Soft semi-linear uniform spaces and their perceptual application

Abstract

In this paper, we use the soft set theory and the concept of semi-linear uniform spaces to introduce the notion of soft semi-linear uniform spaces with its generalization, briefly soft-GSL_US. We investigate some properties of soft topology that induced by soft-GSL_US. Also, we use the members of soft-GSL_US to define a soft proximity space and a soft filter then we establish the relationships between them. Finally, we give the perceptual application of soft semi-linear uniform structures by employing the natural transformation of a soft semi-linear uniform space to a soft proximity.

Keywords

Soft set soft point soft topology soft semi-linear uniform spaces soft proximity

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. $\ddot{O}$ 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 $\tilde{ρ}$ and $\tilde{δ}$ 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;

A soft set H is called absolute soft set, denoted by $\tilde{Z}$ , if H (e) = Z for every e ∈ Ψ;

H is a soft subset of G, denoted by H ⊑ G, if H (e) ⊆ G (e) for every e ∈ Ψ;

H = G, if H ⊑ G and G ⊑ H, and H is a soft proper subset of G, denoted by H ⊏ G, if H ⊑ G and H ≠ G;

A complement of H, denoted by H^c, where H^c : Ψ → P (Z) is a mapping defined by H^c (e) = Z - H (e) for every e ∈ Ψ. Clearly, (H^c) ^c = H;

A soft set F is the union of H and G, denoted by F = H ⊔ G, if F (e) = H (e) ∪ G (e) for every e ∈ Ψ;

A soft set F is the intersection of H and G, denoted by F = H ⊓ G, if F (e) = H (e) ∩ G (e) 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};

A soft point x^e belongs to a soft set H, denoted by $x^{e} \tilde{\in}$ H, if x ∈ H (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)

$φ_{ψ} (H) (k) = {\begin{matrix} ⋃_{e \in ψ^{- 1} (k)} φ (H (e)), if ψ^{- 1} (k) \neq φ \\ φ otherwise \end{matrix}}$ 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 $\tilde{Z}$ and the null soft set Φ are members of ϑ;

The union of an arbitrary number of soft sets in ϑ is a member of ϑ;

The intersection of a finite number of soft sets in ϑ is a member 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 $x^{e} \tilde{\in} G ⊑ H$ .

Definition 1.8. [11] Let (Z, ϑ, Ψ) be a soft topological space. If for every $x^{α} \neq y^{β} \tilde{\in} \tilde{Z}$ there are two soft open sets H and G with $x^{α} \tilde{\in} H$ , $y^{β} \tilde{\in} G$ 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 $x^{e} \tilde{\in} \tilde{Z}$ 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 $x^{e} \tilde{\in} \tilde{Z}$ .

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

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

For every $M \in U$ , there is $N \in U$ with N^-1 ⊑ M, where N^-1 (e) = {(x^α, y^β) : (y^β, x^α) ∈ N (e)} for every e ∈ Ψ;

For every $M \in U$ , there is $N \in U$ with N ∘ N ⊑ M;

If $M, N \in U$ , then $M ⊓ N \in U$ ;

If $M \in U$ and M ⊑ N then $N \in U$ .

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;

For every M ∈ D_SP(Z), M = M^-1.

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 $M (e_{1}) = Δ (e_{1}) \cup {(x_{1}^{e_{1}}, x_{2}^{e_{1}}), (x_{2}^{e_{1}}, x_{1}^{e_{1}})}$ and $M (e_{2}) = Δ (e_{2}) \cup {(x_{1}^{e_{2}}, x_{2}^{e_{2}}), (x_{2}^{e_{2}}, x_{1}^{e_{2}})}$ . 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)};

Define Mⁿ inductively by the formulas M¹ = M and Mⁿ = M^n-1 ∘ M.

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);

Mⁿ ∈ D_SP(Z) holds for every $n \in ℕ$ .

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);

For every M∈ Γ, there is N ∈ Γ with N∘ N ⊑ M ;

$\underset{M \in Γ}{⊓} M = Δ$ ;

For every e ∈ Ψ, $⋃_{M \in Γ} M (e) = SP (Z) \times SP (Z) .$

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 $H_{x} (e) = {\begin{matrix} {x}, & e = x \\ φ, & otherwise \end{matrix}}$ . 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 $M_{e} (x) = {\begin{matrix} G_{e}, & x = e \\ φ, & otherwise \end{matrix}}$ . 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 $x^{α} \tilde{\in}$ $\tilde{Z}$ define $B (x^{α}, M) = {y^{β} \tilde{\in} \tilde{Z} : (x^{α}, y^{β}) \in M (e), \forall e \in Ψ}$ ;

For H ∈ S (Z, Ψ) define $B (H, M) = ⨆_{x^{α} \tilde{\in} H} B (x^{α}, M) .$

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 ;

$\underset{C \in Θ}{⊓} C = Δ .$

Theorem 2.11. If (Z, Γ, Ψ) is a soft-GSL_US, then $ϑ = {O ⊑ \tilde{Z} :$ for every $x^{α} \tilde{\in} O, \exists M \in Γ$ with $B (x^{α}, M) ⊑ O}$

is a soft topology on Z.

Proof: It is obvious that $Φ, \tilde{Z} \in ϑ$ . Let O₁, O₂ ∈ ϑ and $x^{α} \tilde{\in} O_{1} ⊓ O_{2}$ . 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

$y^{β} \tilde{\in} B (x^{α}, M_{1}) ⊓ B (x^{α}, M_{2})$ 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 $y^{β} \tilde{\in} B (x^{α}, M_{1} ⊓ M_{2})$ . Therefore, $B (x^{α}, M_{1} ⊓ M_{2}) ⊑ O_{1} ⊓ O_{2}$ and hence O₁ ⊓ O₂ ∈ ϑ. Finally, let O_i ∈ ϑ and $x^{α} \tilde{\in} \underset{i}{⊔} O_{i}$ . Then there is M ∈ Γ with $B (x^{α}, M) ⊑ O_{i_{\circ}} ⊑ \underset{i}{⊔} O_{i}$ and so $\underset{i}{⊔} O_{i} \in ϑ$ . □

Corollary 2.12. If (Z, Γ, Ψ) is a soft-SL_US, then $ϑ = {O ⊑ \tilde{Z} :$ for every $x^{α} \tilde{\in} O, \exists M \in Γ$ 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 $\tilde{O} = {x^{α} \tilde{\in} \tilde{Z} : \exists M \in Γ$ 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 $x^{α} \tilde{\in} \tilde{O}$ . There are M, P ∈ Γ with $B (x^{α}, M) ⊑ H$ and P ∘ P ⊑ M. Then for every $y^{β} \in B (x^{α}, P)$ , $B (y^{β}, P) ⊑$ $B (x^{α}, M) ⊑ H$ . Hence, $B (x^{α}, P) ⊑ \tilde{O}$ and so $\tilde{O}$ is soft open. Now, since every soft open set O ⊑ H belongs to $\tilde{O}$ 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 $x^{α} \tilde{\in} \tilde{Z}$ and H ∈ S (Z, Ψ), $x^{α} \tilde{\in} \bar{H}$ iff $H ⊓ B (x^{α}, M)$ ≠Φ for every M ∈ Γ;

For every M ∈ Γ and H ∈ S (Z, Ψ) with H × H ⊑ M, then $\bar{H} \times \bar{H} ⊑ M^{3}$ .

Proof: (i) The proof is obvious.

(ii) Let $x^{α}, y^{β} \tilde{\in} \bar{H}$ . Then there are ${x_{1}}^{α^{'}}, {y_{1}}^{β^{'}} \tilde{\in} \tilde{Z}$ with ${x_{1}}^{α^{'}} \tilde{\in} H ⊓ B (x^{α}, M)$ and ${y_{2}}^{β^{'}} \tilde{\in} H ⊓ B (y^{β}, M)$ 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, $\bar{H} \times \bar{H} ⊑ M^{3}$ . □

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 $x^{α} \neq y^{β} \tilde{\in}$ $\tilde{Z} .$ There are M, N ∈ Γ with N ∘ N ⊑ M and (x^α, y^β) ∉ M (e) for some e ∈ Ψ. If $B (x^{α}, N) ⊓ B (y^{β}, N) \neq Φ$ , then there is $z^{γ} \tilde{\in} B (x^{α}, N) ⊓ B (y^{β}, N)$ 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 $\bar{H}$ $= ⊓ {B (H, M) :$ M ∈ Γ} .

Proof: $x^{α} \tilde{\in} \bar{H}$ iff ∃ y^β $\tilde{\in}$ $\tilde{Z}$ with $y^{β} \tilde{\in}$ $B (x^{α}, M)$ and $y^{β} \tilde{\in}$ H for every M ∈ Γ iff ∃ y^β $\tilde{\in} H$ with (x^α, y^β) ∈ M (e) for every e ∈ Ψ and M ∈ Γ iff x^α $\tilde{\in}$ $B (H, M)$ 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 $x^{α} \tilde{\in} B (H, N) ⊓ B (G, N)$ , then there are (x^α, y^β), (x^α, z^γ) ∈ N (e) for every e ∈ Ψ with $y^{β} \tilde{\in} H$ and $z^{γ} \tilde{\in} G$ . Thus, $z^{γ} \tilde{\in} B (H, M)$ , 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

$B (H ⊔ G, M) = B (H, M) ⊔ B (G, M)$

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

$B (H ⊔ G, M) ⊓ B (F, M) = (B (H, M) ⊓ B (F, M)) ⊔ (B (G, M) ⊓ B (F, M))$

$B (H ⊓ G, M) ⊔ B (F, M) ⊑ (B (H, M) ⊔ B (F, M)) ⊓ (B (G, M) ⊔ B (F, M))$ ;

Proof: (i) Let $x^{α} \tilde{\in} B (H ⊔ G, M)$ . Then, there is $y^{β} \tilde{\in} H ⊔ G$ with (x^α, y^β) ∈ M (e) for every e ∈ Ψ. If $y^{β} \tilde{\in} H$ then $x^{α} \tilde{\in} B (H, M) ⊔ B (G, M)$ . 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 $Γ_{(x^{α}, y^{β})} = {M \in Γ : (x^{α}, y^{β}) \tilde{\in} M}$ and ${Γ^{c}}_{(x^{α}, y^{β})} = {M \in Γ : (x^{α}, y^{β}) \tilde{\notin} M}$ Then define $\tilde{ρ}$ and $\tilde{δ}$ on SP (Z) × SP (Z) by $\tilde{ρ} (x^{α}, y^{β}) = ⊓ {M : M \in Γ_{(x^{α}, y^{β})}}$ and $\begin{matrix} \tilde{δ} (x^{α}, y^{β}) = {\begin{matrix} ⨆ {M : M \in Γ_{(x^{α}, y^{β})}^{c}} : x^{α} \neq y^{β} \\ Δ : x^{α} = y^{β} \end{matrix}} \end{matrix}$

Remark 2.21. If $(s^{γ}, t^{λ}) \tilde{\in} \tilde{ρ} (x^{α}, y^{β})$ then $(s^{γ}, t^{λ}) \tilde{\in} M$ for every M ∈ Γ_(x^α,y^β). Also, if $(s^{γ}, t^{λ}) \tilde{\in} \tilde{δ} (x^{α}, y^{β})$ then $(s^{γ}, t^{λ}) \tilde{\in} M$ 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

$\tilde{ρ} (x^{α}, y^{β}) = \tilde{ρ} (y^{β}, x^{α})$ and $\tilde{δ} (x^{α}, y^{β}) = \tilde{δ} (y^{β}, x^{α})$ ;

$\tilde{ρ} (x^{α}, y^{β}) ⊑ M$ for every M ∈ Γ_(x^α,y^β) and $M ⊑ \tilde{δ} (x^{α}, y^{β})$ for every M ∈ Γ^c_(x^α,y^β);

$\tilde{δ} (x^{α}, y^{β}) ⊑ \tilde{ρ} (x^{α}, y^{β})$ ;

If (s^γ, t^λ) ∈ SP (Z) × SP (Z) and $(s^{γ}, t^{λ}) \tilde{\in} \tilde{ρ} (x^{α}, y^{β})$ , then $\tilde{ρ} (s^{γ}, t^{λ}) ⊑ \tilde{ρ} (x^{α}, y^{β})$ ;

If (s^γ, t^λ) ∈ SP (Z) × SP (Z) and $(s^{γ}, t^{λ}) \tilde{\in} \tilde{δ} (x^{α}, y^{β})$ , then $\tilde{δ} (s^{γ}, t^{λ}) ⊑ \tilde{δ} (x^{α}, y^{β})$ ;

If M ∈ Γ with $\tilde{δ} (x^{α}, y^{β}) ⊑ M ⊑ \tilde{ρ} (x^{α}, y^{β})$ , then $\tilde{δ} (x^{α}, y^{β}) = M$ or $\tilde{ρ} (x^{α}, y^{β}) = M$ .

Theorem 2.23. Let (Z, Γ, Ψ) be a soft-SL_US. Then $\tilde{δ} (x^{α}, y^{β}) = \tilde{δ} (s^{γ}, t^{λ})$ iff $\tilde{ρ} (x^{α}, y^{β}) = \tilde{ρ} (s^{γ}, t^{λ})$ .

Proof: First, we show $\tilde{δ} (x^{α}, y^{β}) ⊑ \tilde{δ} (s^{γ}, t^{λ})$ iff $\tilde{ρ} (x^{α}, y^{β}) ⊑ \tilde{ρ} (s^{γ}, t^{λ})$ . Suppose $\tilde{δ} (x α, y^{β}) ⊑ \tilde{δ} (s^{γ}, t^{λ})$ then $Γ_{(s^{γ}, t^{λ})} \subseteq Γ_{(x^{α}, y^{β})}$ and hence $\tilde{ρ} (x^{α}, y^{β}) ⊑ \tilde{ρ} (s^{γ}, t^{λ})$ . Now, if $\tilde{ρ} (x^{α}, y^{β}) ⊑ \tilde{ρ} (s^{γ}, t^{λ})$ then $Γ_{(x^{α}, y^{β})}^{c} \subseteq Γ_{(s^{γ}, t^{λ})}^{c}$ since otherwise there is $M \in Γ_{(x^{α}, y^{β})}^{c}$ and $M \notin Γ_{(s^{γ}, t^{λ})}^{c}$ . So $\tilde{δ} (x^{α}, y^{β}) ⊑ \tilde{ρ} (x^{α}, y^{β}) ⊑ \tilde{ρ} (s^{γ}, t^{λ}) ⊑ M ⊑ \tilde{δ} (x^{α}, y^{β}) ⊑ \tilde{ρ} (x^{α}, y^{β})$ , which is a contradiction. Thus, $Γ_{(x^{α}, y^{β})}^{c} \subseteq Γ_{(s^{γ}, t^{λ})}^{c}$ and hence $\tilde{δ} (x^{α}, y^{β}) ⊑ \tilde{δ} (s^{γ}, t^{λ})$ . By similar technique, $\tilde{ρ} (s^{γ}, t^{λ}) ⊑ \tilde{ρ} (x^{α}, y^{β})$ iff $\tilde{δ} (s^{γ}, t^{λ}) ⊑ \tilde{δ} (x^{α}, y^{β})$ . Therefore, $\tilde{δ} (x^{α}, y^{β}) = \tilde{δ} (s^{γ}, t^{λ})$ iff $\tilde{ρ} (x^{α}, y^{β}) = \tilde{ρ} (s^{γ}, t^{λ})$ . □

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;

There is M ∈ Γ with B (x^α, M) ⊑ H;

There is M ∈ Γ with ${y^{β} \tilde{\in} \tilde{Z} : \tilde{ρ} (x^{α}, y^{β}) ⊑ M} ⊑ H$ .

Proof: The proof is follows from Definition 1.7, Corollary 2.12 and definition of $\tilde{ρ}$ . □

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 $\tilde{ρ} (x^{α}, y^{β}) ⊑ M$ , then $\tilde{ρ} (φ_{ψ} (x^{α}), φ_{ψ} (y^{β})) ⊑ N$ .

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 $x^{α} \tilde{\in} \tilde{Z}$ 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 $\tilde{ρ} (x^{α}, y^{β}) ⊑ M$ , then $\tilde{ρ} (φ_{ψ} (x^{α}), φ_{ψ} (y^{β})) ⊑ N$ . 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 $\tilde{δ} (x^{α}, y^{β}) ⊑ M$ , then $\tilde{δ} (φ_{ψ} (x^{α}), φ_{ψ} (y^{β})) ⊑ N$ .

Proof: Let N ∈ Γ_W. There is P ∈ Γ_Z such that if $\tilde{ρ} (x^{α}, y^{β}) ⊑ P$ , then $\tilde{ρ} (φ_{ψ} (x^{α}), φ_{ψ} (y^{β})) ⊑ N$ . Let M be a proper subset of P and x^α, y^β ∈ SP (Z). If $\tilde{δ} (x^{α}, y^{β}) ⊑ M$ , then $\tilde{δ} (x^{α}, y^{β}) ⊏ P$ and hence $\tilde{ρ} (x^{α}, y^{β}) ⊑ P$ . Therefore, $\tilde{δ} (φ_{ψ} (x^{α}), φ_{ψ} (y^{β})) ⊑ \tilde{ρ} (φ_{ψ} (x^{α}), φ_{ψ} (y^{β})) ⊑ N$ .

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 $\tilde{ρ} (x^{α}, y^{β}) ⊑ M$ , then $\tilde{δ} (φ_{ψ} (x^{α}), φ_{ψ} (y^{β})) ⊑ P ⊏ N$ . Therefore, $\tilde{ρ} (φ_{ψ} (x^{α}), φ_{ψ} (y^{β})) ⊑ N$ .□

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:

$Φ \bar{ξ} H$ ;

If $H \bar{ξ} G$ , then H ⊓ G = Φ;

If HξG, then GξH;

Hξ (G ⊔ F) iff HξG or HξF;

If $H \bar{ξ} G$ , then there is F ∈ S (Z, Ψ) with $H \bar{ξ} F$ and $G \bar{ξ} (\tilde{Z} - F)$ .

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 $H \bar{ξ} G$ .

Theorem 3.2. [12] Let (Z, ξ, Ψ) be a soft proximity space and define the mapping $H \to {\bar{H}}^{ξ}$ by ${\bar{H}}^{ξ} = ⨆ {x^{α} \tilde{\in} \tilde{Z} : x^{α} ξ H}$ . Then, the family $ϑ (ξ) = {H \in S (Z, Ψ) : {\bar{H^{c}}}^{ξ} = H^{c}}$ 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 $B (H, M) ⊓ G \neq Φ$ for every M ∈ Γ.

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

i . Clearly, $Φ \bar{ξ} H$ .

ii . Let M ∈ Γ. If H ⊓ G ≠ Φ, then there is $x^{α} \in \tilde{Z}$ with $x^{α} \tilde{\in} H ⊓ G$ . Since Δ ⊑ M, then $B (H, M) ⊓ G \neq Φ$ .

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 $H \bar{ξ} G$ and $H \bar{ξ} F$ . There are M₁, M₂ ∈ Γ with $(x^{α}, y^{β}) \tilde{\notin} M_{1}$ and $(x^{α}, z^{γ}) \tilde{\notin} M_{2}$ for every $x^{α} \tilde{\in} H, y^{β} \tilde{\in} G$ and $z^{γ} \tilde{\in} F$ . Note that, $(x^{α}, t^{λ}) \tilde{\notin} M_{1} ⊓ M_{2}$ whenever $t^{λ} \tilde{\in} G ⊔ F$ . Therefore, $H \bar{ξ} (G ⊔ F)$ .

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

Corollary 3.4. Let (Z, Γ, Ψ) be a soft-GSL_US. Then, $H \bar{ξ} G$ 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 $B (H, M) ⊓ G \neq Φ$ for every M ∈ Θ;

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

$H \bar{ξ} G$ iff there is M ∈ Θ with $B (H, M) ⊓ B (G, M) = Φ$ .

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

$x^{α} \tilde{\in} {\bar{H}}^{ϑ_{Γ}}$ iff $H ⊓ B (x^{α}, M) \neq Φ$ for every M ∈ Γ

iff M ⊓ (x^α × H) ≠ Φ for every M ∈ Γ

iff x^αξH

iff $x^{α} \tilde{\in} {\bar{H}}^{ϑ (ξ)}$ . □

Proposition 3.7. Let (Z, Γ, Ψ) be a soft-GSL_US. If $H \bar{ξ} G$ , then $G ⊑ Int (\tilde{Z} - H)$ .

Proof: Let $x^{α} \tilde{\in} G$ . Then $x^{α} \bar{ξ} H$ and, by Proposition 2.16, there is M ∈ Γ with $B (x^{α}, M) ⊓ H = Φ$ . Then $B (x^{α}, M) ⊑ (\tilde{Z} - H)$ so by Proposition 2.13, $x^{α} \tilde{\in} Int (\tilde{Z} - H)$ .□

Definition 3.8. [35] Let $\tilde{F} ⊑ S (Z, Ψ)$ . If $\tilde{F}$ satisfy the following:

$Φ \notin \tilde{F}$ ;

For every $G \in \tilde{F}$ and G ⊑ H, $H \in \tilde{F};$

For every $G, H \in \tilde{F}$ , $G ⊓ H \in \tilde{F}$ .

Then $\tilde{F}$ 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 $\tilde{F} (G) = {H \in S (Z, Ψ) : B (G, M) ⊑ H$ for some M ∈ Γ } is a soft filter on Z.

Proof: We need to show that $\tilde{F} (G)$ satisfies axioms of Definition 3.8:

i . If $H \in \tilde{F} (G)$ , then $B (G, M) ⊑ H$ for some M ∈ Γ, hence H ≠ Φ.

ii . If $H \in \tilde{F} (G)$ with H ⊑ K, then $B (G, M) ⊑ H$ for some M ∈ Γ, so $B (G, M) ⊑ K$ and hence $K \in \tilde{F} (G)$ .

iii . If $H, K \in \tilde{F} (G)$ , then $B (G, M) ⊑ H$ and $B (G, N) ⊑ K$ for some M, N ∈ Γ, so $B (G, M ⊓ N) ⊑ H ⊓ K$ and hence $H ⊓ K \in \tilde{F} (G)$ . □

The following proposition, which follows directly from Theorem 3.9, will give some properties of $\tilde{F} (G)$ .

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

If $H \in \tilde{F} (G)$ , then G ⊑ H;

If $H \in \tilde{F} (G)$ , then $(\tilde{Z} - G) \in \tilde{F} (\tilde{Z} - H)$ ;

If G ⊑ H, then $\tilde{F} (H) ⊑ \tilde{F} (G)$ ;

$\tilde{F} (G ⊔ H) = \tilde{F} (G) ⊔ \tilde{F} (H)$ ;

$\tilde{F} (G) ⊓ \tilde{F} (H) ⊑ \tilde{F} (G ⊓ H)$ ;

If $H \in \tilde{F} (G)$ , then there is $K \in \tilde{F} (G)$ with $H \in \tilde{F} (K)$ .

Proposition 3.11. Let (Z, Γ, Ψ) be a soft-GSL_US. Then $G \in \tilde{F} (x^{α})$ iff G ∈ ϑ_Γ.

Proof: Let $x^{α} \tilde{\in} G$ . Then, From Theorems 2.11 and 3.9, $G \in \tilde{F} (x^{α})$ 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 ${B (G, M) : M \in Θ}$ is a base of $\tilde{F} (G)$ .

Proof: For every M ∈ Θ, $B (G, M)$ is a member of $\tilde{F} (G)$ . If $H \in \tilde{F} (G)$ then, by Corollary 3.5, there is M ∈ Θ with $B (G, M) ⊓ (\tilde{Z} - H) = Φ$ . Now, if $x^{α} \tilde{\in} B (G, M)$ , then $(x^{α}, y^{β}) \tilde{\in} M$ for some $y^{β} \tilde{\in} G$ and so $x^{α} \tilde{\notin} (\tilde{Z} - H)$ . 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 $H \in \tilde{F} (G)$ iff $G \bar{ξ} (\tilde{Z} - H)$ .

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 $φ : Z \to P (ℝ)$ is defined by $φ (x) = {φ_{1} (x), φ_{2} (x), \dots, φ_{n} (x)}$ which gives different feature values φ_n (x) of a pixel x. Here, $φ_{1}, φ_{2}, \dots, φ_{n} : Z \to ℝ$ 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 $H = {(x, φ (x)), x \in H, φ (x) \in P (ℝ)},$ 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 $Π : SP (Z) \times SP (Z) \to ℝ$ as: $\begin{matrix} Π (p, q) = \max {∣ φ_{r} (p) - φ_{r} (q) ∣, \\ ∣ φ_{g} (p) - φ_{g} (q) ∣, ∣ φ_{b} (p) - φ_{b} (q) ∣} . \end{matrix}$ Define Θ_ɛ = {(x, y) ∈ SP (Z) × SP (Z) : Π (x, y) < ɛ} ,

Fig. 1

Digital image of a butterfly.

for every $ɛ \in ℝ$ 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: $A = {(x, φ (x)), x \in A, φ (x) \in P (ℝ)},$ and $B = {(x, φ (x)), x \in B, φ (x) \in P (ℝ)},$ 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 ${\bar{ξ}}_{Γ}$ .

5 Conclusion

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 $\tilde{ρ}$ and $\tilde{δ}$ 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 $\tilde{ρ}$ and $\tilde{δ}$ 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.

Footnotes

Acknowledgment

The publication of this paper was supported by Jordan University Research council.

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