Galois correspondences in soft L -topological spaces

1 Introduction

Since Molodtov [22] introduced the notion of soft sets, the soft set theory developed rapidly. This kind of theory can be considered as a new tool for dealing with some uncertainties that traditional tools cannot handle efficiently. At present, works on soft set theory and its applications are progressing rapidly in various fields. Maji et al. [17] combined fuzzy sets and soft sets, and introduced fuzzy soft sets. To continue the investigations on fuzzy soft sets, Ahmad and Kharal [2] presented some more properties of fuzzy soft sets and introduced the notion of a mapping on the class of fuzzy soft sets. Majumdar and Samanta [19] introduced the concept of generalized fuzzy soft sets.

Algebraic structures of fuzzy soft sets and soft sets have been studied by many researchers. In [3], Aktaş and Çaǧman gave the notion of soft groups. Later, Aygünoǧlu and Aygün [4] proposed fuzzy soft groups by using a t-norm. In [11], Feng et al. defined soft semirings and several related concepts.

Topological structures of soft sets and fuzzy soft sets have also been studied by many researchers. In 2011, Gaǧan et al. [5] and Shabir [33] independently introduced the concept of soft topologies. Motivated by fuzzy topology and its related fuzzy spatial structures, such as fuzzy convergence [12, 28], fuzzy convexity [25– 27, 38] and so on, Tanay et al. [34] introduced the topological structure of fuzzy soft sets, which is called fuzzy soft topology. Later, Roy et al. [31] and Varol et al. [35] independently modified the definition of fuzzy soft sets and redefined fuzzy soft topology. From a completely different view point, Varol et al. [36] provided a new approach to soft topology and fuzzy soft topology, and studied the concept of soft compactness and L-fuzzy soft compactness. In a similar way, Cetkin and Aygün [6] introduced the concept of fuzzy soft topogenous structures and studied its properties. Recently, many researchers further studied spatial properties of fuzzy soft topological spaces (See [7– 9, 37]).

In this paper, we will consider the categorical relations between soft topological spaces and soft L-topological spaces in the sense of Varol et al. Moreover, we will introduce the concept of stratified soft L-topological spaces and investigate its relations with soft topological spaces and soft L-topological spaces.

2 Preliminaries

Let (L, ⋁ , ⋀ , ′) be a completely distributive De Morgan algebra. M (L) denotes the set of all non-zero coprime elements in L and P (L) denotes the set of all non-unit prime elements in L. The smallest element and the largest element in L are denoted by ⊥ and ⊤, respectively. Let a, b be elements in L. We say “a is wedge below b” in symbol a ≺ b if for every subset D ⊆ L, ⋁ D ⩾ b implies a ⩽ d for some d ∈ D . We denote β (a) = {b ∣ b ≺ a} and β^* (a) = β (a) ∩ M (L). Thus a = ⋁ β (a) = ⋁ β^* (a) holds for each a ∈ L.

For a nonempty set X, L ^X denotes the set of all L-subsets on X. In particular, 2 ^X denotes the set of all subsets of X. The smallest element and the largest element in L ^X are denoted by ⊥ _ and ⊤ _ , respectively. For any a ∈ L, the constant mapping A : X → L defined by A (x) = a for all x ∈ X, is denoted by a _ . L ^X is also a completely distributive De Morgan algebra when it inherits the structure of the lattice L in a natural way, by defining ⋁,  ⋀ ,   ⩽ and ^′ pointwisely. The set of non-zero coprime elements in L ^X is denoted by J (L ^X ). For a subset U ⊆ X, its characteristic function is denoted by χ _U . For convenience, we give two notations as follows: for each a ∈ L and A ∈ L ^X , let ξ _a  (A) = {x ∣ A (x) ⩽ a} and ι _a  (A) = {x ∣ A (x) ≰ a}.

Let φ : X → Y be a mapping. Define φ^→ : L ^X  → L ^Y and φ^← : L ^Y  → L ^X by φ^→ (A) (y) = ⋁ _φ(x)=yA (x) for A ∈ L ^X and y ∈ Y, and φ^← (B) = B ∘ φ for B ∈ L ^Y , respectively.

Definition 2.1. ([10]) A L-topology on a set X is a subset τ of L ^X which satisfies:

⊥ _ , ⊤ _ ∈ τ ;

The pair (X, τ) is called an L-topological space. It will be called stratified if it satisfies moreover,

(SLT) a _ ∈ τ for all a ∈ L.

Definition 2.2. ([10]) Let (X, τ _X ) and (Y, τ _Y ) be two L-topological spaces and φ : X → Y be a mapping. φ : (X, τ _X ) → (Y, τ _Y ) is called continuous if for all B ∈ L ^Y , B ∈ τ _Y implies φ^← (B) ∈ τ _X .

Definition 2.3. ([36]) A soft topology on a set X with respect to parameters E is a mapping τ : E → 2^{2 X} such that for all e ∈ E, τ (e) ∈2^{2 X} is a classical topology on X. The triple (X, τ, E) is called a soft topological space.

Definition 2.4. ([36]) Let (X, τ _X , E) and (Y, τ _Y , F) be two soft topological spaces. (ψ, φ) : (X, τ _X , E) → (Y, τ _Y , F) is called soft continuous if for all e ∈ E and f = ψ (e) ∈ F, φ : (X, τ _X  (e)) → (Y, τ _Y  (f)) is continuous with respect to the classical topologies τ _X  (e) and τ _Y  (f).

It is easy to check that all soft topological spaces and their soft continuous mappings form a category, denoted by SoTop.

Definition 2.5. ([36]) A soft L-topology on a set X with respect to parameters E is a mapping δ : E → 2 ^{L X} such that for all e ∈ E, δ (e) ∈2 ^{L X} is an L-topology on X. The triple (X, δ, E) is called a soft L-topological space.

Definition 2.6. ([36]) Let (X, δ _X , E) and (Y, δ _Y , F) be two soft L-topological spaces. (ψ, φ) : (X, δ _X , E) → (Y, δ _Y , F) is called soft continuous if for all e ∈ E and f = ψ (e) ∈ F, φ : (X, δ _X  (e)) → (Y, δ _Y  (f)) is continuous with respect to the L-topologies δ _X  (e) and δ _Y  (f).

It is easy to check that all soft L-topological spaces and their soft continuous mappings form a category, denoted by SoL-Top.

For notions related to category theory we refer to [1].

3 A Galois correspondence between SoTop and SoL-Top

In this section, we shall discuss the relations between soft topological spaces and soft L-topological spaces in a categorical sense.

Definition 3.1. Let (X, τ, E) be a soft topological space and A ∈ L ^X . If ξ _a  (A) ∈ τ (e) ′ for all a ∈ L, where τ (e) ′ denotes the set of all closed set in (X, τ (e)), then A is called soft low-semicontinuous with respect to e (e-low-semicontinuous, in short).

Lemma 3.2. Let (X, τ, E) be a soft topological space and A ∈ L ^X . Then A is soft e-low-semicontinuous iff ξ _a  (A) ∈ τ (e) ′ for all a ∈ P (L).

Proof. It suffices to prove the sufficiency. Take any a ∈ L. Since L is a completely distributive De Morgan algebra, there are some prime elements {p _i } _i∈I ⊆ P (L) such that a = ⋀ _i∈Ip _i . Hence, we have A ( x ) ⩽ a ⇔ A ( x ) ⩽ p i for all i ∈ I . Then it follows that ξ a ( A ) = { x ∣ A ( x ) ⩽ a } = ⋂ i ∈ I { x ∣ A ( x ) ⩽ p i } = ⋂ i ∈ I ξ p i ( A ) ∈ τ ( e ) ′ . This shows that A is soft e-low-semicontinuous.□

Proposition 3.3. Let (X, τ, E) be a soft topological space and define δ _τ  : E → 2 ^{L X} as follows: ∀ e ∈ E , δ τ ( e ) = { A ∈ L X ∣ A is soft e - low - semicontinuous } . Then (X, δ _τ , E) is a soft L-topological space.

Proof. For each e ∈ E, we need only show that δ _τ  (e) is an L-topology on X. In fact, δ _τ  (e) is a stratified L-topology on X. Next we verify that δ _τ  (e) satisfies (LT2), (LT3) and (SLT).

(LT2) For each A, B ∈ δ _τ  (e), take any a ∈ P (L). Then it follows that ξ a ( A ∧ B ) = { x ∣ A ( x ) ∧ B ( x ) ⩽ a } = { x ∣ A ( x ) ⩽ a } ∪ { x ∣ B ( x ) ⩽ a } = ξ a ( A ) ∪ ξ a ( B ) ∈ τ ( e ) ′ . By Lemma 3.2, we know A ∧ B ∈ δ _τ  (e).

(LT3) For {A _i  ∣ i ∈ I} ⊆ δ _τ  (e), take any a ∈ P (L). Then it follows that ξ a ( ⋁ i ∈ I A i ) = { x ∣ ⋁ i ∈ I A i ( x ) ⩽ a } = ⋂ i ∈ I { x ∣ A i ( x ) ⩽ a } = ⋂ i ∈ I ξ a ( A i ) ∈ τ ( e ) ′ . This implies that ⋁_i∈IA _i  ∈ δ _τ  (e).

(SLT) For each b ∈ L, take any a ∈ L. Then it follows that ξ a ( b _ ) = { x ∣ b _ ( x ) ⩽ a } = ∅ or X . Hence, ξ a ( b _ ) ∈ τ ( e ) ′ . That is, b _ ∈ δ τ ( e ) .□

Proposition 3.4. If (ψ, φ) : (X, τ _X , E) → (Y, τ _Y , F) is soft continuous, then so is (ψ, φ) : (X, δ _{τ X} , E) → (Y, δ _{τ Y} , F).

Proof. Take any e ∈ E, f = ψ (e) ∈ F and A ∈ δ _{τ Y}  (f). By definition, it follows that ξ _a  (A) ∈ τ _Y  (f) ′ for all a ∈ L. Since (ψ, φ) : (X, τ _X , E) → (Y, τ _Y , F) is soft continuous, it follows that φ ← ( ξ a ( A ) ) ∈ τ X ( e ) ′ . Then we obtain

ξ a ( φ ← ( A ) ) = { x ∣ φ ← ( A ) ( x ) ⩽ a } = φ ← ( { y ∣ A ( y ) ⩽ a } ) = φ ← ( ξ a ( A ) ) ∈ τ X ( e ) ′ . This implies that φ^← (A) ∈ δ _{τ X}  (e). Hence, we obtain the soft continuity of (ψ, φ) : (X, δ _{τ X} , E) → (Y, δ _{τ Y} , F).□

By Propositions 3.3 and 3.4, we obtain a functor ω as follows: ω : { SoTop → So L - Top , ( X , τ , E ) ↦ ( X , δ τ , E ) , ( ψ , φ ) ↦ ( ψ , φ ) .

Conversely, we shall construct a functor from SoL-Top to SoTop. In order to construct this functor, we first give the following two results.

Proposition 3.5.Let (X, δ, E) be a soft L-topological space and define L δ : E → 2 2 X as follows: ∀ e ∈ E , L δ ( e ) = { ι a ( A ) ∣ a ∈ L , A ∈ δ ( e ) } . Then L δ ( e ) is a subbase of a classical topology.

Proof. Take a =⊥ and A = ⊤ _ . Then ι a ( A ) = X ∈ L δ ( e ) . Hence, L δ ( e ) is a subbase.□

By Proposition 3.5, the following proposition is straightforward and the proof is omitted.

Proposition 3.6.Let (X, δ, E) be a soft L-topological space and denote T δ ( e ) by the topology generated by L δ ( e ) . Then ( X , T δ , E ) is a soft topological space.

Proposition 3.7. If (ψ, φ) : (X, δ _X , E) → (Y, δ _Y , F) is soft continuous, then so is ( ψ , φ ) : ( X , T δ X , E ) → ( Y , T δ Y , F ) .

Proof. Take any e ∈ E, f = ψ (e) ∈ F and U ∈ T δ Y ( f ) . Since U ∈ T δ Y ( f ) , there exist A ∈ δ _Y  (f) and a ∈ L such that U = ι _a  (A). By the soft continuity of (ψ, φ) : (X, δ _X , E) → (Y, δ _Y , F), it follows that φ^← (A) ∈ δ _X  (e). Then we have φ ← ( U ) = φ ← ( ι a ( A ) ) = ι a ( φ ← ( A ) ) ∈ L δ X ( e ) ⊆ T δ X ( e ) . This shows the soft continuity of ( ψ , φ ) : ( X , T δ X , E ) → ( Y , T δ Y , F ) .□

By Propositions 3.6 and 3.7, we obtain a functor ι as follows: ι : { So L - Top → SoTop , ( X , δ , E ) ↦ ( X , τ δ , E ) , ( ψ , φ ) ↦ ( ψ , φ ) .

Proposition 3.8. If (X, τ, E) is a soft topological space and (X, δ, E) is soft L-topological space, then τ _{δ τ}  (e) = τ (e) and δ (e) ⊆ δ _{τ δ}  (e) for all e ∈ E.

Proof. Take any U ∈ L δ τ ( e ) . Then there exist a ∈ L and A ∈ δ _τ  (e) such that U = ι _a  (A). Since A ∈ δ _τ  (e), it follows that ξ _a  (A) ∈ τ (e) ′. Hence, we have U = ι _a  (A) = X ∖ ξ _a  (A) ∈ τ (e). This implies that L δ τ ( e ) ⊆ τ ( e ) . Since L δ τ ( e ) is a subbase of T δ τ ( e ) , we have τ _{δ τ}  (e) ⊆ τ (e).

Take any U ∈ τ (e) and let A = χ _U and a =⊥. Then for each b ∈ L, it follows that ξ _b  (A) = X ∖ U or X. Hence, ξ _b  (A) ∈ τ (e) ′ for each b ∈ L. This implies A ∈ δ _τ  (e). Further, it follows that U = ι a ( A ) ∈ L δ τ ( e ) ⊆ τ δ τ ( e ) . Hence, τ (e) ⊆ τ _{δ τ}  (e). Therefore, we obtain τ _{δ τ}  (e) = τ (e).

Take any A ∈ δ (e). Then ι a ( A ) ∈ L δ ( e ) ⊆ τ δ ( e ) and ξ _a  (A) = X ∖ ι _a  (A) ∈ τ _δ  (e) ′ for each a ∈ L. By definition, it follows that A ∈ δ _{τ δ}  (e). By the arbitrariness of A, we obtain δ (e) ⊆ δ _{τ δ}  (e), as desired.□

By the above results, we obtain the main result in this section.

Theorem 3.9. (ω, ι) is a Galois correspondence between SoTop and SoL-Top.

Corollary 3.10. SoTop can be embedded in SoL-Top as a coreflective subcategory.

4 Stratified soft L-topological spaces

In this section, we will propose stratified soft L-topological spaces and study its relations with soft topological spaces and soft L-topological spaces in a categorical sense.

Definition 4.1. A stratified soft L-topology on a set X with respect to parameters E is a mapping δ : E → 2 ^{L X} such that for all e ∈ E, δ (e) ∈2 ^{L X} is a stratified L-topology on X. The triple (X, δ, E) is called a stratified soft L-topological space.

Let SSoL-Top denote the full subcategory of SoL-Top, consisted of stratified soft L-topological spaces.

By Proposition 3.3, we can see the functor ω :SoTop→SoL-Top is in fact a functor from SoTop to SSoL-Top. Moreover, all the results in the former section with respect to SoL-Top still hold with respect to SSoL-Top. Hence, we obtain the following results easily.

Theorem 4.2. (ω, ι) is a Galois correspondence between SoTop and SSoL-Top.

Corollary 4.3. SoTop can be embedded in SSoL-Top as a coreflective subcategory.

Next we construct another functor from SSoL-Top to SoTop. In order to do this, the following lemma is necessary.

Lemma 4.4. Let (X, δ, E) be a stratified soft L-topological space. If χ_{ξ a (A)} ∈ δ (e) ′ for all a ∈ L, then A ∈ δ (e).

Proof. Suppose that χ_{ξ a (A)} ∈ δ (e) ′ for all a ∈ L. Then we need only show that A ′ ¯ ⩽ A ′ . Take any x _λ  ∈ J (L ^X ) such that x _λ ≰A′. Then λ = ⋁ β^* (λ) ≰A′ (x). Hence, there exists μ ∈ β^* (λ) such that μ≰A′ (x). Put E = {y ∣ μ ⩽ A′ (y)} = {y ∣ A (y) ⩽ μ′} = ξ_μ′ (A). Then x ∉ E and x _λ ≰χ _E  = χ_ξμ′(A) ∈ δ (e) ′. Put b = ⋁ {A′ (y) ∣ y ∉ E}. Then b _ ∨ χ E ∈ δ ( e ) ′ and x λ ≰ b _ . This implies x λ ≰ b _ ∨ χ E . By definition of E, we can easily obtain that A ′ ⩽ b _ ∨ χ E . This shows there exists a closed remote neighborhood b _ ∨ χ E of x _λ such that A ′ ⩽ b _ ∨ χ E . Therefore, x λ ≰ A ′ ¯ . By the arbitrariness of x _λ , we obtain that A ′ ¯ ⩽ A ′ , as desired.□

Proposition 4.5. Let (X, δ, E) be s stratified soft L-topological space and define τ δ * : E → 2 2 X as follows: ∀ e ∈ E , τ δ * ( e ) = { U ∈ 2 X ∣ χ U ∈ δ ( e ) } . Then ( X , τ δ * , E ) is a soft topological space.

Proof. It is easy to verify that τ δ * ( e ) is a classical topology on X for all e ∈ E. Hence the proof is omitted.□

Proposition 4.6. If (ψ, φ) : (X, δ _X , E) → (Y, δ _Y , F) is soft continuous, then so is ( ψ , φ ) : ( X , τ δ X * , E ) → ( Y , τ δ Y * , F ) .

Proof. Take any e ∈ E, f = ψ (e) ∈ F and U ∈ τ δ Y * ( f ) . Then it follows that χ _U  ∈ δ _Y  (f). Since (ψ, φ) : (X, δ _X , E) → (Y, δ _Y , F) is soft continuous, we have φ^← (χ _U ) = χ_φ^←(U) ∈ δ _X  (e). This implies that φ ← ( U ) ∈ τ δ X * ( e ) . Therefore, ( ψ , φ ) : ( X , τ δ X * , E ) → ( Y , τ δ Y * , F ) is soft continuous.□

By Propositions 4.5 and 4.6, we obtain a functor [ ] as follows: [ ] : { So L - Top → SoTop , ( X , δ , E ) ↦ ( X , τ δ * , E ) , ( ψ , φ ) ↦ ( ψ , φ ) .

Proposition 4.7. If (X, δ, E) is a stratified L-topological space and (X, τ, E) is a soft topological space, then τ δ τ * ( e ) = τ ( e ) and δ τ δ * ( e ) ⊆ δ ( e ) .

Proof. Take any U ∈ 2 ^X . Then U ∈ τ δ τ * ( e ) ⇔ χ U ∈ δ τ ( e ) ⇔ ∀ a ∈ L , ξ a ( χ U ) ∈ τ ( e ) ′ ⇔ ∀ a ∈ L , ι a ( χ U ) ∈ τ ( e ) ⇔ U ∈ τ ( e ) . This means τ δ τ * ( e ) = τ ( e ) .

Take any A ∈ L ^X . Then A ∈ δ τ δ * ( e ) ⇔ ∀ a ∈ L , ξ a ( A ) ∈ τ δ * ( e ) ′ ⇔ ∀ a ∈ L , χ ξ a ( A ) ∈ δ ( e ) ′ ⇒ A ∈ δ ( e ) . ( by Lemma 4.4 ) □

By Proposition 4.7, we obtain one of the main results in this section.

Theorem 4.8. ([ ], ω) is a Galois correspondence between SSoL-Top and SoTop.

Corollary 4.9. SoTop can be embedded in SSoL-Top as a reflective subcategory.

Next we will discuss the stratification of soft L-topological spaces.

Let (X, δ, E) be a soft L-topological space and define ϕ _δ  : E → 2 ^{L X} as follows: ∀ e ∈ E , ϕ δ ( e ) = δ ( e ) ∪ { a _ ∣ a ∈ L } . It is easy to see that ϕ _δ  (e) is a subbase of a stratified L-topology, which will be denoted by S δ ( e ) . Subsequently, ( X , S δ , E ) is a stratified soft L-topological space.

Definition 4.10. Let (X, δ, E) be a soft L-topological space. The stratified soft L-topological space ( X , S δ , E ) is called the stratification of (X, δ, E).

Proposition 4.11. If (ψ, φ) : (X, δ _X , E) → (Y, δ _Y , F) is soft continuous, then so is ( ψ , φ ) : ( X , S δ X , E ) → ( Y , S δ Y , F ) .

Proof. For each e ∈ E and f = ψ (e) ∈ F, take A ∈ ϕ δ Y ( f ) = δ Y ( f ) ∪ { a _ ∣ a ∈ L } . If A ∈ δ _Y  (f), then by the soft continuity of (ψ, φ) : (X, δ _X , E) → (Y, δ _Y , F), we have φ ← ( A ) ∈ δ X ( e ) ⊆ ϕ δ X ( e ) ⊆ S δ X ( e ) . If A = a _ for some a ∈ L, then φ ← ( a _ ) = a _ ∈ ϕ δ X ( e ) ⊆ S δ X ( e ) . This means that φ ← ( A ) ∈ S δ X ( e ) for all A ∈ ϕ _{δ Y}  (f). Therefore, ( ψ , φ ) : ( X , S δ X , E ) → ( Y , S δ Y , F ) is soft continuous.□

By the above results, we obtain a functor S as follows: ω : { So L - Top → SSo L - Top , ( X , δ , E ) ↦ ( X , S δ , E ) , ( ψ , φ ) ↦ ( ψ , φ ) .

Since SSoL-Top is a full subcategory of SoL-Top, there is an embedding functor I : SSoL-Top→ SoL-Top. Obviously, the following result holds.

Proposition 4.12. If (X, δ, E) is a soft L-topological space and ( X , S , E ) is a stratified soft L-topological space, then δ ( e ) ⊆ S δ ( e ) and S S ( e ) = S ( e ) .

By Proposition 4.12, we have the following results.

Theorem 4.13. ( I , S ) is a Galois correspondence between SSoL-Top and SoL-Top.

Corollary 4.14. SSoL-Top can be embedded in SoL-Top as a coreflective subcategory.

5 Conclusions

In this paper, we investigated several categorical relations among some subcategories of the category of soft L-topological spaces in the sense of Varol et al. Actually, in the framework of this kind of soft fuzzy topological spaces, we can also consider the relations among soft L-fuzzifying topological spaces, soft L-topologies and soft L-fuzzy topologies in the future.