Soft isomorphism for digital images and computational topological applications

Abstract

In this paper the concept of soft continuity is focused on for digital images by using soft sets which is defined on κ - adjacent digital images. Also the definitions of digital soft isomorphism and digital soft retraction are given. Some theorems are obtained deal with soft isomorphism and soft retraction for digital images and some numerical examples are presented in dimension 2. Finally digital soft retraction is obtained as a soft topological invariant.

Keywords

Digital image soft set soft continuous function soft retraction

1 Introduction

The literature of digital topology contains numerous theorems and results related with digitally continuous functions, digital homeomorphism, digital retractions, digital homotopy, digital fundamental groups, digital homology etc. For more details see [17 , 40–45]. In 1986, Rosenfeld [48] gave the notion of continuity for digital pictures. In addition, Boxer investigated some properties of continuous functions, homeomorphism (isomorphism) and homotopy for digital images in [22, 23].

On the other hand soft set theory was studied by several mathematicians [1–16 , 49–53]. The definition of soft set was given by Molodsov [39] in 1999. Various operations on soft sets was investigated by Maji [37]. In addition many properties of the soft set operations are investigated in [1, 5]. Soft mappings which are great importance in soft set theory are redefined in [4]. Also continuity of soft mappings was studied in [14, 53].

The motivation for this work is to construct the soft set structure in digital image and to define the soft continuity for soft sets over κ - adjacent digital images. The study of the concept of soft continuous functions occupied a wide range of their interests. However soft set theory had not been studied in digital images. So we prefer to focus on to investigate the properties and applications of soft continuity in digital images such as soft homeomorphism (in digital images, named soft isomorphism) and soft retraction.

The advantage of this paper is investigating the basic and algebraic topological properties by using soft structure in digital images unlike other studies. It will be a new tool and a new area both soft set theory and digital topology researchers.

This paper is composed of four sections. After the Introduction, in Section 2, we give some basic definitions related to digital images and soft set theory which will be used in this paper. Then, in Section 3, we present the soft set operations in digital images and in Section 4, give the soft isomorphism and soft retraction in digital images.

In this paper we present soft theoretical concepts for digital images. Through this study, suppose that $ℤ^{n}$ is an n- dimensional Euclidean spaces. A finite subset of $ℤ^{n}$ with an adjacency relation is considered as a digital image. Generally, in examples, we focus on 1- and 2- dimensional digital images by using κ = 2 and κ = 4, κ = 8 adjacency relations, respectively.

2 Prelimineries

In this section we recall some basic definitions and theorems from digital topology and soft set theory. At first we mention the adjacency relation which is used for digital images and then we recall the concept of digital image. Secondly we state the basic notions from soft set theory.

Definition 2.1 [31, 35]

Two points p and q in $ℤ$ are 2 - adjacent if |p - q|=1.

Two points p and q in $ℤ^{2}$ are 8 - adjacent if they are distinct and differ by at most 1 in each coordinate.

Two points p and q in $ℤ^{2}$ are 4 - adjacent if they are 8 - adjacent and differ by exactly one coordinate.

Two points p and q in $ℤ^{3}$ are 26 - adjacent if they are distinct and differ by at most 1 in each coordinate.

Two points p and q in $ℤ^{3}$ are 18 - adjacent if they are 26 - adjacent and differ in at most two coordinates.

Two points p and q in $ℤ^{3}$ are 6 - adjacent if they are 18 - adjacent and differ by exactly one coordinate.

Fig. 1

2 - adjanceny in $ℤ$ .

Fig. 2

4 - adjanceny and 8 - adjanceny in $ℤ^{2}$ .

Fig. 3

6 - adjanceny, 18 - adjanceny and 26 - adjanceny in $ℤ^{3}$ .

Definition 2.2 [35] Let $ℤ$ denote the set of integers. Then $ℤ^{n}$ is the set of lattice points in Euclidean n-dimensional space. A (binary) digital image is a finite subset of $ℤ^{n}$ with the adjacency relation κ.

Definition 2.3 [48] A k - path from x to y in X is a sequence $(x = x_{0}, x_{1}, x_{2}, \dots, x_{m - 1}, x_{m} = y)$ in X such that each point x_i is k - adjacent to x_i+1 for m ≥ 1 and 1 ≤ i ≤ m. The number m is called the length of this path.

Suppose that κ is an adjacency relation defined on $ℤ^{n}$ . A digital image $X \subset ℤ^{n}$ is κ - connected if and only if for every pair of points {x, y} ⊂ X, x ≠ y, there is a set {x₀, x₁, …, x_c} ⊂ X such that x = x₀, x_c = y and x_i and x_i+1 are κ - neighbors, i ∈ {0, 1, …, c - 1} [31].

Definition 2.4 [22, 23] Let X and Y be digital images such that $X \subset ℤ^{n_{0}}$ , $Y \subset ℤ^{n_{1}}$ . Then the digital function f : X → Y is a function which is defined between digital images.

Definition 2.5 [22, 23] Let X and Y be digital images such that $X \subset ℤ^{n_{0}}$ , $Y \subset ℤ^{n_{1}}$ . Assume that f : X → Y be a function. Let κ_i be an adjacency relation defined on $ℤ^{n_{i}}$ , i ∈ {0, 1}. f is called to be (κ₀, κ₁) - continuous if the image under f of every κ₀ - connected subset of X is κ₁ - connected.

Proposition 1 [22, 23] Let X and Y be digital images. Then the function f : X → Y is said to be (κ₀, κ₁) - continuous if and only if for every {x₀, y₀} ⊂ X such that x₀ and x₁ are κ₀ - adjacent, either f (x₀) = f (x₁) or f (x₀) and f (x₁) are κ₁ - adjacent.

Definition 2.6 [22] Let $a, b \in ℤ$ with a < b. A set of the form ${[a, b]}_{ℤ} = {z \in Z | a \leq z \leq b}$ is called a digital interval.

Definition 2.7 [22, 23] Let κ₀ and κ₁ be adjacency relation on X and Y, respectively. Two digital images (X, κ₀) and (Y, κ₁) are (κ₀, κ₁) - homeomorphic if there is a one-to-one and onto function f : X → Y that is (κ₀, κ₁) - continuous such that the inverse function f^-1 : Y → X is (κ₁, κ₀) - continuous .

In digital topology,it is suggested to use the terminology ‘isomorphic digital images’ rather than ‘homeomorphic digital images’ because, for example digital models of homeomorphic Euclidean sets need not be digitally homeomorphic. Also, all Euclidean simple closed curves are homeomorphic, but digital simple closed curves of differing cardinalities are not even of the same digital homotopy types [38].

Definition 2.8 [39] A pair (F, A) is said to be a soft set over the universe X, where F is a mapping given by F : A → P (X) and A ⊆ E. Any soft set (F, A) can be extended to a soft set of type (F, E), where F (e)≠ ∅ for all e ∈ A and F (e) =∅ for all e ∈ E ∖ A. S (X, E) indicates the family of all soft sets over X .

Definition 2.9 [1, 5, 37] Assume that (F, E) and (F′, E′) be two soft sets over X . It is said that (F, E) is a soft subset of (F′, E′) and it is denoted by $(F, E) \tilde{\subset} (F^{'}, E^{'})$ if

(1) E ⊆ E′, and

(2) F (a) ⊆ F′ (a) , for all a ∈ E .

(F, E) is said to be a soft super set of (F′, E′) , if (F′, E′) is a soft subset of (F, E) . It is symbolized by $(F, E) \tilde{\supset} (F^{'}, E^{'}) .$

Definition 2.10 [37] Let (F, E) and (F′, E′) be two soft sets over X . The union of (F, E) and (F′, E′) is soft set (F″, E″) , where E″ = E ∪ E′ and for all c ∈ E″, $f (x) = {\begin{matrix} F (c), & c∈ E-E′ \\ G (c), & c∈ E′ -E \\ F (c) \cup G (c), & c∈ E∩ E′ \end{matrix}$ It is written $(F, E) \tilde{\cup} (F^{'}, E^{'}) = (F^{″}, E^{″}) .$

Definition 2.11 [37] Let (F, E) and (F′, E′) be two soft sets over X . The intersection of (F, E) and (F′, E′) is a soft set (F″, E″) , where E″ = E ∩ E′ and for all c ∈ E″, F″ (c) = F (c) ∩ F′ (c) . It is written $(F, E) \overset{ˇ}{\cap} (F^{'}, E^{'}) = (F^{″}, E^{″}) .$

Definition 2.12 [5, 37] Let (F, E) be soft set over X . The relative complement of (F, E) is denoted by (F, E) ^c and is defined (F, E) ^c = (F^c, E) , where F^c : E → P (X) is a mapping given by F^c (e) = X - F (e) for all e ∈ E .

Definition 2.13 [37] Let (F, E) and (F′, E′) be two soft sets over X . Then (F, E) - (F′, E′) is a soft set (e, F (e)) : F (e) ∉ F′ (E′) , e ∉ E′ .

Definition 2.14 [37] Let (F, E) be soft set over X . Then

(1) (F, E) is called to be null soft set denoted by $\overset{ˇ}{\emptyset}$ if for every e∈ E, F (e) = ∅.

(2) (F, E) is called to be absolute soft set denoted by $\overset{ˇ}{E}$ if for every e ∈ E, F (e) = X .

3 Soft set structures in digital images

In this section we reorganize some notions from [32, 41] and [42].

Definition 3.1 Let κ be an adjacency relation on $ℤ^{n}$ and E be the set of parameters. If F is a mapping from E to all κ - adjacent subset of X, then the pair (F, E) _κ is called to be a κ - adjacent soft set over $X \subset ℤ^{n}$ , that is $F : E \to P (X),$ where P (X) denotes the power set of X . A κ - adjacent digital soft set will be denoted by (F, E) _κ .

Example 1 Consider the 2- adjacency relation on $ℤ$ . Let E = {e₁, e₂} be the parameter set and $X = {0, 1, 2} \subset ℤ$ be 2- adjacent digital image. Define the map F : E → P (X) by $\begin{matrix} F (e_{1}) & = & {0, 1, 2}, \\ F (e_{2}) & = & {{0, 1}, {1, 2}} . \end{matrix}$ Then 2- digital soft set (F, E) ₂ occurs as the following: $(F, E)_{2} = {(e_{1}, {0, 1, 2}), (e_{2}, {{0, 1}, {1, 2}})} .$

Example 2 Consider the digital image $\begin{matrix} X = {(0, 0), (1, 0), (2, 0), (0, 1), (0, - 1), (1, 1), \\ (2, 1), (2, - 1)} \subset ℤ^{2} \end{matrix}$ and, κ₁ = 4 and κ₂ = 8 adjacency relations on $ℤ^{2}$ . Let E = {e₁, e₂, e₃} be the parameter set. Define the function F_i : E → P (X) , i = 1, 2 by $\begin{matrix} F_{1} {(e_{1})}_{4} & = & {(1, 0), (2, 0)}, \\ F_{1} {(e_{2})}_{4} & = & {(0, 0), (1, 0)}, \\ F_{1} {(e_{3})}_{4} & = & {(1, 0)}; \end{matrix}$ $\begin{matrix} F_{2} {(e_{1})}_{8} & = & {(1, 0), (0, 1)}, \\ F_{2} {(e_{2})}_{8} & = & {(1, 0), (2, - 1)}, \\ F_{2} {(e_{3})}_{8} & = & {(1, 0), (2, - 1)} . \end{matrix}$ Then 4 - adjacent and 8 - adjacent digital soft sets are given by $\begin{matrix} {(F_{1}, E)}_{4} = {(e_{1}, {(1, 0), (2, 0)}), \\ (e_{2}, {(0, 0), (1, 0)}), (e_{3}, {(1, 0)})}, \\ {(F_{2}, E)}_{8} = {(e_{1}, {(1, 0), (0, 1)}), \\ (e_{2}, {(1, 0), (2, - 1)}), (e_{3}, {(1, 0), (2, - 1)})} . \end{matrix}$

Definition 3.2 Let $X \subset ℤ^{n}$ be a κ - adjacent digital image. If F (a) =∅ for every a ∈ A with (F, A) _κ is a digital soft set over digital image X, then (F, A) _κ is said to be a digital soft null set and symbolized by Φ_κ.

Definition 3.3 Let $X \subset ℤ^{n}$ be a κ - adjacent digital image. Assume that (F, A) _κ and (G, B) _κ be two κ - adjacent soft sets over X. Then the soft intersection of (F, A) _κ and (G, B) _κ is κ - adjacent soft set (H, C) _κ with C = A ∩ B and H (c) = F (c) ∩ G (c) for all c ∈ c. This soft intersection is demonstrated by ${(F, A)}_{κ} \tilde{\cap} {(G, B)}_{κ} = {(H, C)}_{κ}$ .

Example 3 Let X = {(0, 0) , (1, 0) , (-1, 0) , (0, 1) , (0, - 1)} be a 8 - adjacent digital image in $ℤ^{2}$ . Define the digital soft sets (F, A) ₈ and (G, B) ₈ with the parameter set E = {e₁, e₂, e₃, e₄} as follows: $\begin{matrix} {(F, A)}_{8} = {(e_{1}, {(0, 0), (1, 0)}), \\ (e_{2}, {(0, - 1), (- 1, 0)}), (e_{4}, {(- 1, 0), (1, 0)})}, \\ {(G, B)}_{8} = {(e_{1}, {(1, 0), (- 1, 0)}), \\ (e_{2}, {(0, 1), (- 1, 0)}), (e_{3}, {(0, 1), (0, 0)})} . \end{matrix}$ Thus C = A ∩ B and for all c ∈ C we have $\begin{matrix} F (e_{1}) \cap G (e_{2}) = H (e_{1}) = (1, 0), \\ F (e_{2}) \cap G (e_{2}) = H (e_{2}) = (0, - 1) . \end{matrix}$ Then the digital soft set ${(H, C)}_{8} = {(e_{1}, {(1, 0)}), (e_{2}, {(0, - 1)})}$ is the soft intersection of digital soft sets (F, A) ₈ and (G, B) ₈.

Definition 3.4 Let $X \subset ℤ^{n}$ be a κ - adjacent digital image. The soft union of two digital soft sets (F, A) _κ and (G, B) _κ over X is digital soft set (H, C) _κ over X, where C = A ∪ B; $F : A \to P (X)$ , $G : B \to P (X)$ and $H : C \to P (X)$ and for all c ∈ C $H (c) = {\begin{matrix} F (c), & c∈ A-B \\ G (c), & c∈ B-A \\ F (c) \cup G (c), & c∈ A∩ B \end{matrix}$ The soft union of two digital soft sets is demonstrated by ${(F, A)}_{κ} \tilde{\cup} {(G, B)}_{κ} = {(H, C)}_{κ} .$

Example 4 Consider the digital image $\begin{matrix} X = {(0, 0), (0, 1), (0, - 1), (1, 0), (- 1, 0), \\ (1, - 1), (1, 1), (- 1, - 1), (- 1, 1)} \end{matrix}$ with 8 - adjacency. Define the digital soft sets (F, A) ₈ and (G, B) ₈ as follows: $\begin{matrix} {(F, A)}_{8} = {(e_{1}, {(0, 1), (0, 0)}), \\ (e_{2}, {(0, - 1), (1, 0)}), (e_{3}, {(- 1, 1), (1, 1)})}, \\ {(G, B)}_{8} = {(e_{3}, {(1, 1), (0, 0)}), \\ (e_{4}, {(- 1, 1), (- 1, 0)}), (e_{5}, {(1, 0), (- 1, - 1)})} \end{matrix}$

where E = {e₁, e₂, e₃, e₄, e₅} is the parameter set, A, B ⊂ E, $F : A \to P (X)$ ,

$G : B \to P (X) .$

On the other hand define the function $H : C \to P (X)$ where

C = A ∪ B = {e₁, e₂, e₃, e₄, e₅} as the following: $\begin{matrix} H (e_{1}) & = & F (e_{1}) \cup G (e_{1}); e_{1} \in A - B \\ H (e_{1}) & = & {(0, 1), (0, 0)}, \\ H (e_{2}) & = & F (e_{2}) \cup G (e_{2}); e_{2} \in A - B \\ H (e_{1}) & = & {(0, - 1), (1, 0)}, \\ H (e_{3}) & = & F (e_{3}) \cup G (e_{3}); e_{3} \in A \cap B \\ H (e_{3}) & = & {(1, 1))}, \\ H (e_{4}) & = & F (e_{4}) \cup G (e_{4}); e_{4} \in B - A \\ H (e_{4}) & = & {(- 1, 1), (- 1, 0)}, \\ H (e_{5}) & = & F (e_{5}) \cup G (e_{5}); e_{5} \in B - A \\ H (e_{5}) & = & {(1, 0), (- 1, - 1)} . \end{matrix}$ Then

(H, C) ₈ = {(e₁, {(0, 1) , (0, 0)}) , (e₂, {(0, - 1) , (1, 0)}) , (e₃, {, (1, 1)}) , (e₄, {(-1, 1) , (-1, 0)}) , (e₅, {(-1, - 1))} is the soft union of digital soft sets (F, A) ₈ and (G, B) ₈.

Remark 1 Definition 3.3 and Definition 3.4 is valid for digital soft sets which have same κ - adjacency relation on a κ - adjacent digital image X. Definition 3.3 and Definition 3.4 may not been satisfied for the soft sets which have different adjacency relations or which over different adjacent digital images.

Definition 3.5 Let κ be an adjacency relation on $ℤ^{n} .$ Suppose that (F, E) _κ be a digital soft set over $X \subseteq ℤ^{n} .$ Then (F, E) _κ is called to be a digital soft connected set, if for every pair of points {x_e, y_e} ⊆ (F, E) _κ, x_e ≠ y_e, there exist a set ${x_{0} (e), x_{1} (e), \dots, x_{c} (e)} \subseteq {(F, E)}_{κ}$ such that x (e) = x₀ (e), x_c (e) = y (e) and, x_i (e) and x_i+1 (e) are κ - neighbors, i ∈ {0, 1, …, c - 1} .

Definition 3.6 Let κ₁ and κ₂ be two adjacency relations on $ℤ^{n}$ and E be the set of parameters. Assume that (F_i, E) _κ₁,(i = 1, 2, …, n) are soft sets over $X \subseteq ℤ^{n}$ and (G_j, E) _κ₂,(j = 1, 2, …, n) are soft sets over $Y \subseteq ℤ^{n}$ . If the image under f of each digital soft connected set defined on X is a digital soft connected set defined on Y, then the function f : X → Y is called to be digitally soft continuous.

Example 5 Consider the 4 - adjacency relation on $ℤ^{2}$ .

Let $\begin{matrix} X = {(0, 1), (0, 0), (0, - 1), (- 1, 0), (1, 0)}, \\ Y = {(2, 1), (3, 0), (2, - 1), (2, 0), (1, 0)} \subseteq ℤ^{2} \end{matrix}$ and E = {e₁, e₂} be the parameter set. Assume that (F₁, E) ₄, (F₂, E) ₄ are digital soft sets defined on X and (G₁, E) ₄, (G₂, E) ₄ are digital soft sets defined on Y. Firstly define the function $F_{i} : E \to P (X),$ i = 1, 2 as the following: $\begin{matrix} F_{1} {(e_{1})}_{4} = {(1, 0), (0, 0)}, \\ F_{1} {(e_{2})}_{4} = {(0, 0), (0, - 1), (- 1, 0)}; \\ F_{2} {(e_{1})}_{4} = {(0, 0), (0, - 1), (0, 1)}, \\ F_{2} {(e_{2})}_{4} = {(- 1, 0), (0, 0), (1, 0)} . \end{matrix}$ Put the (F₁, E) ₄, (F₂, E) ₄ digital soft sets such that $\begin{matrix} {(F_{1}, E)}_{4} = {(e_{1}, {(1, 0), (0, 0)}), \\ (e_{2}, {(0, 0), (0, - 1), (- 1, 0)})}, \\ {(F_{2}, E)}_{4} = {(e_{1}, {(0, 0), (0, - 1), (0, 1)}), \\ (e_{2}, {(0, 0), (1, 0), (- 1, 0)})} . \end{matrix}$ Secondly, define the functions $G_{i} : E \to P (Y), i = 1, 2$ by $\begin{matrix} G_{1} {(e_{1})}_{4} & = & {(2, 0)}, \\ G_{1} {(e_{2})}_{4} & = & {(2, 0), (2, - 1), (3, 0)}; \\ G_{2} {(e_{1})}_{4} & = & {(2, 1), (2, - 1), (2, 0)}, \\ G_{2} {(e_{2})}_{4} & = & {(2, 0), (3, 0)} . \end{matrix}$ Put the digital soft sets (G₁, E) ₄, (G₂, E) ₄ such that $\begin{matrix} {(G_{1}, E)}_{4} = {(e_{1}, {(2, 0)}), \\ (e_{2}, {(2, 0), (2, - 1), (3, 0)})}, \\ {(G_{2}, E)}_{4} = {(e_{1}, {(2, 1), (2, - 1), (2, 1)}), \\ (e_{2}, {(2, 0), (3, 0)})} . \end{matrix}$ On the other hand define the function f : X → Y as follows: $\begin{matrix} f ((0, 0)) & = & f ((1, 0)) = (2, 0), \\ f ((0, - 1)) & = & (2, - 1), \\ f ((- 1, 0)) & = & (3, 0) \\ f ((0, 1)) & = & (2, 1) . \end{matrix}$ Then f is digitally soft continuous, because $\begin{matrix} f ({(F_{1}, E)}_{4}) = ({(G_{1}, E)}_{4}), \\ f ({(F_{2}, E)}_{4}) = ({(G_{2}, E)}_{4}) . \end{matrix}$

Example 6 Consider the 8 - adjacency relation on $ℤ^{2}$ . Let $\begin{matrix} X = {(1, 0), (0, 1), (1, 1), (- 1, 1), (- 1, 0), \\ (- 1, - 1), (0, - 1), (1, - 1)} \end{matrix}$ and

$\begin{matrix} Y = {(- 1, 0), (0, 0), (0, 1), (0, - 1), (- 1, - 1), (- 1, 1), \\ (- 2, 0), (- 2, - 1), (- 2, 1)} \end{matrix}$ be subsets of $ℤ^{2}$ . Put the set E = {e₁, e₂, e₃, e₄} as parameter set. Assume that (F₁, E) ₈, (F₂, E) ₈, (F₃, E) ₈ (F₄, E) ₈ are digital soft sets defined on X and (G₁, E) ₈, (G₂, E) ₈, (G₃, E) ₈, (G₄, E) ₈ are digital soft sets defined on Y. Define the 8 - adjacent digital soft sets on X as the following $\begin{matrix} {(F_{1}, E)}_{8} = {(e_{1}, {(1, 0), (1, 1), (1, - 1)}), \\ (e_{2}, \emptyset), (e_{3}, \emptyset), (e_{4}, \emptyset)}, \\ {(F_{2}, E)}_{8} = {(e_{1}, \emptyset), (e_{2}, \\ {(0, 1), (0, 0), (0, - 1)}), (e_{3}, \emptyset), (e_{4}, \emptyset)}, \\ {(F_{3}, E)}_{8} = {(e_{1}, \emptyset), (e_{2}, \\ \emptyset), (e_{3}, {(- 1, 0), (- 1, - 1), (- 1, 1)}), (e_{4}, \emptyset)}, \\ {(F_{4}, E)}_{8} = {(e_{1}, \emptyset), (e_{2}, \\ \emptyset), (e_{3}, \emptyset), (e_{4}, {(0, 0), (- 1, 0), (- 1, 1)})} \end{matrix}$ where $F_{i} : E \to P (X)$ , i = 1, 2, 3, 4 defined as follows

F₁ (e₁) = {(-2, 1) , (-2, 0) , (-2, - 1)} , F₁ (e₂) = ∅ , F₁ (e₃) = ∅ , F₁ (e₄) = ∅ .

F₂ (e₁) = ∅ , F₂ (e₂) = {(-1, 1) , (-1, 0) , (-1, - 1)} , F₂ (e₃) = ∅ , F₂ (e₄) = ∅ .

F₃ (e₁) = ∅ , F₃ (e₂) = ∅ , F₃ (e₃) = {(0, 1) , (0, 0) , (0, - 1)} , F₃ (e₄) = ∅ .

F₄ (e₁) = ∅ , F₄ (e₂) = ∅ , F₄ (e₃) = ∅ , F₄ (e₄) = {(0, 1) , (0, 0) , (0, - 1)} .

On the other hand define the 8 - adjacent digital soft sets on Y by the following: $\begin{matrix} {(G_{1}, E)}_{8} = {(e_{1}, {(- 2, 1), (- 2, 0), (- 2, - 1)}), \\ (e_{2}, \emptyset), (e_{3}, \emptyset), (e_{4}, \emptyset)}, \\ {(G_{2}, E)}_{8} = {(e_{1}, \emptyset), (e_{2}, \\ {(- 1, 1), (- 1, 0), (- 1, - 1)}), (e_{3}, \emptyset), (e_{4}, \emptyset)}, \\ {(G_{3}, E)}_{8} = {(e_{1}, \emptyset), (e_{2}, \\ \emptyset), (e_{3}, {(0, 1), (0, 0), (0, - 1)}), (e_{4}, \emptyset)}, \\ {(G_{4}, E)}_{8} = {(e_{1}, \emptyset), (e_{2}, \\ \emptyset), (e_{3}, \emptyset), (e_{4}, {(0, - 1), (0, 0), (- 1, 0)})} \end{matrix}$ where $G_{i} : E \to P (X)$ . Define the function f : X → Y as follows: $\begin{matrix} f ((1, 0)) & = & (- 2, 0), \\ f ((0, 1)) & = & (- 1, 1), \\ f ((1, 1)) & = & (- 2, 1), \\ f ((- 1, 1)) & = & (0, 1), \\ f ((- 1, 0)) & = & (0, 0), \\ f ((- 1, - 1)) & = & (0, - 1), \\ f ((0, - 1)) & = & (- 1, - 1), \\ f ((1, - 1)) & = & (- 2, - 1) . \end{matrix}$ Then f is 8- digital soft continuous. Indeed if we calculate the images under f of digital soft sets on X, we have $\begin{matrix} f ({(F_{1}, E)}_{8}) = {(e_{1}, {(- 2, 1), (- 2, 0), (- 2, - 1)}), \\ (e_{2}, \emptyset), (e_{3}, \emptyset), (e_{4}, \emptyset)} = ({(G_{1}, E)}_{8}), \\ f ({(F_{2}, E)}_{8}) = {(e_{1}, \emptyset), (e_{2}, {(- 1, 1), (- 1, 0), \\ (- 1, - 1)}), (e_{3}, \emptyset), (e_{4}, \emptyset)} = ({(G_{2}, E)}_{8}), \\ f ({(F_{3}, E)}_{8}) = {(e_{1}, \emptyset), (e_{2}, \emptyset), \\ (e_{3}, {(0, 1), (0, 0), (0, - 1)}), (e_{4}, \emptyset)} = ({(G_{3}, E)}_{8}), \\ f ({(F_{4}, E)}_{8}) = {(e_{1}, \emptyset), (e_{2}, \emptyset), (e_{3}, \emptyset), \\ (e_{4}, {(0, - 1), (0, 0), (- 1, 0)})} = ({(G_{4}, E)}_{8}) . \end{matrix}$ Thus we obtained that the image of 8 - connected digital soft sets under f are 8 - connected digital soft sets. Hence f is digitally soft continuous due to the 8 - adjacency relation on $ℤ^{2}$ .

On the other hand f is not 4- digital soft continuous. consider the soft set (F₂, E) ₈ on X. (F₂, E) ₈ is 4- digital soft connected, but f ((F₂, E) ₈) is the soft set (G₂, E) ₈ which is not 4- digital soft connected on Y.

Theorem 3.1 The composition of digitally soft continuous functions is digitally soft continuous, that is let κ₀, κ₁, κ₂ be the adjacency relations on the digital images $X, Y, Z \subseteq ℤ^{n}$ , respectively. If f : X → Y and g : Y → Z are digitally soft continuous, then g ∘ f : X → Z is digitally soft continuous.

Proof: Let (F, E) _κ₀ be any κ₀ - connected digital soft set over X . Then f ((F, E) _κ₀) is a κ₁ - connected digital soft set over Y, because f is digitally soft continuous. Thus g ∘ f ((F, E) _κ₀) = g (f ((F, E) _κ₀)) and g (f ((F, E) _κ₀)) is a κ₂ - connected over Z for the κ₁ - connected digital soft set f ((F, E) _κ₀) over Y, since g is digitally soft continuous. Hence g ∘ f is digitally soft continuous.□

Remark 2 The sum of digitally soft continuous functions does not need to be digitally continuous.

We can give an example for the above remark.

Consider the 4 - adjacency relation on $ℤ^{2}$ . Let $X = {(0, 0), (1, 0), (2, 0), (1, 1), (1, - 1)}$ and $Y = {(0, 0), (- 1, 0), (- 2, 0), (- 1, 1), (- 1, - 1)}$ be digital images on $ℤ^{2}$ and let E = {e₁, e₂} be the parameter set. Assume that (F₁, E) ₄, (F₂, E) ₄, (F₃, E) ₄ are digital soft sets over X such that F_i : E → P (X) , i = 1, 2, 3; and (G₁, E) ₄, (G₂, E) ₄, (G₃, E) ₄, (G₄, E) ₄, (G₅, E) ₄, (G₅, E) ₄ are digital soft sets over Y such that G_i : E → P (Y) , i = 1, 2, 3, 4, 5, 6. Consider the following soft sets: $\begin{matrix} (F_{1}, E)_{4} & = & {(e_{1}, {(1, 0), (1, - 1)}), (e_{2}, {(0, 0), (1, 0)})} \\ (F_{2}, E)_{4} & = & {(e_{1}, {(2, 0)}), (e_{2}, {(1, 0), (2, 0)})} \\ (F_{3}, E)_{4} & = & {(e_{1}, {(1, 1)}), (e_{2}, {(1, 1), (0, 1)})}; \end{matrix}$ $\begin{matrix} (G_{1}, E)_{4} = {(e_{1}, {(- 1, 0), (- 2, 0)}), \\ (e_{2}, {(0, 0), (- 1, 0)})} \\ (G_{2}, E)_{4} = {(e_{1}, {(- 1, 1)}), (e_{2}, {(- 1, 0), (- 1, 1)})} \\ (G_{3}, E)_{4} = {(e_{1}, {(- 1, - 1)}), \\ (e_{2}, {(- 1, - 1), (- 1, 0)})} \\ (G_{4}, E)_{4} = {(e_{1}, {(- 1, 0), (- 1, - 1)}), \\ (e_{2}, {(- 1, 1), (- 1, 0)})} \\ (G_{5}, E)_{4} = {(e_{1}, {(0, 0)}), (e_{2}, {(- 1, 0), (0, 0)})} \\ (G_{6}, E)_{4} = {(e_{1}, {(- 2, 0)}), (e_{2}, {(- 2, 0), (- 1, 0)})} . \end{matrix}$ Define the map f : X → Y as $\begin{matrix} f ((0, 0)) & = & (0, 0) \\ f ((1, 0)) & = & (- 1, 0) \\ f ((2, 0)) & = & (- 1, 1) \\ f ((1, 1)) & = & (- 1, - 1) \\ f ((1, - 1)) & = & (- 2, 0) \end{matrix}$ Then f is digitally continuous, since $\begin{matrix} f ((F_{1}, E)_{4})) & = & (G_{1}, E)_{4}) \\ f ((F_{2}, E)_{4})) & = & (G_{2}, E)_{4}) \\ f ((F_{3}, E)_{4})) & = & (G_{3}, E)_{4}) . \end{matrix}$

On the other hand, define the map g : X → Y such that $\begin{matrix} g ((0, 0)) & = & (- 1, 1) \\ g ((1, 0)) & = & (- 1, 0) \\ g ((2, 0)) & = & (0, 0) \\ g ((1, 1)) & = & (- 2, 0) \\ g ((1, - 1)) & = & (- 1, - 1) . \end{matrix}$ Then g is digitally continuous, since $\begin{matrix} g ((F_{1}, E)_{4})) & = & (G_{4}, E)_{4}) \\ g ((F_{2}, E)_{4})) & = & (G_{5}, E)_{4}) \\ g ((F_{3}, E)_{4})) & = & (G_{6}, E)_{4}) . \end{matrix}$ Thus

(f + g) (1, 1) = (-1, - 1) + (-2, 0) = (-3, - 1) ∉ Y .

Hence the sum of two digital soft continuous functions is not digital soft continuous.

Theorem 3.2 The identity map is digitally soft continuous.

Proof: Let κ be an adjacency relation on $ℤ^{n} .$ Suppose that (F, E) _κ is any κ - connected digital soft set over X . Thus, I ((F, E) _κ) = (F, E) _κ and therefore I ((F, E) _κ) is κ - connected digital soft set, since (F, E) _κ is κ - connected digital soft set. Hence the identity map I : X → X is digitally soft continuous.□

4 Soft isomorhism and soft retraction in digital images

Definition 4.1 Consider the digital images $X, Y \subseteq ℤ^{n}$ with κ - adjancency relation. Let (F_i, E) _κ, (G_i, E) _κ, i = 1, 2, …, n be digital soft sets defined on X and Y, respectively. Assume that f : X → Y is a digital continuous function, that is one to one and onto. If f^-1 : Y → X is digital soft continuous function, then f is called a digital soft isomorphism and we say X and Y are digital soft isomorphic.

We will prefer to use statement ‘isomorphism’ rather than ‘homeomorphism’, because we are working on digital images.

Example 7 Let X = [-1, 1] ₂, Y = [0, 2] ₂ be 2 - adjancency digital images. Put the parameter set E = {e₁, e₂}. Consider the digital soft sets (F₁, E) ₂, (F₂, E) ₂ defined on the digital image X and let (G₁, E) ₂, (G₂, E) ₂ be two digital soft sets defined on the digital image Y as follows: At first define the functions F_i : E → P (X) , i = 1, 2 by $\begin{matrix} F_{1} {(e_{1})}_{2} & = & {0}, \\ F_{1} {(e_{2})}_{2} & = & {- 1, 0, 1}, \\ F_{2} {(e_{1})}_{2} & = & {1}, \\ F_{2} {(e_{2})}_{2} & = & {0} \end{matrix}$ and define the functions G_i : E → P (X) , i = 1, 2 by $\begin{matrix} G_{1} {(e_{1})}_{2} = {1}, \\ G_{1} {(e_{2})}_{2} = {0, 1, 2}, \\ G_{2} {(e_{1})}_{2} = {2}, \\ G_{2} {(e_{2})}_{2} = {1} . \end{matrix}$ Then the function f (x) = x + 1 is a digital soft isomorphism from X to Y . Indeed, the image of all 2-digital soft connected sets defined on X under f is 2-digital soft connected. $\begin{matrix} f (F_{1}) {(e_{1})}_{2} = {1}, \\ f (F_{1}) {(e_{2})}_{2} = {0, 1, 2}, \\ f (F_{2}) {(e_{1})}_{2} = {2}, \\ f (F_{2}) {(e_{2})}_{2} = {1}; \\ f ({(F_{1}, E)}_{2}) = {(e_{1}, {1}), (e_{2}, {0, 1, 2})} = {(G_{1}, E)}_{2}, \\ f ({(F_{2}, E)}_{2}) = {(e_{1}, {2}), (e_{2}, {1})} = {(G_{2}, E)}_{2} . \end{matrix}$ Therefore f is digitally soft continuous. On the other hand f is one to one and onto. f^-1 (x) = x - 1 is digitally soft continuous, because $\begin{matrix} f^{- 1} (G_{1}) {(e_{1})}_{2} = {0}, \\ f^{- 1} (G_{1}) {(e_{2})}_{2} = {- 1, 0, 1}, \\ f^{- 1} (G_{2}) {(e_{1})}_{2} = {1}, \\ f^{- 1} (G_{2}) {(e_{2})}_{2} = {0}; \\ f^{- 1} ({(G_{1}, E)}_{2}) = {(e_{1}, {0}), \\ (e_{2}, {- 1, 0, 1})} = {(F_{1}, E)}_{2}, \\ f^{- 1} ({(G_{2}, E)}_{2}) = {(e_{1}, {1}), \\ (e_{2}, {0})} = {(F_{2}, E)}_{2} . \end{matrix}$ Hence f is digital soft isomorphism and the digital images X and Y are digitally soft isomorphic.

Theorem 4.1 Digitally soft isomorphism is an equivalence relation.

Proof: Consider the digital soft sets (F_i, E) _κ,(G_i, E) _κ and (H_i, E) _κ i = 1, 2, …, n κ - adjancent defined on $X, Y, Z \subseteq ℤ^{n}$ , respectively.

Digital soft isomorphism is reflexive since I : X → X identity map is a digital soft isomorphism.

Let f : X → Y be digital soft isomorphism, then f is digital soft continuous, one to one and onto function and f^-1 : Y → X is digital soft continuous because f is digital soft isomorphism. Therefore it is easily seen that the function f^-1 : Y → X is digitally soft isomorphism. Hence symmetry property is satisfied.

Let f : X → Y and g : Y → Z be digital soft isomorphism. Then f and g are digitally soft continuous, one to one and onto functions since f and g are digital soft isomorphisms. Furthermore f^-1 and g^-1 are digitally soft continuous. The composition g ∘ f is digitally soft continuous since f and g are digital soft continuous. On the other hand the composition f^-1 ∘ g^-1 is digitally continuous since f^-1 and g^-1 are digital soft continuous. Therefore g ∘ f : X → Z is digital soft isomorphism since g ∘ f and f^-1 ∘ g^-1 = (g ∘ f) ^-1 are digitally soft continuous. Hence the transitivity property is satisfied.

Thus digitally soft isomorphism is an equivalence relation.□

Definition 4.2 Let X₀, X be κ - adjancent digital images in $ℤ^{n}$ with X₀ ⊆ X and let E be a parameter set. Assume that (F_i, E) _κ, i = 1, 2, …, n are digital soft sets on X and (G_i, E) _κ, i = 1, 2, …, n are digital soft sets on X₀ such that (G_i, E) _κ ⊆ (F_i, E) _κ . The function r : X → X₀ defined by r (x) = x for all x ∈ X₀ is called digital soft retraction. X₀ is said to be the digital soft retract of X.

Example 8 Consider 8 - adjancency relation in $ℤ^{2}$ . Let

$X_{0} = {(0, 0), (1, 0), (2, 0), (1, 1), (1, - 1)}$ and $\begin{matrix} X = {(0, 0), (0, 1), (1, 0), (- 1, 0), (1, - 1), (1, 1), \\ (2, 0), (2, 1), (2, - 1)} \end{matrix}$ be digital images and let E = {e₁, e₂} be a parameter set. Consider digital soft sets $\begin{matrix} {(F_{1}, E)}_{8} = {(e_{1}, {(0, 0), (1, 1), (1, 0)}), \\ (e_{2}, {(1, 0), (0, 1), (1, 1)})}, \\ {(F_{2}, E)}_{4} = {(e_{1}, {(1, 0), (2, 0), (2, 1)}), \\ (e_{2}, {(1, 0), (1, - 1), (2, - 1)})} \end{matrix}$ on X and $\begin{matrix} {(G_{1}, E)}_{4} = {(e_{1}, {(0, 0), (1, 0)}), (e_{2}, {(1, 0), (1, 1)})}, \\ {(G_{2}, E)}_{8} = {(e_{1}, {(2, 0), (1, 0)}), (e_{2}, {(1, 0), (1, - 1)})} \end{matrix}$ on X₀ with F_i : E → P (X) , G_i : E → P (X₀) , i = 1, 2 . Define the function r : X → X₀ as follows: $\begin{matrix} r ((0, 0)) & = & r ((- 1, 0)) = (0, 0), \\ r ((1, 0)) & = & (1, 0), \\ r ((2, 0)) & = & r ((2, 1)) = (2, 0), \\ r ((1, 1)) & = & r ((0, 1)) = (1, 1), \\ r ((1, - 1)) & = & r ((2, - 1)) = (1, - 1) . \end{matrix}$ Then r is digital soft retraction since for ∀i ∈ {0, 1}, (G_i, E) _κ ⊆ (F_i, E) _κ and r (x) = x for all x ∈ X₀.

Theorem 4.2 Digital soft retraction is digitally soft continuous.

Proof: Consider the κ - adjancent digital images X₀ and X in $ℤ^{n}$ with X₀ ⊂ X. Let (G_i, E) _κ be digital soft set on X₀ and (F_i, E) _κ be digital soft set on X such that (G_i, E) _κ ⊂ (F_i, E) _κ, (i = 1, 2, …, n). Suppose that r : X → X₀ is a digital soft retraction. Let (F_i, E) _κ be any κ - connected soft set such that r (x) = x for all x ∈ X₀. Then r ((F, E) _κ) = (G, E) _κ and it follows that r ((F_i, E) _κ) is a κ - connected soft set on X₀. Therefore r is digitally soft continuous since the image of every κ - connected digital soft set over X is κ - connected digital soft set over Y. □

Theorem 4.3 Let X₀, X and Y be κ - adjancent digital images on $ℤ^{n}$ with X₀ ⊂ X and E be a parameter set. Suppose that X₀ is digital soft retraction of X and f : X → Y is digital soft isomorphism. Then f (X₀) is digital soft retraction of Y.

Proof: Assume that (F_i, E) _κ, i = 1, 2, …, n are digital soft sets on X and (G_i, E) _κ, i = 1, 2, …, n are digital soft sets on X₀ such that (G_i, E) _κ ⊆ (F_i, E) _κ. Let r : X → X₀ be digital soft retraction. Then the composition f ∘ r ∘ r^-1 : Y → f (X₀) is digital soft retraction. Indeed, as the composition of digitally soft continuous functions is digitally soft continuous, by taking composition of the digitally soft continuous functions $r \circ f^{- 1} : Y \to X_{0}$ and $f : X \to Y$ , we obtain the digital soft continuous function $f \circ r \circ f^{- 1} : X \to X_{0}$ It follows that f ((G_i, E) _κ) ⊆ f ((F_i, E) _κ), since (G_i, E) _κ ⊆ (F_i, E) _κ. Hence f ((F_i, E) _κ) is a digital soft set over X. Now examine whether if X₀ ⊂ X, then f (X₀) ⊂ Y. Consider the functions $\begin{matrix} r & : & X \to X_{0} \\ r_{1} & = & f \circ r \circ f^{- 1} : Y \to f (X_{0}) \end{matrix}$ i) X₀ ⊂ X and f is one to one and onto because f is digitally soft isomorphism. Hence it is clear that f (X₀) ⊂ f (X) = Y .

ii) $\begin{matrix} f \circ r (f^{- 1} (y)) & = & f (r (f^{- 1} (y))) r is digital retraction \\ = & f (f^{- 1} (y)) f f^{- 1} is identity \\ = & y . \end{matrix}$ Consequently f (X₀) is digital soft retract of Y.□

Example 9 Let X₀ = [0, 1] ₂, X = [-1, 2] ₂ and Y = [0, 3] ₂ be 2 - adjancent digital images in $ℤ$ and r : X → X₀ be a digital soft retraction. Define the digital soft sets (F₁, E) ₂, (F₂, E) ₂ on X; (G₁, E) ₂, (G₂, E) ₂ on X₀ and (H₁, E) ₂, (H₂, E) ₂ on Y as follows; $\begin{matrix} {(F_{1}, E)}_{2} = {(e_{1}, {- 1, 0}), (e_{2}, {0, 1, 2})}, \\ {(F_{2}, E)}_{2} = {(e_{1}, {- 1, 0, 1}), (e_{2}, {1, 2})}, \\ {(G_{1}, E)}_{2} = {(e_{1}, {0}), (e_{2}, {0, 1})}, \\ {(G_{2}, E)}_{2} = {(e_{1}, {0, 1}), (e_{2}, {1})}, \\ {(H_{1}, E)}_{2} = {(e_{1}, {0, 1}), (e_{2}, {1, 2, 3})}, \\ {(H_{2}, E)}_{2} = {(e_{1}, {0, 1, 2}), (e_{2}, {2, 3})} . \end{matrix}$ Define the digital soft retraction r : X → X₀ such that $\begin{matrix} r (- 1) = r (0) = 0, \\ r (1) = r (2) = 1 . \end{matrix}$ On the other hand define the digital soft isomorphism f : X → Y by $f (- 1) = 0, f (0) = 1, f (1) = 2, f (2) = 3 .$ Then f ∘ r ∘ f^-1 : Y → f (X₀) is a digital soft retraction as shown in the following: $\begin{matrix} f \circ r \circ f^{- 1} (0) = f \circ r \circ f^{- 1} (1) = 1, \\ f \circ r \circ f^{- 1} (2) = f \circ r \circ f^{- 1} (3) = 2 . \end{matrix}$ Indeed, the digital soft sets f ((G₁, E) ₂) and f ((G₂, E) ₂) on f (X₀) are $\begin{matrix} f ((G_{1}, E)) = {(e_{1}, {1}), (e_{2}, {1, 2})}, \\ f ((G_{2}, E)) = {(e_{1}, {1, 2}), (e_{2}, {2})} . \end{matrix}$ Then $\begin{matrix} f \circ r \circ f^{- 1} ({(H_{1}, E)}_{2}) = f ({(G_{1}, E)}_{2}), \\ f \circ r \circ f^{- 1} ({(H_{2}, E)}_{2}) = f ({(G_{2}, E)}_{2}) . \end{matrix}$ Hence f ∘ r ∘ f^-1 is a digital soft retraction.

Corollary 1 Digital soft isomorphism preserves the digital soft retraction.

5 Conclusion

Soft sets and soft continuous functions are studied for digital images in current manuscript. The theory is stated by using κ - adjancency relation for digital images on $ℤ^{n}$ . Theorems are supported by various examples. We mostly preferred to use the 4 - adjancency and 8 - adjancency relations in examples. The soft isomorphism and the soft retraction are defined in digital images and it is concluded that digital retraction is digitally soft continuous and digital soft isomorphism preserves digital soft retraction. Soft continuous mappings have great importance for various areas of mathematics. For future work different properties and applications of soft continuous mappings can be investigated for digital images. However different adjacency relations can be used for higher dimensional digital images in order to examine new properties. Also, some basic concepts in algebraic topology can be expressed in soft digital spaces as a new tool for future works.

Footnotes

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

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and Çakır

, On continuity of softmappings, Applied Mathematics and Information Sciences 9(1) (2015), 403–409.