On Normalistic Vague Soft Groups and Normalistic Vague Soft Group Homomorphism

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.


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 Aktas ¸and C ¸aǧman 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, ) is a vague soft group that satisfies the 2 Advances in Fuzzy Systems commutative law given by  F (⋅) =  F (⋅) and 1 −  F (⋅ ) = 1 −  F ( ⋅ ).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.

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 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 5 (see [3]).Let  be a group and let (, ) be a nonnull soft set over .Definition 7 (see [6]).A pair ( F, ) is called a vague soft set over  where F is a mapping given by F :  → () 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 F() for all  ∈ , from this family, may be considered as the set of -approximate elements of the vague soft set ( F, ).Hence the vague soft set ( F, ) can be viewed as consisting of a collection of approximations of the following form: for all  ∈  and for all  ∈ .
The support of ( F, ) denoted by Supp( F, ) is defined as for 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 ( F, ) is said to be nonnull if Supp( F, ) ̸ = 0.
Then for every ,  ∈ [0, 1], where  ≤ , the (, )-cut or the vague soft cut of ( F, ) is a subset of  which is defined as follows: ( F, ) for every  ∈ .
Definition 13 (see [10]).Let ( F, ) be a vague soft set over  and let  be a nonnull subset of .Then ( F, )  is called a vague soft characteristic set of  in [0, 1] and the lower bound and the upper bound of ( F )  are defined as follows: where ( F )  is a subset of ( F, )  ,  ∈ , ,  ∈ [0, 1], and  > .

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.
In other words, for every  ∈ , F is a vague subgroup in Rosenfeld's sense.
Proposition 15 (see [7]).Let ( F, ) be a vague soft group over  and let  be the identity element of .Then for every  ∈  and  ∈ , Proposition 16 (see [7]).Let ( F, ) be a vague soft set.Then

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.
Then the following statements are equivalent for each  ∈  and for every ,  ∈ : Proof.Let ( F, ) be a normalistic vague soft group over .Then F is a normal vague subgroup of  for every  ∈ Supp( F, ).Since  ⊂ , ( F, ) is a vague soft subgroup of ( F, ).Therefore F is a normal vague subgroup of ( F, ) for every  ∈ Supp( F, ).This implies that F is a normal vague subgroup of  for every  ∈ Supp( F, ), because ( F, ) is a vague soft subgroup of ( F, ).Then ( F, ) is a normalistic vague soft group over .
Proposition 20.Let ( F, ) be a normalistic vague soft group over  and let ( F, ) + be a nonnull vague soft set over  which is as defined below: for every  ∈ Supp( F, ) and  ∈  while  is the identity element of group .Then ( F, ) + is a normalistic vague soft group over .
Proof.F is a normal vague subgroup of  for each  ∈ Supp( F, ).Now let ,  ∈ ( F, ) + and  ∈ Supp( F, ) Thus the following is obtained: and Similarly, it can be proven that 1 −  ( F ) . Therefore ( F ) + is vague subgroup of .Next we prove normality: Hence it has been proven that ( F ) + is a normal vague subgroup of .As such, ( F, ) + is a normalistic vague soft group over .
Theorem 21.Let  be a nonnull subset of , let ( F, ) be a normalistic vague soft group over , let ( F, )  be a vague soft characteristic set over , and let ( F, ) + be a vague soft set over  as defined in Proposition 20.If ( F, ) + is a normalistic vague soft set over , then  is a normal subgroup of .
Proof.Suppose that  is a nonnull subset of  and ( F, ) + is a normalistic vague soft group over .Then ( F ) + is a normal vague subgroup of  for every  ∈ Supp( F, ) + .Now let ,  ∈  and  ∈ Supp( F, ) + .Thus we obtain Since ( F ) + is a normal vague subgroup of , we obtain the following: Similarly, it can be proven that 1 −  ( F ) +  ( −1 ) ≥  too.This means that ( F ) +  ( −1 ) ≥  and therefore  −1 ∈ .Hence

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Proof.The proofs are straightforward and are therefore omitted.
Theorem 25.Let  :  →  be a group epimorphism and let ( F, ) be a normalistic vague soft group over .Then (( F), Supp( F, )) is a normalistic vague soft group over .
Proof.The proof is similar to that of Theorem 25 and is therefore omitted.
Theorem 27.Let ( F, ) and ( Ĝ, ) be normalistic vague soft groups over  and , respectively, and let  :  →  be a group epimorphism.Then consider the following.
(ii) If ( F, ) is an absolute normalistic vague soft group over , then (( F), Supp( F, )) is an absolute 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 F() = Ker() for every  ∈ Supp( F, ).Then for every  ∈ Supp( F, ) and  ∈ , it follows that ( F)() = ( F()) = () = {  }.Then the truth and false membership function of (( F), Supp( F, )) are as given below: for every  ∈  and  ∈ .As such, (( F), Supp( F, )) is a trivial normalistic vague soft group over .
(ii) The proof is similar to the proof of part (i) and is therefore omitted.
(iv) The proof is similar to the proof of part (iii) and is therefore omitted.

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  ∈ .As such, Ĝ() is a normal vague subgroup of  for all  ∈ Supp( Ĝ, ).

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.
[9]inition 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   :  → [0, 1] and a false membership function   :  → [0, 1].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 [0, 1] with each point in , where   () +   () ≤ 1.This approach bounds the grade of membership of  to a subinterval [  (), 1 −   ()] of [0, 1].Hence a vague set is a form of fuzzy set.
Then (, ) is called a normalistic soft group over  if () is a normal subgroup of  for all  ∈ Supp(, ).
and 1 −  F (⋅) = 1 −  F (⋅); that is, F ( ⋅ ) = F ( ⋅ ).If condition (iii) is satisfied, then ( F, ) is said to be an Abelian vague soft set over .Definition 18.Let  be a group and let ( F, ) be a nonnull vague soft group over .Then ( F, ) is called a normalistic vague soft group over  if, for every  ∈ Supp( F, ) and for every ,  ∈ , either one of the following conditions is satisfied:(i)