φ-1G^,Supp(G^,B)) is an absolute normalistic vague soft group over X with truth and false membership function as given below:(25)tφ-1G^by=1ifb∈Supp⁡G^,B,0otherwise,fφ-1G^by=0ifb∈Supp⁡G^,B,1otherwisefor every y∈Y. Hence (φ-1(G^),Supp(G^,B)) is an absolute normalistic vague soft group over X.
(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]).
Letφ:X→Y and ψ:A→B be two functions, where A and B are the set of parameters for the classical sets X and Y, respectively. Let (F^,A) and (G^,B) be vague soft groups over X and Y, respectively. Then the ordered pair (φ,ψ) is called avague soft function from (F^,A) to G^,B, denoted as φ,ψ:F^,A→G^,B.Definition 29 (see [10]).
Let(F^,A) and (G^,B) be two vague soft groups over X and Y, respectively. Let φ,ψ:F^,A→(G^,B) be a vague soft function.(i)
Theimage of (F^,A) under the vague soft function φ,ψ, denoted as φ,ψF^,A, is a vague soft set over Y, which is defined as (26)φ,ψF^,A=φF^,ψA,whereφ(F^a)φx=(φ(F^))ψ(a)y for every a∈A, x∈X, and y∈Y.(ii)
Thepreimage of (G^,B) under the vague soft function φ,ψ, denoted as (φ,ψ)-1(G^,B), is a vague soft set over X, which is defined as (27)φ,ψ-1G^,B=φ-1G^,ψ-1B,whereφ-1G^bφ-1y=(φ-1(G^))ψ-1b(x) for every b∈B, x∈X, and y∈Y.If φ and ψ are injective (surjective), then the vague soft function (φ,ψ) 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(F^,A) and (G^,B) be normalistic vague soft groups over X and Y, respectively. Then the vague soft function φ,ψ:F^,A→(G^,B) is called as follows.
(a) It is callednormalistic vague soft group homomorphism if the following conditions are satisfied:(i)
φ is a homomorphism from X to Y,(ii)
ψ is a function from A into B,(iii)
φ(F^x)=G^(ψx) for every x∈A.Then (F^,A) is said to benormalistic vague soft homomorphic to (G^,B) and this is denoted as F^,A~NG^,B.
(b) It is callednormalistic vague soft group isomorphism if the following conditions are satisfied:(i)
φ is an isomorphism from X to Y,(ii)
ψ is a bijective mapping from A to B,(iii)
φ(F^x)=G^(ψx) for every x∈A.Then (F^,A) is said to benormalistic vague soft isomorphic to G^,B and this is denoted as F^,A≅NG^,B.Theorem 31.
Let(F^,A) and (G^,B) be normalistic vague soft groups over X and Y, respectively, and let φ,ψ:F^,A→(G^,B) be a normalistic vague soft group homomorphism. Then consider the following:(i)
(φ,ψ)(F^,A) is a normalistic vague soft group over Y.(ii)
(φ,ψ)-1(G^,B) is a normalistic vague soft group over X.Proof.
The proofs follow from Definitions18 and 30 and are therefore omitted.Theorem 32.
LetX, Y, and Z be groups, let F^,A, (G^,B), and J^,C be normalistic vague soft groups over X, Y, and Z, respectively, and let φ,ψ:F^,A→(G^,B) and φ∗,ψ∗:G^,B→(J^,C) be normalistic vague soft group homomorphisms. Then (φ∗∘φ,ψ∗∘ψ) is a normalistic vague soft group homomorphism from F^,A to J^,C.Proof.
Suppose that(φ,ψ) represents a normalistic vague soft homomorphism from F^,A to (G^,B). Then (φ,ψ) is a vague soft function such that φ:X→Y is a group homomorphism from X into Y and ψ:A→B is a function from A to B. Also (φ,ψ) satisfies the following condition:(28)φF^x=G^ψx∀x∈Supp⁡F^,A.Now suppose that (φ∗,ψ∗) represents a normalistic vague soft homomorphism from (G^,B) to (J^,C). Then (φ∗,ψ∗) is a vague soft function such that φ∗:Y→Z represents a group homomorphism from Y to Z, ψ∗:B→C is a function from A into B, and the following condition is satisfied: (29)φ∗G^x=J^ψ∗x∀x∈Supp⁡G^,B.Therefore we have φ∗∘φ:X→Z which is a group homomorphism from X into Z and the composition ψ∗∘ψ:A→C represents a function from A into C. Therefore it follows that (φ∗∘φ,ψ∗∘ψ) is a vague soft function from F^,A to (J^,C). Furthermore for all x∈Supp(F^,A) we have (30)φ∗∘φF^x=φ∗φF^x=φ∗G^ψx=J^ψ∗ψx=J^ψ∗∘ψx.Hence (φ∗∘φ,ψ∗∘ψ) is a normalistic vague soft group homomorphism from F^,A to (J^,C).Theorem 33.
LetX and Y be groups and let (F^,A) and (G^,B) be vague soft sets over X and Y, respectively. If (F^,A) is a normalistic vague soft group over X and F^,A≅NG^,B, then G^,B is a normalistic vague soft group over Y.Proof.
SinceF^,A≅NG^,B, there exists a vague soft group isomorphism from F^,A to G^,B. This means that there exists an isomorphism φ from X to Y and a bijective mapping ψ from A to B which satisfies φF^x=G^ψx for every x∈A. Now suppose that F^,A is a normalistic vague soft group over X. Then F^x is a normal vague subgroup of X for all x∈SuppF^,A. Thus φF^x is a normal vague subgroup of Y for all x∈SuppF^,A. Since ψ is a bijective mapping, for all y∈SuppG^,B⊆B, there exists an x∈A such that y=ψx. Hence we have φF^x=G^(y)=G^ψx for all x∈A. As such, G^(y) is a normal vague subgroup of Y for all y∈SuppG^,B.Corollary 34.
Let(F^,A) and (G^,B) be normalistic vague soft groups over X and Y, respectively, and F^,A≅NG^,B. If F^(x) is a normal vague subgroup of X, then G^ψx is a normal vague subgroup of Y and F^x≅NG^ψx.


## 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.



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