Soft Topological Transformation Groups

Abstract

The aim of the present study is to introduce the concept of soft topological transformation groups by examining the topological transformation groups, which are the core subject of algebraic topology under the soft approach. Actions of soft topological groups on soft topological spaces are studied, and the category of soft topological transformation groups is constructed. Also, a translation and conjugation of the soft topological groups are described. Finally, the definitions of soft orbit spaces and soft homogeneous spaces are given, and some of the properties of these concepts are examined in detail.

1. Introduction

Topology and algebra are two main areas of mathematics that play complementary roles. Topology deals with continuity and convergence, while algebra examines all kinds of operations. Although these two areas tend to develop independently, they are in a natural interaction with other areas of mathematics such as functional analysis, representation theory, dynamical systems. In particular, most of the important objects of mathematics contain a blend of topological and algebraic structures. Topological transformation groups are objects of this kind. Historically, the study of topological transformation groups were originated in the work of Montgomery and Zippin, which contains a solution of Hilbert’s fifth problem [1]. This subject has reached a wide range of study potential and continues to be one of the most dominant topics in mathematics nowadays [2].

The other central theme in the work is the soft set theory. This theory proposed by Molodtsov is a recent mathematical approach for managing vagueness and uncertainty [3]. Researches based on the soft set theory continues to develop rapidly in many fields such as engineering, medicine, economics, environmental sciences. In recent years, this theory has been combined with some important mathematical theories such as algebra and topology. Many different algebraic properties of soft sets were presented in the literature by Maji, Aktaş, Jun and Feng [4,5,6,7]. Maji et al. defined new operations on soft sets [4]. In [5], Aktaş and Çağman improved the definitions of soft group and soft subgroup, and proved their new properties. Recently, Oguz et al. studied the actions of soft groups and obtained some important properties [8]. In addition, other algebraic properties of soft sets were studied in [9,10,11,12]. In the meantime, the pioneers of topological studies on soft sets are Shabir and Naz [13]. They investigated the notion of soft topological spaces. Later, Nazmul and Samanta presented the definition of a soft topological group [14]. Other than those, many topological studies have emerged [15,16,17,18,19,20,21,22,23,24].

In this paper, by using the concept of soft continuous we are going to define the action of soft topological groups and investigate some of their properties. Consequently, we are going to introduce the notion of soft topological transformation groups by studying the combination between the topological transformation groups and soft set theory. Finally, we are going to present some investigations about soft orbit spaces and soft homogeneous spaces. 2. Preliminaries

In this section, we review some definitions and results about the soft set theory used throughout the paper. For more details, we refer the reader to [3,4].

Throughout this paper, we assume that X is an initial universe set and E is a nonempty set of parameters. LetP(X)denote the power set of X andA⊂E.

Definition 1

([3]). A pair(F,A)is called a soft set over X, where F is a mapping defined by

F:A⟶P(X)

In general, one can see that a soft set over X is a parametrized family of subsets of the universe X. We will sometimes use the representation(X,F,A)instead of soft set(F,A)over X.

Definition 2

([4]). A soft set(X,F,A)is called a null soft set indicated by Φ, ifF(ε)=∅for allε∈A.

Definition 3

([4]). A soft set(X,F,A)is called an absolute soft set indicated byA˜, ifF(ε)=Xfor allε∈A.

Definition 4

([4]). Let(F,A)and(G,B)be two soft sets over the common universe X. Then(F,A)is said to be a soft subset of(G,B), denoted by(F,A)⊂˜(G,B)if the following conditions are satisfied:

i.

A⊂B.

ii.

∀ε∈A,f(ε)andG(ε)are identical approximations.

Definition 5

([4]). Let(F,A)and(G,B)be two soft sets over the common universe X. Their intersection is a soft set(H,C)such thatC=A∩BandH(ε)=F(ε)∩G(ε),∀ε∈C.

It is denoted by(F,A)∩˜(G,B)=(H,C).

Definition 6

([4]). Let(F,A)and(G,B)be two soft sets over the common universe X. Their union is a soft set(H,C), whereC=A∪B, and∀ε∈C,

H(e)=F(a),ifε∈A−BG(a),ifε∈B−AF(a)∪G(a),ifε∈A∩B

This situation is denoted by(F,A)∪˜(G,B)=(H,C).

Below, we give soft group and related concepts.

Definition 7

([5]). Let G be a group and A be a nonempty set. For the soft set(F,A)over G, it is said that(F,A)is a soft group over G if and only ifF(ε)<Gfor allε∈A.

As it can be seen from the definition, the soft group(F,A)is actually a parameterized family of subgroups of the group G. From now on, the triplet(G,F,A)stands for the soft group(F,A)over G.

Example 1

([5]). Assume thatG=A=_S3={e,(12),(13),(23),(123),(132)}, andF(e)={e},F(12)={e,(12)},F(13)={e,(13)},F(23)={e,(23)},F(123)=F(132)={e,(123),(132)}. Then(F,A)is a soft group over G, sinceF(ε)is a subgroup of G for allε∈A.

Definition 8

([5]). Let(F,A)be a soft set over G. Then, for allε∈A

i.

(F,A)is said to be an identity soft group over G ifF(ε)={e}, where e is the identity element of G.

ii.

(F,A)is said to be an absolute soft group over G ifF(ε)=G.

Here we state the action of a soft group, introduced by Oguz et al.

Definition 9

([8]). LetG=(G,F,A)be a soft group and letX=(X,^F′,A)be a soft set. Then, a left soft (group) action of G on X is defined by the binary operation

_Πε:F(ε)×^F′(ε)⟶^F′(ε)

satisfying the following properties for allε∈A:

i.

_Πε(e,x)=xfor allx∈^F′(ε).

ii.

_Πε(g,_Πε(h,x))=_Πε(gh,x)for allx∈^F′(ε)andg,h∈F(ε).

The following example of soft (group) action seems to be illustrative.

Example 2

([8]). SupposeA=G=_S3is the group of permutations on theS={1,2,3}. Consider the soft groupG=(G,F,A)as follows

F(e)={e},

F(12)={e,(12)},

F(13)={e,(13)},

F(23)={e,(23)},

F(123)=F(132)={e,(123),(132)}.

Also, the soft setX=(X,F,A)withX={1,2,3}is given by

^F′(e)={1},

^F′(12)={1,2},

^F′(13)={1,3},

^F′(23)={2,3},

F(123)=F(132)={1,2,3}.

For alla∈A, the mapping

_Πε:F(ε)×^F′(ε)⟶^F′(ε)(g,x)↦_Πa(g,x)=σ(x)

is a left (group) action such thate∈Gis the identity element and_Πε(e,x)=xas desired. Moreover, it is obvious that_Πε(g,_Πε(h,x))=_Πε(gh,x)forg,h∈F(ε).

Similar to Molodtsov’s perspective, the definition of a soft topology proposed by Kandemır is as follows:

Definition 10

([16]). Let the initial universe set X be a topological space with a topology τ and let(F,A)be a soft set defined over X. Then,(F,A)is called a soft topology over X ifF(ε)is a subspace of X with respect to the topology_τF(ε)induced by τ for allε∈A. Also,(F,A,τ)is called a soft topological space over X.

Now, we shall recall the definition of a soft topological group.

Definition 11

([18]). Let G be a group that is furnished with a topology τ and let(F,A)be a non-null soft set described over G. Then, the structure(F,A,τ)is defined as a soft topological group over G if the following statements are satisfied for allε∈A:

i.

F(ε)is a subgroup of G.

ii.

The mapping(x,y)⟼x−yof the topological spaceF(ε)×F(ε)ontoF(ε)is continuous.

Example 3

([18]). ConsiderG=_S3={e,(12),(13),(23),(123),(132)}as a group,A={_e1,_e2,_e3}andF(_e1)={e},F(_e2)={e,(12)},F(_e3)={e,(123),(132)}. LetB={{e},{(12)},{(123)},{(132)}}be a base for the topology τ. It is easy to check thatF(a)is a subgroup of G for allε∈A. In addition, the second condition of above definition is provided. This means that(F,A,τ)is a soft topological group.

Definition 12

([18]). Let(F,A,τ)and(^F′,B,^τ′)be two soft topological groups over G and^G′, respectively. Letf:G⟶^G′andg:A⟶Bbe two mappings. Then, the pair(f,g)is called a soft topological group homomorphism if

i. f is a group epimorphism and g is a surjection.

ii.

f(F(ε))=^F′(g(ε))for allε∈A.

iii.

_fε:(F(ε),_τF(ε))⟶(^F′(g(ε)),_{τ^F′(g(ε))′})is continuous for allε∈A.

It is worth noting that(F,A,τ)is called soft topologically homomorphic to(^F′,B,^τ′)and written as(F,A,τ)∼(^F′,B,^τ′).

3. Actions of Soft Topological Groups In this section, after introducing the actions of soft topological groups we define the concept of a soft topological transformation group. We first begin by giving the following definitions.

Definition 13.

Let(F,A,τ)be a soft topological group over G and(^F′,A,^τ′)a soft topological space over X. Then, a left soft (continuous) action of(F,A,τ)on(^F′,A,^τ′)is a continuous map

_θε:F(ε)×^F′(ε)⟶^F′(ε)

such that it has the following properties for allε∈A:

i.

_θε(e,x)=xfor allx∈^F′(ε).

ii.

_θε(g,_θε(h,x))=_θε(gh,x)for allx∈^F′(ε)andg,h∈F(ε).

Also, the soft topological space(^F′,A,^τ′)is called a softG−space. It is clear that each_Πεis a left (continuous) action mapping for allε∈A. In a similar way, a right soft (continuous) action is defined.

Definition 14.

Let the soft topological space(^F′,A,^τ′)be a softG−space. Then,(F,A,τ)is called a soft topological transformation group on(^F′,A,^τ′).

As an illustration, let us consider the following examples.

Example 4.

Let G be a soft topological group andX=G. Then, define the mapping

_θε:F(ε)×F(ε)⟶F(ε)(g,h)↦_θε(g,h)=h.

It is easy to verify that_θεis continuous action for allε∈A. Thus, every soft topological group G acts on itself by translation as above.

Example 5.

Consider a soft topological group G and letX=G. In this case, the soft (continuous) action of G on itself by multiplication is as follows. For allε∈A,

_θε:F(ε)×F(ε)⟶F(ε)(g,h)↦_θε(g,h)=gh.

We have already seen that each mapping_Πεis continuous action for allε∈A.

Example 6.

Every soft topological group G acts on itself by conjugation as follows. LetX=G. For allε∈A, define the mapping

_θε:F(ε)×F(ε)⟶F(ε)(g,h)↦_θε(g,h)=gh^g−1.

It is easy to notice that each_Πεis a continuous action.

Apart from these examples, a different construction is as follows.

Example 7.

Let(^F′,A,^τ′)be a soft topological space over X and(F,A,τ)be a soft topological group acting on the soft topological space(^F′,A,^τ′)as given below

_θε:F(ε)×^F′(ε)⟶^F′(ε)(g,x)↦_θε(g,x).

Suppose thatf:(X,^τ′)⟶(Y,^τ″)is an injective continuous function. Now defining^F″=f(^F′), we establish

_θε′:F(ε)×f(^F′(ε))⟶f(^F′(ε))(g,f(x))↦_θε′(g,f(x))=f(_θε(g,x))

for allε∈A. So using Definition 13, one can easily see that_Πε′is a soft (continuous) action of(F,A,τ)on(^F″,A,^τ″).

Proposition 1.

Let(^F′,A,^τ′)be a soft topological space over X and(^F″,A,^τ″)be a soft topological space over Y. If(^F′,A,^τ′)and(^F″,A,^τ″)are two softG−space, then(^F′×^F″,A,^τ′×^τ″)is a softG−space.

Proof.

Let(F,A,τ)be a soft topological group over G. For allε∈A, consider a softG−space(^F′,A,^τ′)with the soft (continuous) action

_θε:F(ε)×^F′(ε)⟶^F′(ε)(g,x)↦_θε(g,x)

and a softG−space(^F″,A,^τ″)with the soft (continuous) action

_ϕε:F(ε)×^F″(ε)⟶^F″(ε)(g,y)↦_ϕε(g,y).

Then, we can establish a soft (continuous) action denoted as

_φε:F(ε)×(^F′(ε)×^F″(ε))⟶(^F′(ε)×^F″(ε))(g,(x,y))↦_φε(g,(x,y))=(_θε(g,x),_ϕε(g,y)).

It is easy to verify that_φεis a soft (continuous) action of(F,A,τ)on(^F′×^F″,A,^τ′×^τ″). This gives us a softG−space of(^F′×^F″,A,^τ′×^τ″).  □

Let us present some types of soft (continuous) action here.

Definition 15.

A soft (continuous) action of the soft topological group(F,A,τ)on the soft topological space(^F′,A,^τ′)is said to be

i.

transitive if for each pairx,yin^F′(ε)there exists an element g inF(ε)such that_θε(g,x)=x.

ii.

effective (or faithful) if for every two distinct elementsg,hinF(ε)there is an element x in^F′(ε)such that_θε(g,x)≠_θε(h,x).

iii.

free if giveng,hinF(ε), the existence of an element x in^F′(ε)with_θε(g,x)=_θε(h,x)impliesg=h.

By this definition, we get the following desired result.

Proposition 2.Every free soft (continuous) action is faithful.

Definition 16.

A soft (continuous) action of the soft topological group(F,A,τ)on the soft topological space(^F′,A,^τ′)is said to be regular if it is both transitive and free or equivalently, if for every elementx,yin^F′(ε)there exists only one element g inF(ε)such that_θε(g,x)=y.

Example 8.

The soft (continuous) action of any soft topological group(F,A,τ)on itself by multiplication is both regular and faithful.

Let us continue with soft homogeneous spaces.

Definition 17.A soft G-space is called soft homogeneous if its soft (continuous) action is transitive.

Example 9.

Choose(^F′,B,τ)over^G′as a soft topological subgroup group of(F,A,τ). Then(F,A,τ)is a soft homogeneous^G′−spacewith the soft (continuous) action given by

_θϵ:^F′(ϵ)×F(ϵ)⟶F(ϵ)(x,g)↦_θϵ(x,g)=xg

for allϵ∈B.

On the frame of this example, let(^F′,B,τ)be a soft topological space over the quotient spaceG/^G′. We can define a soft (continuous) action

_ψϵ:F(ϵ)×F(ϵ)/^F′(ϵ)⟶F(ϵ)/^F′(ϵ)(g,_g1x)↦_ψϵ(g,_g1x)=(g_g1)x

for allϵ∈B. Thus, we say that(^F′,B,τ)is a soft homogeneousG−space.

Below, we present the category of softG−spaces as a new category.

Definition 18.

Let(^F′,A,^τ′)and(^F″,A,^τ″)be two soft topological spaces over X and Y, respectively. Also, let(^F′,A,^τ′)and(^F″,A,^τ″)are softG−spaces with the soft actions_θεand_θε′. Then a map_fε:^F′(ε)⟶^F″(ε), defined by_fε(_θε(g,x))=_θε′(g,_fε(x))for everyε∈A, is said to be a softG−equivariant.

Notice that softG−spaces together with softG−equivariants between them form a category.

We shall now obtain some important characterizations giving concepts such as the fixed point set, centralizer and normalizer related to the soft (continuous) action.

Definition 19.

Let(^F′,A,^τ′)be aG−soft space. Then, for any soft subset(H,^F″,B)⊆˜(X,^F′,A), we define the fixed point set as follows

Fi_x^F″ (ε)={g∈F(ε):_θε(g,x)=x,x∈^F″(ε)}.

In particular, for allε∈Aandx∈^F′(ε), the fixed point set x is defined by

Sta_b^F′ (ε)={g∈F(ε):_θε(g,x)=x}.

At this point, we can give a result.

Proposition 3.

The setsFi_x^F″ (ε)andSta_b^F′ (ε)defined above are soft topological groups over G.

Proof.

We have the map

Fi_x^F″ :A⟶P(G)ε↦Fi_x^F″ (ε).

It is clear thatFi_x^F″ (ε)is a topological subgroup with respect to the topologies induced by G for allε∈A. Then, the pair(Fi_x^F″ ,A)is a soft topological group over G. Analogously, we consider the map

Sta_b^F′ :A⟶P(G)ε↦Sta_b^F′ (ε)

which implies thatSta_b^F′ (ε))is a topological subgroup with respect to the topologies induced by G for allε∈A. Therefore, the pair(Sta_b^F′ ,A)is a soft topological group over G.  □

Remark 1.

It is well-known that the fixed point sets are subgroups of the topological group G in the classical theory. Also, the soft analog of this statement is true. In other words,Sta_bG(x)andFi_xGYare soft topological subgroups of the soft topological group G.

Proposition 4.

ForFi_x^F″ (ε)andSta_b^F′ (ε))as given above, the following equality holds:

Fi_x^F″ (ε)=_{∩x∈^F″(ε)}Sta_b^F′ (ε).

Proof.

For proof, considerg∈Fi_x^F″ (ε). Then_θε(g,x)=xfor allx∈^F″(ε), which implies thatg∈Sta_b^F′ (ε). That is,g∈_{∩x∈^F″(ε)}Sta_b^F′ (ε)and soFi_x^F″ (ε)⊆_{∩x∈^F″(ε)}Sta_b^F′ (ε)is obtained.

Conversely, takeg∈_{∩x∈^F″(ε)}Sta_b^F′ (ε). Theng∈Sta_b^F′ (ε)for allx∈^F″(ε)and so_θε(g,x)=x. Thus, it is readily verified thatg∈Fi_x^F″ (ε). By this newly obtained fact, the proof is completed.  □

Definition 20.

Let(^F′,A,^τ′)be aG−soft space. Then, for allx∈^F′(ε)the soft orbit of^F′(ε)underF(ε)is defined as

_Orb^F′ (ε)={_θε(g,x):g∈F(ε)}.

It is not difficult to show that the pair(_Orb^F′ ,A)is a soft set over X with the mapping

_Orb^F′ :A⟶P(X).

Also, it is clear that^F′(ε)and^F′(ϵ)belong to the same soft orbit if_Orb^F′ (ε)=_Orb^F′ (ϵ).

Proposition 5.

For theG−soft space(^F′,A,^τ′), ifx∈^F′(ε),g∈F(ε), andy=_θε(g,x), thenx=_θε(^g−1,y). Moreover, ifx≠^x′then_θa(g,x)≠_θε(g,^x′).

Proof.

Lety=_θε(g,x)such that

_θε(^g−1,y)=_θε(^g−1,_θε(g,x))=_θε(^g−1g,x)=_θε(e,x)=x.

Assumex≠^x′, so_θε(g,x)=_θε(g,^x′). When^g−1is applied to both sides, it follows that

_θε(^g−1,_θε(g,x))=_θε(^g−1,_θε(g,^x′))

_θε(^g−1g,x)=_θε(^g−1g,^x′)

_θε(e,x)=_θε(e,^x′)

x=^x′

This contradicts by the hypothesisx≠^x′. Hence,_θε(g,x)≠_θε(g,^x′).  □

Definition 21.

Let the soft topological group(F,A,τ)acts on itself by conjugation. For allε∈Aandh∈F(ε), the soft centralizer ofF(ε)is a set defined by

_CF(ε)={g∈F(ε):_θε(g,h)=_θε(h,g)}.

Then, we have the followings:

Proposition 6.

The soft centralizer_CF(ε)given above is a soft topological group over G.

Proof.

For allε∈Aconsider

_CF:A⟶P(G)ε↦_CF(ε).

Then one can easily check that_CF(ε)is a topological subgroup of G. Thus, the pair(_CF,A)is a soft topological group over G.  □

Definition 22.

Let the soft topological group(F,A,τ)act on itself by conjugation and let(H,^F″,B)be a soft subset of(G,F,A). The soft stability subgroup is defined

_N^F″ (ε)={g∈F(ε):_θε(g,h)=h,h∈^F″(ε)}

and is said to be a soft normalizer of(H,^F″,B).

From this definition, we can get the following:

Proposition 7.

The soft normalizer_N^F″ (ε)given above is a soft topological group over G.

Proof.

For allε∈Awe define

_N^F″ :A⟶P(G)ε↦_N^F″ (ε).

Then it is easy to verify that_N^F″ (ε)is a topological subgroup of G. Hence, the pair(_N^F″ ,A)is a soft topological group over G.  □

Lastly, we can directly obtain the following result, which completes this study.

Corollary 1.

If(H,^F″,B)is a soft topological subgroup of(G,F,A), then(H,^F″,B)is a soft topological subgroup of(G,_N^F″ ,A). Furthermore, if(H,^F″,B)is a normal soft topological subgroup of(G,F,A),(G,_N^F″ ,A)is the largest soft topological subgroup of(G,F,A).

4. Conclusions

In summary, it can be said that this study is the beginning of a new structure between topological transformation groups and soft set theory, which are one of the important topics of algebraic topology. Several fundamental properties of soft topological transformation groups are also investigated. At the same time, some notions such as stabilizer, centralizer and normalizer related to the soft (continuous) action are described, and the characterizations between them are presented. Furthermore, by defining softG−equivariants, the category of softG−spaces is established. To extend our work in this paper, by using the concept of soft (continuous) action defined on soft topological groups, soft topological crossed modules can be defined so that by constructing the category of soft topological crossed modules, some novel categorical equivalents between category theory and soft set theory can be studied.

Funding

This research received no external funding.

Acknowledgments

The author thanks to the reviewers for the constructive and helpful suggestions on the revision of this paper.

Conflicts of Interest

The author declares no conflict of interest.

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