Soft semi-linear uniform spaces and their perceptual application

1 Introduction

Soft set theory is a generalization of fuzzy set theory that was introduced in 1999 by Molodtsov [18] as a new mathematical technique to deal with uncertainties and solve numerous practical problems in fields including engineering, social science, economics, medical science that traditional methods had failed to handle. Maji et al. ([16, 17]) provided the first practical application of soft sets in decision-making problems and he offered theoretical knowledge of soft sets and studied subset, operation such as intersection and union, and equality of soft sets. Çağman and Enginoğlu [6] developed a soft matrix theory and a successful application of soft sets to decision-making problems in 2010. In soft topological spaces, Aygünoğlu and Aygün [3] established soft product topology, soft compactness and generalized Tychonoff theorem. Among the importance occupied by the soft set theory many authors used the notion of soft sets to present a number of important research in a various branches ([2, 34]).

Uniformity is an effective tool for investigation of topology and it can be considered as a link between metric and topology since it is striking parallels with metrics, but its application is much wider. Çaetkin and Aygün [7] introduced an extension of uniform structures by using soft sets. They defined the concept of a soft uniformity consisting of soft sets on Z × Z by depending on the axioms proposed by Bourbaki [5] where Z is an initial universe. O ̈ zbakir and Demir [20] defined a soft uniformity as a family of soft sets on SP (Z) × SP (Z) with the parameters set Ψ that is different to Çetkin and Aygün’s definition where SP (Z) is the family of all soft points over Z. They looked at some of the fundamental properties of a soft uniformity and compared soft uniformities to soft metric and soft topology. In [9] Demir et al. established a new type of uniform space known as se-uniform spaces and offered some of their basic properties, then the concept of a soft E-distance in se-uniform spaces was presented. Also, they constructed some fixed soft element theorems for various mappings on se-uniform spaces using the soft E-distance. In [10] Demir et al. looked into soft proximity spaces and studied how proximity spaces and soft proximity spaces are related. Then, they developed the concept of a soft neighborhood in the context of soft proximity spaces, which provide a different perspective on the study of soft proximity spaces. Finally, they illustrated how to derive a soft proximity space from a soft uniform space.

This manuscript contributes to this direction by employing the concept of semi-linear uniform space that is defined by Tallafha and Khalil [30]. A semi-linear uniform space is defined by reformulating the terms of uniform spaces to present a new mixture spaces of analysis and topology, many authors ([24, 29]) have contributed to several semi-linear uniform spaces investigations, one of the most essential works in this field [26] is the negative answer to the question "Does f have a unique fixed point if it is a contraction from a complete semi-linear uniform space to itself?" In this work, we define the concept of a soft semi-linear uniform space by depending on the axioms proposed by Tallafha and Khalil [30] and we establish a link between the generalized soft semi-linear uniform spaces and the soft proximity spaces.

We recall some definitions and properties of soft sets then this work is organized as follows: In Section 2, we introduce the notion of soft semi-linear uniform spaces and we present some properties of this space. Also, through the axioms of this space we are able to introduce the two soft mapping ρ ˜ and δ ˜ on SP (Z) × SP (Z) through which we present some of the properties of this space. In section 3, we employ the members of soft-GSL _US to introduce a soft proximity space. In section 4, we employ the natural transformation of a soft semi-linear uniform space to a soft proximity to give the perceptual application of soft semi-linear uniform structures.

Definition 1.1. [3, 18] Let P (Z) be the power set of Z. A soft set H on the universe Z with the parameters set Ψ is defined by H = {(e, H (e)) : e ∈ Ψ, H (e) ∈ P (Z)} where H is a mapping H : Ψ → P (Z).

The family of all soft sets over Z with the parameters set Ψ is denoted by S (Z, Ψ).

Definition 1.2. ([1, 21]) Let H, G ∈ S (Z, Ψ). Then:

A soft set H is called null soft set, denoted by Φ, if H (e) = φ for every e ∈ Ψ;

Definition 1.3. ([8, 19]) Let Ψ be a parameters set. Then:

A soft set P over Z is called a soft point, denoted by x ^e , if there is e ∈ Ψ with P (e) = {x} for some x ∈ Z and P (e^∗) = φ for all e^∗ ∈ Ψ - {e};

The family of all soft points over Ψ is denoted by SP (Z) and the family of all soft sets over SP (Z) × SP (Z) with the parameters set Ψ is denoted by S (SP (Z) × SP (Z) , Ψ).

Definition 1.4. [4] Let S (Z, Ψ _Z ) and S (W, Ψ _W ) be the families of all soft sets over Z and W, respectively. Then the Cartesian product H × G for H ∈ S (Z, Ψ _Z ) and G ∈ S (W, Ψ _W ) is defined by:

(H × G) (e, k) = H (e) × G (k),   for every (e, k) ∈ Ψ _Z  × Ψ _W .

Definition 1.5. [13] Let S (Z, Ψ _Z ) and S (W, Ψ _W ) be the families of all soft sets over Z and W, respectively, and φ : Z → W and ψ : Ψ _Z  → Ψ _W be two mappings. Then the soft mapping φ _ψ : S (Z, Ψ _Z ) → S (W, Ψ _W ) is defined by:

for H ∈ S (Z, Ψ _Z )

φ ψ ( H ) ( k ) = { ⋃ e ∈ ψ - 1 ( k ) φ ( H ( e ) ) , if ψ - 1 ( k ) ≠ φ φ otherwise } for every k ∈ Ψ _W . Where φ _ψ (H) is called a soft image of a soft set H;

Moreover, for G ∈ S (W, Ψ _W ), φ ψ - 1 ( G ) ( e ) = φ - 1 ( G ( ψ ( e ) ) ) for every e ∈ Ψ _Z . Where φ ψ - 1 ( G ) is called a soft inverse image of a soft set G.

The soft mapping φ _ψ is called injective, if φ and ψ are injective. The soft mapping φ _ψ is called surjective, if φ and ψ are surjective ([3, 36]).

Definition 1.6. [27] A family ϑ of soft sets of Z with a parameters set Ψ is called a soft topology on Z if:

The absolute soft set Z ˜ and the null soft set Φ are members of ϑ;

Then (Z, ϑ, Ψ) is called a soft topological space where each member in ϑ is soft open and its complement is soft closed.

Definition 1.7. [19] Let (Z, ϑ, Ψ) be a soft topological space. A soft set H is a soft neighborhood of the soft point x ^e if there is a soft open set G with x e ∈ ˜ G ⊑ H .

Definition 1.8. [11] Let (Z, ϑ, Ψ) be a soft topological space. If for every x α ≠ y β ∈ ˜ Z ˜ there are two soft open sets H and G with x α ∈ ˜ H , y β ∈ ˜ G and H ⊓ G = Φ, then (Z, ϑ, Ψ) is called a soft T₂-space.

Definition 1.9. [20] Let (Z, ϑ₁, Ψ _Z ) and (W, ϑ₂, Ψ _W ) be two soft topological spaces and φ _ψ : (Z, ϑ₁, Ψ _Z ) → (W, ϑ₂, Ψ _W ) be a soft mapping. Then φ _ψ is soft continuous at x e ∈ ˜ Z ˜ if for every soft neighborhood G of φ _ψ (x ^e ) in W, there is a soft neighborhood H of x ^e in Z with φ _ψ (H) ⊑ G. A soft mapping φ _ψ is called soft continuous on Z if it is soft continuous at each x e ∈ ˜ Z ˜ .

Definition 1.10. [20] The non-empty family U ⊆ S ( SP ( Z ) × S P ( Z ) , Ψ ) is called a soft uniformity for Z if the following are satisfied:

For every M ∈ U , then Δ ⊑ M, where Δ (e) = {(x ^α, x ^α) : x ^α ∈ SP (Z)} for every e ∈ Ψ;

Then ( Z , U , Ψ ) is called soft uniform spaces on Z.

2 Soft semi-linear uniform spaces

In this section, we reframe the axioms of soft uniform spaces to introduce the notion of soft semi-linear uniform spaces. Then we present some properties of this space and we show that there is a soft topology induced by soft semi-linear uniform space.

Definition 2.1. Let D_SP(Z) be the family which contains all members of S (SP (Z) × SP (Z) , Ψ) that satisfy the following:

For every M ∈ D_SP(Z), Δ ⊑ M;

If Δ ∉ D_SP(Z), we will use D SP ( Z ) * .

Example 2.2. Let Z = {x₁, x₂, x₃} with parameters set Ψ = {e₁, e₂}. Clearly, Δ ( e 1 ) = Δ ( e 2 ) = { ( x 1 e 1 , x 1 e 1 ) , ( x 1 e 2 , x 1 e 2 ) , ( x 2 e 1 , x 2 e 1 ) , ( x 2 e 2 , x 2 e 2 ) , ( x 3 e 1 , x 3 e 1 ) , ( x 3 e 2 , x 3 e 2 ) } . Let M ( e 1 ) = Δ ( e 1 ) ∪ { ( x 1 e 1 , x 2 e 1 ) , ( x 2 e 1 , x 1 e 1 ) } and M ( e 2 ) = Δ ( e 2 ) ∪ { ( x 1 e 2 , x 2 e 2 ) , ( x 2 e 2 , x 1 e 2 ) } . Then Δ, M ∈ D_SP(Z).

Definition 2.3. Let M, N ∈ D_SP(Z). Then,

The composition of M and N, denoted by M ∘ N, is defined by for every e ∈ Ψ, (M ∘ N) (e) = {(x ^α, y ^β) : thereissome  z ^γ ∈ SP (Z) with (x ^α, z ^γ) ∈ N (e)  and  (z ^γ, y ^β) ∈ M (e)};

Note that the composition is associative, i . e, (M ∘ N) ∘ P = M ∘ (N ∘ P). However, the composition is not commutative.

Proposition 2.4. For any M, N ∈ D_SP(Z), we have

M ⊓ N ∈ D_SP(Z);

Definition 2.5. Let Γ be a subfamily of D SP ( Z ) * that satisfy the following:

Γ is a chain (i . e, for every M, N ∈ Γ either M ⊑ N or N ⊑ M);

Then (Z, Γ, Ψ) is called soft semi-linear uniform spaces on Z, briefly soft-SL _US .

Example 2.6. Let Z = Ψ = ℝ and H _x  : Ψ → P (Z) be defined by H x ( e ) = { { x } , e = x φ , otherwise } . Then H _x  (e) is a soft point, denoted by x ^x , and so (x ^x , y ^y ) ∈ SP (Z) × SP (Z) for every x, y ∈ Z. Set G _e  = {(x ^x , y ^y ) : x² + y² < |e|} ⊔ Δ then for every e > 0, G _e  ⊆ SP (Z) × SP (Z). Let M _e  : Ψ → P (SP (Z) × SP (Z)) be defined by M e ( x ) = { G e , x = e φ , otherwise } . Then M _e  ∈ S (SP (Z) × SP (Z) , Ψ) for every e > 0. Therefore, (Z, Γ, Ψ) is soft-SL _US where Γ = {M _e  : e > 0}.

Definition 2.7. If (i) of Definition 2.5 is replaced by the following condition: M ⊓ N ∈ Γ for every M, N ∈ Γ, then (Z, Γ, Ψ) is called generalized soft semi-linear uniform space on Z, briefly soft-GSL _US .

Note that if Γ is a chain then for every M, N ∈ Γ, M ⊓ N either M or N and hence every soft-SL _US is soft-GSL _US .

Definition 2.8. Let (Z, Γ, Ψ) be a soft-GSL _US and M ∈ Γ. Then:

For x α ∈ ˜ Z ˜ define B ( x α , M ) = { y β ∈ ˜ Z ˜ : ( x α , y β ) ∈ M ( e ) , ∀ e ∈ Ψ } ;

Definition 2.9. Let (Z, Γ, Ψ) be a soft-GSL _US . A subfamily Θ of Γ is called a soft base of Γ if for every M ∈ Γ there is C ∈ Θ with C ⊑ M .

Remark 2.10. Any soft base Θ for a soft-GSL _US has the following properties:

For every C ∈ Θ, there is O ∈ Θ with O∘ O ⊑ U ;

Theorem 2.11. If (Z, Γ, Ψ) is a soft-GSL _US , then ϑ = { O ⊑ Z ˜ : for every x α ∈ ˜ O , ∃ M ∈ Γ with B ( x α , M ) ⊑ O }

is a soft topology on Z.

Proof: It is obvious that Φ , Z ˜ ∈ ϑ . Let O₁, O₂ ∈ ϑ and x α ∈ ˜ O 1 ⊓ O 2 . There are M₁, M₂ ∈ Γ with B ( x α , M 1 ) ⊑ O 1 and B ( x α , M 2 ) ⊑ O 2 . Note that B ( x α , M 1 ) ⊓ B ( x α , M 2 ) = B ( x α , M 1 ⊓ M 2 ) since

y β ∈ ˜ B ( x α , M 1 ) ⊓ B ( x α , M 2 ) iff (x ^α, y ^β) ∈ M₁ (e) and (x ^α, y ^β) ∈ M₂ (e), ∀e ∈ Ψ iff (x ^α, y ^β) ∈ M₁ (e) ∩M₂ (e), ∀e ∈ Ψ iff y β ∈ ˜ B ( x α , M 1 ⊓ M 2 ) . Therefore, B ( x α , M 1 ⊓ M 2 ) ⊑ O 1 ⊓ O 2 and hence O₁ ⊓ O₂ ∈ ϑ. Finally, let O _i  ∈ ϑ and x α ∈ ˜ ⊔ i O i . Then there is M ∈ Γ with B ( x α , M ) ⊑ O i ∘ ⊑ ⊔ i O i and so ⊔ i O i ∈ ϑ . □

Corollary 2.12. If (Z, Γ, Ψ) is a soft-SL _US , then ϑ = { O ⊑ Z ˜ : for every x α ∈ ˜ O , ∃ M ∈ Γ with B ( x α , M ) ⊑ O }

is a soft topology on Z.

The soft topology in Theorem 2.11 and Corollary 2.12 which is induced by (Z, Γ, Ψ) will be denoted by (Z, ϑ _Γ, Ψ).

Proposition 2.13. Let (Z, Γ, Ψ) be a soft-GSL _US . For the soft topological space (Z, ϑ _Γ, Ψ), if H ∈ S (Z, Ψ) then O ˜ = { x α ∈ ˜ Z ˜ : ∃ M ∈ Γ with B ( x α , M ) ⊑ H } is equal to Int (H). Where Int (H) is defined as the union of all soft open sets contained in H.

Proof: Let x α ∈ ˜ O ˜ . There are M, P ∈ Γ with B ( x α , M ) ⊑ H and P ∘ P ⊑ M. Then for every y β ∈ B ( x α , P ) , B ( y β , P ) ⊑ B ( x α , M ) ⊑ H . Hence, B ( x α , P ) ⊑ O ˜ and so O ˜ is soft open. Now, since every soft open set O ⊑ H belongs to O ˜ we completed the proof.□

Corollary 2.14. Let (Z, Γ, Ψ) be a soft-GSL _US . For the soft topological space (Z, ϑ _Γ, Ψ) we have the following:

For every x α ∈ ˜ Z ˜ and H ∈ S (Z, Ψ), x α ∈ ˜ H ¯ iff H ⊓ B ( x α , M ) ≠Φ for every M ∈ Γ;

Proof: (i) The proof is obvious.

(ii) Let x α , y β ∈ ˜ H ¯ . Then there are x 1 α ′ , y 1 β ′ ∈ ˜ Z ˜ with x 1 α ′ ∈ ˜ H ⊓ B ( x α , M ) and y 2 β ′ ∈ ˜ H ⊓ B ( y β , M ) for every M ∈ Γ. Therefore (x ^α, x₁ ^α′), (y ^β, y₁ ^β′), (x₁ ^α′, y₁ ^β′) ∈ M (e) for every e ∈ Ψ and so (x ^α, y ^β) ∈ M³ (e) for every e ∈ Ψ. Hence, H ¯ × H ¯ ⊑ M 3 . □

Proposition 2.15. Let (Z, Γ, Ψ) be a soft-GSL _US . The soft topology (Z, ϑ _Γ, Ψ) is soft T₂-space.

Proof: Let x α ≠ y β ∈ ˜ Z ˜ . There are M, N ∈ Γ with N ∘ N ⊑ M and (x ^α, y ^β) ∉ M (e) for some e ∈ Ψ. If B ( x α , N ) ⊓ B ( y β , N ) ≠ Φ , then there is z γ ∈ ˜ B ( x α , N ) ⊓ B ( y β , N ) and so (x ^α, z ^γ), (y ^β, z ^γ) ∈ N (e) for every e ∈ Ψ. Hence (x ^α, y ^β) ∈ (N ∘ N) (e) ⊆ M (e) for every e ∈ Ψ, which is a contradiction. □

Proposition 2.16. Let (Z, Γ, Ψ) be a soft-GSL _US . For the soft topological space (Z, ϑ _Γ, Ψ), if H ∈ S (Z, Ψ) then H ¯ = ⊓ { B ( H , M ) : M ∈ Γ} .

Proof: x α ∈ ˜ H ¯ iff ∃ y ^β ∈ ˜ Z ˜ with y β ∈ ˜ B ( x α , M ) and y β ∈ ˜ H for every M ∈ Γ iff ∃ y ^β ∈ ˜ H with (x ^α, y ^β) ∈ M (e) for every e ∈ Ψ and M ∈ Γ iff x ^α ∈ ˜ B ( H , M ) for every M ∈ Γ . □

Theorem 2.17. Let (Z, Γ, Ψ) be a soft-GSL _US and H, G ∈ S (Z, Ψ). Then, there is M ∈ Γ with B ( H , M ) ⊓ G = Φ iff there is M ∈ Γ with B ( H , M ) ⊓ B ( G , M ) = Φ .

Proof: Let M ∈ Γ with B ( H , M ) ⊓ G = Φ . Then there is N ∈ Γ with N ∘ N ⊑ M. If x α ∈ ˜ B ( H , N ) ⊓ B ( G , N ) , then there are (x ^α, y ^β), (x ^α, z ^γ) ∈ N (e) for every e ∈ Ψ with y β ∈ ˜ H and z γ ∈ ˜ G . Thus, z γ ∈ ˜ B ( H , M ) , which is a contradiction. Hence B ( H , M ) ⊓ B ( G , M ) = Φ .

To prove the sufficiency, let G ∈ S (Z, Ψ), then G ⊑ B ( G , M ) for every M ∈ Γ and so B ( H , M ) ⊓ G = Φ . □

Corollary 2.18. Let (Z, Γ, Ψ) be a soft-GSL _US with a soft base Θ and H, G ∈ S (Z, Ψ). Then, there is M ∈ Θ with B ( H , M ) ⊓ G = Φ iff there is M ∈ Θ with B ( H , M ) ⊓ B ( G , M ) = Φ .

Proposition 2.19. Let (Z, Γ, Ψ) be a soft-GSL _US . If H, G, F ∈ S (Z, Ψ), then for every M ∈ Γ

B ( H ⊔ G , M ) = B ( H , M ) ⊔ B ( G , M )

B ( H ⊓ G , M ) ⊑ B ( H , M ) ⊓ B ( G , M ) ;

Proof: (i) Let x α ∈ ˜ B ( H ⊔ G , M ) . Then, there is y β ∈ ˜ H ⊔ G with (x ^α, y ^β) ∈ M (e) for every e ∈ Ψ. If y β ∈ ˜ H then x α ∈ ˜ B ( H , M ) ⊔ B ( G , M ) . Hence B ( H ⊔ G , M ) ⊑ B ( H , M ) ⊔ B ( G , M ) . We can prove B ( H , M ) ⊔ B ( G , M ) ⊑ B ( H ⊔ G , M ) by similar technique. The prove of other part is obvious.

(ii) The prove follows from i. □

Definition 2.20. Let (Z, Γ, Ψ) be a soft-SL _US . For (x ^α, y ^β) ∈ SP (Z) × SP (Z), set Γ ( x α , y β ) = { M ∈ Γ : ( x α , y β ) ∈ ˜ M } and Γ c ( x α , y β ) = { M ∈ Γ : ( x α , y β ) ∉ ˜ M } Then define ρ ˜ and δ ˜ on SP (Z) × SP (Z) by ρ ˜ ( x α , y β ) = ⊓ { M : M ∈ Γ ( x α , y β ) } and δ ˜ ( x α , y β ) = { ⨆ { M : M ∈ Γ ( x α , y β ) c } : x α ≠ y β Δ : x α = y β }

Remark 2.21. If ( s γ , t λ ) ∈ ˜ ρ ˜ ( x α , y β ) then ( s γ , t λ ) ∈ ˜ M for every M ∈ Γ_{(x ^α,y ^β)}. Also, if ( s γ , t λ ) ∈ ˜ δ ˜ ( x α , y β ) then ( s γ , t λ ) ∈ ˜ M for some M ∈ Γ ^c _{(x ^α,y ^β)}.

Proposition 2.22. Let (Z, Γ, Ψ) be a soft-SL _US . If x ^α, y ^β ∈ SP (Z) then

ρ ˜ ( x α , y β ) = ρ ˜ ( y β , x α ) and δ ˜ ( x α , y β ) = δ ˜ ( y β , x α ) ;

Theorem 2.23. Let (Z, Γ, Ψ) be a soft-SL _US . Then δ ˜ ( x α , y β ) = δ ˜ ( s γ , t λ ) iff ρ ˜ ( x α , y β ) = ρ ˜ ( s γ , t λ ) .

Proof: First, we show δ ˜ ( x α , y β ) ⊑ δ ˜ ( s γ , t λ ) iff ρ ˜ ( x α , y β ) ⊑ ρ ˜ ( s γ , t λ ) . Suppose δ ˜ ( x α , y β ) ⊑ δ ˜ ( s γ , t λ ) then Γ ( s γ , t λ ) ⊆ Γ ( x α , y β ) and hence ρ ˜ ( x α , y β ) ⊑ ρ ˜ ( s γ , t λ ) . Now, if ρ ˜ ( x α , y β ) ⊑ ρ ˜ ( s γ , t λ ) then Γ ( x α , y β ) c ⊆ Γ ( s γ , t λ ) c since otherwise there is M ∈ Γ ( x α , y β ) c and M ∉ Γ ( s γ , t λ ) c . So δ ˜ ( x α , y β ) ⊑ ρ ˜ ( x α , y β ) ⊑ ρ ˜ ( s γ , t λ ) ⊑ M ⊑ δ ˜ ( x α , y β ) ⊑ ρ ˜ ( x α , y β ) , which is a contradiction. Thus, Γ ( x α , y β ) c ⊆ Γ ( s γ , t λ ) c and hence δ ˜ ( x α , y β ) ⊑ δ ˜ ( s γ , t λ ) . By similar technique, ρ ˜ ( s γ , t λ ) ⊑ ρ ˜ ( x α , y β ) iff δ ˜ ( s γ , t λ ) ⊑ δ ˜ ( x α , y β ) . Therefore, δ ˜ ( x α , y β ) = δ ˜ ( s γ , t λ ) iff ρ ˜ ( x α , y β ) = ρ ˜ ( s γ , t λ ) . □

Theorem 2.24. Let (Z, Γ, Ψ) be a soft-SL _US . For the soft topological space (Z, ϑ _Γ, Ψ) the following are equivalent:

H is soft neighborhood of the soft point x ^α;

Proof: The proof is follows from Definition 1.7, Corollary 2.12 and definition of ρ ˜ . □

Definition 2.25. Let φ _ψ : (Z, Γ _Z , Ψ _Z ) → (W, Γ _W , Ψ _W ). A soft mapping φ _ψ is soft semi-linear uniformly continuous iff for every N ∈ Γ _W , there is M ∈ Γ _Z with for every x ^α, y ^β ∈ SP (Z), if ρ ˜ ( x α , y β ) ⊑ M , then ρ ˜ ( φ ψ ( x α ) , φ ψ ( y β ) ) ⊑ N .

Theorem 2.26. Every soft semi-linear uniformly continuous mapping is soft continuous.

Proof: Let φ _ψ : (Z, Γ _Z , Ψ _Z ) → (W, Γ _W , Ψ _W ) be soft semi-linear uniformly continuous. Suppose x α ∈ ˜ Z ˜ and B ( φ ψ ( x α ) , N ) be a soft neighborhood of φ _ψ (x ^α) with respect to ϑ _{Γ W} . By soft semi linear uniform continuity there is M ∈ Γ _Z such that if ρ ˜ ( x α , y β ) ⊑ M , then ρ ˜ ( φ ψ ( x α ) , φ ψ ( y β ) ) ⊑ N . Therefore, φ ψ ( B ( x α , M ) ) ⊑ B ( φ ψ ( x α ) , N ) . □

Proposition 2.27. Let φ _ψ : (Z, Γ _Z , Ψ _Z ) → (W, Γ _W , Ψ _W ). A soft mapping φ _ψ is soft semi-linear uniformly continuous iff for every N ∈ Γ _W there is M ∈ Γ _Z with for every x ^α, y ^β ∈ SP (Z), if δ ˜ ( x α , y β ) ⊑ M , then δ ˜ ( φ ψ ( x α ) , φ ψ ( y β ) ) ⊑ N .

Proof: Let N ∈ Γ _W . There is P ∈ Γ _Z such that if ρ ˜ ( x α , y β ) ⊑ P , then ρ ˜ ( φ ψ ( x α ) , φ ψ ( y β ) ) ⊑ N . Let M be a proper subset of P and x ^α, y ^β ∈ SP (Z). If δ ˜ ( x α , y β ) ⊑ M , then δ ˜ ( x α , y β ) ⊏ P and hence ρ ˜ ( x α , y β ) ⊑ P . Therefore, δ ˜ ( φ ψ ( x α ) , φ ψ ( y β ) ) ⊑ ρ ˜ ( φ ψ ( x α ) , φ ψ ( y β ) ) ⊑ N .

To prove the sufficiency, Let N ∈ Γ _W . Take P a proper subset of N, by assumption, there is M ∈ Γ _Z with for every x ^α, y ^β ∈ SP (Z), if ρ ˜ ( x α , y β ) ⊑ M , then δ ˜ ( φ ψ ( x α ) , φ ψ ( y β ) ) ⊑ P ⊏ N . Therefore, ρ ˜ ( φ ψ ( x α ) , φ ψ ( y β ) ) ⊑ N .□

3 Soft proximity spaces

In this section, we employ the members of soft-GSL _US to introduce a soft proximity space and a soft filter then we give a relation between them.

Definition 3.1. [12] Let ξ be a binary relation on S (Z, Ψ). If for every H, G, F ∈ S (Z, Ψ) the following are satisfied:

Φ ξ ¯ H ;

Then ξ is called a proximity of soft sets on Z and (Z, ξ, Ψ) is soft proximity space. If the soft sets H, G ∈ S (Z, Ψ) are ξ-related we will denoted by HξG, otherwise we will denoted by H ξ ¯ G .

Theorem 3.2. [12] Let (Z, ξ, Ψ) be a soft proximity space and define the mapping H → H ¯ ξ by H ¯ ξ = ⨆ { x α ∈ ˜ Z ˜ : x α ξ H } . Then, the family ϑ ( ξ ) = { H ∈ S ( Z , Ψ ) : H c ¯ ξ = H c } is a soft topology on Z.

Theorem 3.3. Let (Z, Γ, Ψ) be a soft-GSL _US and H, G ∈ S (Z, Ψ). Then, we define a proximity of soft sets on Z by: HξG iff B ( H , M ) ⊓ G ≠ Φ for every M ∈ Γ.

Proof: We need to show that ξ satisfies axioms of Definition 3.1:

i . Clearly, Φ ξ ¯ H .

ii . Let M ∈ Γ. If H ⊓ G ≠ Φ, then there is x α ∈ Z ˜ with x α ∈ ˜ H ⊓ G . Since Δ ⊑ M, then B ( H , M ) ⊓ G ≠ Φ .

iii . It follows from the fact M = M^-1 for every M ∈ Γ.

iv . It is clear that if Hξ (G ⊔ F) then HξG or HξF. To prove the sufficiency, suppose that H ξ ¯ G and H ξ ¯ F . There are M₁, M₂ ∈ Γ with ( x α , y β ) ∉ ˜ M 1 and ( x α , z γ ) ∉ ˜ M 2 for every x α ∈ ˜ H , y β ∈ ˜ G and z γ ∈ ˜ F . Note that, ( x α , t λ ) ∉ ˜ M 1 ⊓ M 2 whenever t λ ∈ ˜ G ⊔ F . Therefore, H ξ ¯ ( G ⊔ F ) .

v . If H ξ ¯ G , there are M, N ∈ Γ with N ∘ N ⊑ M and B ( H , M ) ⊓ G = Φ . Put F = Z ˜ - B ( H , N ) . To show H ξ ¯ F , let y β ∈ ˜ B ( H , N ) ⊓ F . Then, there is x α ∈ ˜ H with ( x α , y β ) ∈ ˜ N , which is a contradiction. Now, to show G ξ ¯ ( Z ˜ - F ) , let y β ∈ ˜ B ( G , N ) ⊓ ( Z ˜ - F ) . Then there is x α ∈ ˜ G with ( x α , y β ) ∈ ˜ N . Since ( t λ , y β ) ∈ ˜ N for some t λ ∈ ˜ H , then x α ∈ ˜ B ( H , M ) . Therefore, B ( H , M ) ⊓ G ≠ Φ , which is a contradiction.

Corollary 3.4. Let (Z, Γ, Ψ) be a soft-GSL _US . Then, H ξ ¯ G iff there is M ∈ Γ with B ( H , M ) ⊓ B ( G , M ) = Φ .

Proof: It follows directly from Theorems 2.17 and 3.3. □

Corollary 3.5. Let (Z, Γ, Ψ) be a soft-GSL _US with a soft base Θ and H, G ∈ S (Z, Ψ). Then:

HξG iff B ( H , M ) ⊓ G ≠ Φ for every M ∈ Θ;

Theorem 3.6. Let (Z, Γ, Ψ) be a soft-GSL _US . Then ϑ (ξ) = ϑ _Γ.

Proof: By Corollary 2.14 and Theorem 3.2 we show that the soft closed set in two topologies are coincides. Let H ∈ S (Z, Ψ), then

x α ∈ ˜ H ¯ ϑ Γ iff H ⊓ B ( x α , M ) ≠ Φ for every M ∈ Γ

iff M ⊓ (x ^α × H) ≠ Φ for every M ∈ Γ

iff x ^αξH

iff x α ∈ ˜ H ¯ ϑ ( ξ ) . □

Proposition 3.7. Let (Z, Γ, Ψ) be a soft-GSL _US . If H ξ ¯ G , then G ⊑ Int ( Z ˜ - H ) .

Proof: Let x α ∈ ˜ G . Then x α ξ ¯ H and, by Proposition 2.16, there is M ∈ Γ with B ( x α , M ) ⊓ H = Φ . Then B ( x α , M ) ⊑ ( Z ˜ - H ) so by Proposition 2.13, x α ∈ ˜ Int ( Z ˜ - H ) .□

Definition 3.8. [35] Let F ˜ ⊑ S ( Z , Ψ ) . If F ˜ satisfy the following:

Φ ∉ F ˜ ;

Then F ˜ is called a soft filter on Z.

Theorem 3.9. Let (Z, Γ, Ψ) be a a soft-GSL _US . If Φ ≠ G ∈ S (Z, Ψ), then F ˜ ( G ) = { H ∈ S ( Z , Ψ ) : B ( G , M ) ⊑ H for some M ∈ Γ } is a soft filter on Z.

Proof: We need to show that F ˜ ( G ) satisfies axioms of Definition 3.8:

i . If H ∈ F ˜ ( G ) , then B ( G , M ) ⊑ H for some M ∈ Γ, hence H ≠ Φ.

ii . If H ∈ F ˜ ( G ) with H ⊑ K, then B ( G , M ) ⊑ H for some M ∈ Γ, so B ( G , M ) ⊑ K and hence K ∈ F ˜ ( G ) .

iii . If H , K ∈ F ˜ ( G ) , then B ( G , M ) ⊑ H and B ( G , N ) ⊑ K for some M, N ∈ Γ, so B ( G , M ⊓ N ) ⊑ H ⊓ K and hence H ⊓ K ∈ F ˜ ( G ) . □

The following proposition, which follows directly from Theorem 3.9, will give some properties of F ˜ ( G ) .

Proposition 3.10. Let (Z, Γ, Ψ) be a soft-GSL _US . Then:

If H ∈ F ˜ ( G ) , then G ⊑ H;

Proposition 3.11. Let (Z, Γ, Ψ) be a soft-GSL _US . Then G ∈ F ˜ ( x α ) iff G ∈ ϑ _Γ.

Proof: Let x α ∈ ˜ G . Then, From Theorems 2.11 and 3.9, G ∈ F ˜ ( x α ) iff there is M ∈ Γ with B ( x α , M ) ⊑ G iff G ∈ ϑ _Γ. □

Proposition 3.12. Let (Z, Γ, Ψ) be a soft-GSL _US with the soft base Θ. For Φ ≠ G ∈ S (Z, Ψ), then { B ( G , M ) : M ∈ Θ } is a base of F ˜ ( G ) .

Proof: For every M ∈ Θ, B ( G , M ) is a member of F ˜ ( G ) . If H ∈ F ˜ ( G ) then, by Corollary 3.5, there is M ∈ Θ with B ( G , M ) ⊓ ( Z ˜ - H ) = Φ . Now, if x α ∈ ˜ B ( G , M ) , then ( x α , y β ) ∈ ˜ M for some y β ∈ ˜ G and so x α ∉ ˜ ( Z ˜ - H ) . Therefore, B ( G , M ) ⊑ H . □

The proof of the following proposition follows directly from Theorems 3.9 and 3.3.

Proposition 3.13. Let (Z, Γ, Ψ) be a a soft-GSL _US . If Φ ≠ G ∈ S (Z, Ψ), then H ∈ F ˜ ( G ) iff G ξ ¯ ( Z ˜ - H ) .

4 Application of soft semi-linear uniform spaces in the classification of digital images

In this section, we investigate the natural transformation of a soft semi-linear uniform space to a soft proximity to introduce the perceptual application of soft semi-linear uniform structures. Peters et al. ([22, 23]) introduced the notion of perceptual nearness via probe map and employed it in the classification of digital images. Let ℝ (the set of all real numbers) be the initial universe and Z be a set of pixels of a digital picture. A probe map φ : Z → P ( ℝ ) is defined by φ ( x ) = { φ 1 ( x ) , φ 2 ( x ) , ⋯ , φ n ( x ) } which gives different feature values φ _n  (x) of a pixel x. Here, φ 1 , φ 2 , ⋯ , φ n : Z → ℝ are mappings that describe a pixel x perceptually (see [23]). Two sets are perceptually near iff they are never empty and they contain pairs of perceived objects that have descriptions within some tolerance of each other.

Perceptual objects that have the same appearance are considered perceptually near each other, i . e ., objects with matching descriptions. A description is a set of values of mappings representing features of an object. To clarify that, assume the description of an object consists of one mapping value. For example, let b ∈ L, b′ ∈ L′ be books contained in two libraries L, L′ and φ (b) = number of pages in b, where φ is a sample map representing book length and book length is a feature of a book. If φ (b) = φ (b′), then book b is near book b′ (see [22]).

Let H ⊆ Z. Then H can be viewed as a soft set on the universe ℝ by H = { ( x , φ ( x ) ) , x ∈ H , φ ( x ) ∈ P ( ℝ ) } , where φ (x) = {φ₁ (x) , φ₂ (x) , ⋯ , φ _n  (x)} .

In the next example, we illustrate how we can construct visual soft sets of a digital image and investigate soft semi-linear uniform space to classify digital images. A similar approach to classify digital images via rough sets can be found in ([28, 31–33]).

Example 4.1. Consider the image of a butterfly in Fig. 1(A). An extracted part of it is shown in Fig. 1(B), which we will consider on the universe ℝ . Assume that the feature value-color for classification of images and the set of pixels in Fig. 1(B) is Z. The color strength of each pixel x can be described as φ (x) = {φ _r  (x) , φ _g  (x) , φ _b  (x)} , where φ _r  (x) , φ _g  (x) , φ _b  (x) represent the red, green and blue intensity values of the pixel x, respectively. Each intensity value is on a scale of 0 to 255. The probe mapping of each pixel represents its RGB value. Define a map Π : SP ( Z ) × SP ( Z ) → ℝ as: Π ( p , q ) = max { ∣ φ r ( p ) - φ r ( q ) ∣ , ∣ φ g ( p ) - φ g ( q ) ∣ , ∣ φ b ( p ) - φ b ( q ) ∣ } . Define Θ _ɛ = {(x, y) ∈ SP (Z) × SP (Z) : Π (x, y) < ɛ} ,

OPEN IN VIEWER

Fig. 1

Digital image of a butterfly.

for every ɛ ∈ ℝ and 0 < ɛ < 2 .

Then {Θ _ɛ : 0< ɛ < 2 } is a soft base for a soft semi-linear uniform Γ on Z. Two subsets H and G of SP (Z) are called perceptually near (Hξ _ΓG) if the following holds: Hξ _ΓG iff (H × G) ⊓ Θ _ɛ ≠ ∅ , for all Θ _ɛ ∈ Γ.

Here ξ _Γ is a soft proximity on Z.

Let A and B denote two sets of pixels, as shown in Fig. 1(B). Then A and B are two soft sets on the universe ℝ with: A = { ( x , φ ( x ) ) , x ∈ A , φ ( x ) ∈ P ( ℝ ) } , and B = { ( x , φ ( x ) ) , x ∈ B , φ ( x ) ∈ P ( ℝ ) } , where φ (x) = {φ _r  (x) , φ _g  (x) , φ _b  (x)} is the probe mapping which gives color strength of x ∈ A, x ∈ B and φ _r  (x) , φ _g  (x) , φ _b  (x) ∈ {0, 1, 2, 3, …, 255}. We have chosen RGB values as a feature values of elements (pixels). In Fig. 1(B), we may verify that the sets B and C are perceptually near. Also, the sets A and C are perceptually far from each other as the RGB values of every element in C have a difference more than 2 from RGB values of each element of B . In this way, we can classify digital images by using different feature values. We can distinguish different kinds of objects in the picture on a digital platform by using far (not near) relation ξ ¯ Γ .

In this manuscript, we learned about the concept of soft-SL _US and its generalization, as well as some of its most important properties. The definition of soft-SL _US allowed us to define the two soft maps ρ ˜ and δ ˜ on SP (Z) × SP (Z) then we employed them to define when a soft mapping φ _ψ is soft semi-linear uniformly continuous. Following that, we investigated the members of soft-GSL _US to introduce a soft proximity space then we distinguished this work from previous research in soft uniformity by demonstrating the perceptual application of soft semi-linear uniform structures via the natural transformation of a soft-SL _US to a soft proximity.

Since the fixed point theory is important in many areas of mathematics and applied sciences including mathematical models, optimization and economic theories, we intend to investigate the soft maps ρ ˜ and δ ˜ to define the soft contraction for studying the fixed point.

Declarations

Conflict of intersect. The authors declare that there is no conflict of interests regarding the publication of this article.