New Insights into Rough Set Theory: Transitive Neighborhoods and Approximations

Abstract

Rough set theory is a methodology that defines the definite or probable membership of an element for exploring data with uncertainty and incompleteness. It classifies data sets using lower and upper approximations to model uncertainty and missing information. To contribute to this goal, this study presents a newer approach to the concept of rough sets by introducing a new type of neighborhood called j-transitive neighborhood or j-TN. Some of the basic properties of j-transitive neighborhoods are studied. Also, approximations are obtained through j-TN, and the relationships between them are investigated. It is proven that these approaches provide almost all the properties provided by the approaches given by Pawlak. This study also defines the concepts of lower and upper approximations from the topological view and compares them with some existing topological structures in the literature. In addition, the applicability of the j-TN framework is demonstrated in a medical scenario. The approach proposed here represents a new view in the design of rough set theory and its practical applications to develop the appropriate strategy to handle uncertainty while performing data analysis.

1. Introduction

In classical set theory, an element belongs or does not belong to a set. Such a binary distinction is inadequate to deal with many uncertain real-life situations. Different generalized set concepts are defined to handle such situations, among them fuzzy sets [1], soft sets [2], intuitionistic fuzzy sets [3], and rough sets [4,5].

Rough set theory addresses classification problems when the exact membership of elements is uncertain. The technique uses equivalence relations that group elements in equivalence classes, which are also called indistinguishable classes. Then, the target set is approximated by creating two basic approaches:

• Lower approximation: Set of equivalence classes covered by the target set. Elements belonging to such equivalence classes are assumed to be members of the set with certainty; there is no doubt about their membership.
• Upper approximation: This is a set consisting of equivalence classes that have a nonempty intersection with the target set. The elements in these classes are highly probable members of the set, but there is uncertainty about their exact membership.

The difference between the lower and upper approximations is called the boundary region and reflects the unreliability in classification. If the boundary region is not an empty set, the target set is a rough set. Additionally, an “accuracy value”, with values ranging from 0 to 1, is used to measure the effectiveness of classification within a data set [6].

However, defining an equivalence relation for every uncertain scenario is not always feasible. For this reason, a binary relation was used instead of an equivalence relation, and the j-neighborhood concepts were defined instead of an equivalence class [7,8].

Since the accuracy value in rough set theory represents a measure of success, achieving an accuracy value closer to “1” has become a primary objective for researchers. As a result, numerous generalizations of j-neighborhood have been developed to improve classification performance and address various complexities in data analysis. Some of these are $C_j$-neighborhoods [9], subset neighborhoods [10], maximal neighborhoods [11,12], and $E_j$-neighborhoods [13]. The new defined neighborhood concepts are used to reduce uncertainty. For example, the authors developed previous models inspired by $E_j$-neighborhoods using the concept of ideals in [13]. They demonstrated that their proposed models expand the amount of confirmed information and reduce ambiguity, leading to more accurate decisions.

Topology is one of the tools used in studying rough set models due to the similarity between topological operators and the approximation operators of these models. Many studies have explored these concepts and the relationships between them in both domains. Additionally, some topological concepts, such as generalizations of open sets, have been applied to the study of these models, as demonstrated in studies [14,15,16,17,18]. Furthermore, topological generalizations have been employed to address practical problems in these models. For instance, the authors of [19,20] utilized the concepts of supratopology and infratopology to describe rough set models.

Rough set theory is particularly useful in fields such as data mining [21], pattern recognition [22], artificial intelligence [23], and decision making to prevent the spread of some epidemics, such as COVID-19 [24,25,26,27], where clear decisions are difficult due to the complexity or fuzziness of the data.

Although many types of neighborhoods have been defined as generalizations of rough sets in the literature, there has not been much work specifically addressing situations where elements are not directly related to each other. This research introduces a new concept in rough set theory called the j-transitive neighborhood (abbreviated as j-TN) to achieve more effective results in such cases. In previous studies, the hybridization of rough neighborhoods focused on the same types of neighborhoods. This study, however, links different types of neighborhoods, a novel approach that allows for the definition of new rough set models. These models will provide tools to address certain practical problems that appear in medical science and social media

The sections of this study are as follows. The first section is the introduction. The second section of this study is dedicated to providing the necessary foundational knowledge on rough set theory. In the third section, j-TN is defined, its properties are examined, and examples are provided to support this. Furthermore, various analyses are performed on the sets of transitive elements that ensure that two elements are j-TN, and certain results are obtained based on the properties of the binary relation. In the fourth section, the concepts of lower and upper approximations are defined using the j-TN concept. Consequently, the accuracy values of a set are derived, and these values are compared in tables with the approximation and accuracy values obtained from j-neighborhood and j-TN. In the fifth section, the lower and upper approximation concepts are redefined on the topological structure obtained through j-TN. Furthermore, comparisons are made with previously defined topological structures in the literature, evaluated through examples. In the sixth section, the concept j -TN is applied to identify indirect contacts and classify disease risks to take the necessary precautions to break the transmission chain in an infectious disease. Finally, the seventh section is dedicated to the conclusion and future work.

2. Preliminaries

This section introduces the fundamental concepts and findings used throughout this paper.

Definition 1. 

A binary relation $\mathcal{R}$ on a set X is a subset of $X\times X.$ The relation $\mathcal{R}$ is called:

(i) 

reflexive, if $(x,x)\in \mathcal{R},\forall x\in X$

(ii) 

symmetric, if $(x,y)\in \mathcal{R}\iff (y,x)\in \mathcal{R}$

(iii) 

transitive, if $(x,z)\in \mathcal{R}$ whenever $(x,y),(y,z)\in \mathcal{R}$

(iv) 

antisymmetric, if $x=y$ whenever $(x,y),(y,x)\in \mathcal{R}$

(v) 

equivalence, if it is reflexive, symmetric, and transitive.

(vi) 

partial order, if it is reflexive, antisymmetric, and transitive.

• If $\mathcal{R}$ is an equivalence relation on X, then the set $\{y\in X\mid (x,y)\in \mathcal{R}\}$ is called the equivalence class of x and denoted by ${\left[x\right]}_{\mathcal{R}}.$ Inverse relation of $\mathcal{R}$, denoted by ${\mathcal{R}}^{-1}$, is defined as $(x,y)\in {\mathcal{R}}^{-1}$ whenever $(y,x)\in \mathcal{R}.$

Pawlak introduced the concept of rough sets, given in the following, by defining approximation concepts to address managing uncertain and incomplete information.

Definition 2 

([20]). Let $\mathcal{R}$ be an equivalence relation on a nonempty set X. Then, the pair $(X,\mathcal{R})$ is called an approximation space. Moreover, the sets

$$\underline{\mathcal{P}}\left(A\right)=\bigcup \{{\left[x\right]}_{\mathcal{R}}:x\in X,{\left[x\right]}_{\mathcal{R}}\subseteq A\}$$

$$\overline{\mathcal{P}}\left(A\right)=\bigcup \{{\left[x\right]}_{\mathcal{R}}:x\in X,{\left[x\right]}_{\mathcal{R}}\cap A\ne \varnothing \}$$

are called lower and upper approximations of a set $A\subseteq X$, respectively. If the lower and upper approximations of a set A are not equal, then A is referred to as a rough set. A rough set is denoted by $(\underline{\mathcal{P}}\left(A\right),\overline{\mathcal{P}}\left(A\right))$.

The subsequent proposition presents the features of approximations.

Proposition 1 

([4]). Consider an approximation space $(X,\mathcal{R})$ and let $A,B\subseteq X$. The following properties hold:

(P1) 

$\underline{\mathcal{P}}\left(A\right)\subseteq A\subseteq \overline{\mathcal{P}}\left(A\right)$

(P2) 

$\underline{\mathcal{P}}(\varnothing )=\overline{\mathcal{P}}(\varnothing )=\varnothing$

(P3) 

If $A\subseteq B$, then $\underline{\mathcal{P}}\left(A\right)\subseteq \underline{\mathcal{P}}\left(B\right)$ and $\overline{\mathcal{P}}\left(A\right)\subseteq \overline{\mathcal{P}}\left(B\right)$.

(P4) 

$\underline{\mathcal{P}}(A\cup B)\supseteq \underline{\mathcal{P}}\left(A\right)\cup \underline{\mathcal{P}}\left(B\right)$ and $\overline{\mathcal{P}}(A\cup B)=\overline{\mathcal{P}}\left(A\right)\cup \overline{\mathcal{P}}\left(B\right)$

(P5) 

$\underline{\mathcal{P}}(A\cap B)=\underline{\mathcal{P}}\left(A\right)\cap \underline{\mathcal{P}}\left(B\right)$ and $\overline{\mathcal{P}}(A\cap B)\subseteq \overline{\mathcal{P}}\left(A\right)\cap \overline{\mathcal{P}}\left(B\right)$

(P6) 

$\underline{\mathcal{P}}\left(A^c\right)={\left(\overline{\mathcal{P}}\left(A\right)\right)}^c$ and $\overline{\mathcal{P}}\left(A^c\right)={\left(\underline{\mathcal{P}}\left(A\right)\right)}^c$

(P7) 

$\underline{\mathcal{P}}\left(\underline{\mathcal{P}}\left(A\right)\right)=\underline{\mathcal{P}}\left(A\right)$ and $\overline{\mathcal{P}}\left(\overline{\mathcal{P}}\left(A\right)\right)=\overline{\mathcal{P}}\left(A\right).$

Definition 3. 

The boundary and accuracy of approximations are defined as, respectively:

$${\mathcal{B}}_{\mathcal{R}}\left(A\right)=\overline{\mathcal{P}}\left(A\right)-\underline{\mathcal{P}}\left(A\right)$$

$${\alpha }_{\mathcal{R}}\left(A\right)={\displaystyle \frac{|\underline{\mathcal{P}}\left(A\right)|}{|\overline{\mathcal{P}}\left(A\right)|}},\mathrm{where}\left|\overline{\mathcal{P}}\left(A\right)\right|\ne \varnothing$$

Consequently, ${\mathcal{B}}_{\mathcal{R}}\left(A\right)\ne \varnothing$ and ${\alpha }_{\mathcal{R}}\left(A\right)\ne 1$ hold for a rough set A.

A lot of generalizations of the concept of rough set are defined with the help of the concepts of $j$-neighborhoods, defined as follows:

Definition 4 

([14]). Let X be a set and $\mathcal{R}$ be a binary relation on X. The j-neighborhoods of an element $x\in X$ are defined as:

(i) 

$r$-neighborhood: $N_r\left(x\right)=\{y\in X\mid (x,y)\in \mathcal{R}\}$.

(ii) 

$l$-neighborhood: $N_l\left(x\right)=\{y\in X\mid (y,x)\in \mathcal{R}\}$.

(iii) 

$i$-neighborhood: $N_i\left(x\right)=N_r\left(x\right)\cap N_l\left(x\right)$.

(iv) 

$u$-neighborhood: $N_u\left(x\right)=N_r\left(x\right)\cup N_l\left(x\right)$.

(v) 

$<r>$-neighborhood: $N_{<r>}\left(x\right)={\bigcap }_{x\in N_r\left(y\right)}{N}_r\left(y\right)$.

(vi) 

$<l>$-neighborhood: $N_{<l>}\left(x\right)={\bigcap }_{x\in N_l\left(y\right)}{N}_l\left(y\right)$.

(vii) 

$<i>$-neighborhood: $N_{<i>}\left(x\right)=N_{<r>}\left(x\right)\cap N_{<l>}\left(x\right)$.

(viii) 

$<u>$-neighborhood: $N_{<u>}\left(x\right)=N_{<r>}\left(x\right)\cup N_{<l>}\left(x\right)$.

Remark 1. 

Throughout this paper, $J=\{r,l,i,u,<r>,<l>,<i>,<u>\}$.

Definition 5 

([14]). Let $2^X$ denotes the power set of X and ${\sigma }_j:X\to 2^X$ be a mapping which assigns to each x in X its j-neighborhood in $2^X$ for $j\in J$. The triple $(X,\mathcal{R},{\sigma }_j)$ is termed a j-neighborhood space.

Definition 6 

([20]). Let $(X,\mathcal{R},{\sigma }_j)$ be a j-neighborhood space. The lower and upper approximation of $A\subseteq X$ respect to ${\sigma }_j$ are, respectively, defined as follows:

$$\begin{array}{c}\hfill \underline{{\mathcal{N}}_j}\left(A\right)=\{x\in X\mid N_j\left(x\right)\subseteq A\}\end{array}$$

$$\begin{array}{c}\hfill \overline{{\mathcal{N}}_j}\left(A\right)=\{x\in X\mid N_j\left(x\right)\cap A\ne \varnothing \}\end{array}$$

The j-boundary and j-accuracy of A are defined, respectively, as follows:

$$\begin{array}{c}\hfill {\mathcal{B}}_{{\mathcal{N}}_j}\left(A\right)=\overline{{\mathcal{N}}_j}\left(A\right)-\underline{{\mathcal{N}}_j}\left(A\right)\end{array}$$

(1)

$$\begin{array}{c}\hfill {\mathcal{A}}_{{\mathcal{N}}_j}\left(A\right)={\displaystyle \frac{|\underline{{\mathcal{N}}_j}\left(A\right)\bigcap A|}{|\overline{{\mathcal{N}}_j}\left(A\right)\bigcup A|}}\end{array}$$

(2)

where $A\ne \varnothing .$

Definition 7 

([28]). Let $(X,\mathcal{R},{\sigma }_j)$ denote a j-neighborhood space. In this context, $T_j=\{U\subseteq X\mid N_j\left(a\right)\subseteq U,\mathit{for}\mathit{each}a\in U\}$ forms a topology on X. The space $(X,\mathcal{R},T_j)$ is called a j-topological space.

The j-lower and j-upper approximations of a set $A\subseteq X$ are defined as follows:

$${\underline{T}}_j\left(A\right)=\bigcup \{U\in T_j\mid U\subseteq A\}$$

$${\overline{T}}_j\left(A\right)=\bigcap \{H\mid H^c\in T_j\mathrm{and}A\subseteq H\}$$

It is evident that ${\underline{T}}_j\left(A\right)$ and ${\overline{T}}_j\left(A\right)$ correspond to the interior and closure of the set A, respectively.

The regions and measures of subsets are derived using ${\underline{T}}_j\left(A\right)$ and ${\overline{T}}_j\left(A\right)$, analogous to the corresponding definitions (1) and (2) in Definition 6.

3. j-Transitive Neighborhood

This section presents the concept of j-transitive neighborhood (abbreviated j-TN).

Definition 8. 

Let $R\subseteq X\times X.$ The j-transitive neighborhoods of a point $x\in X$ are defined as follows:

$T{N}_r\left(x\right)=\{y\in X\mid N_r\left(x\right)\cap N_l\left(y\right)\ne \varnothing \}$.

$T{N}_l\left(x\right)=\{y\in X\mid N_l\left(x\right)\cap N_r\left(y\right)\ne \varnothing \}$.

$T{N}_i\left(x\right)=T{N}_l\left(x\right)\cap T{N}_r\left(x\right)$.

$T{N}_u\left(x\right)=T{N}_l\left(x\right)\cup T{N}_r\left(x\right)$.

$T{N}_{<r>}\left(x\right)=\{y\in X\mid N_{<r>}\left(x\right)\cap N_{<l>}\left(y\right)\ne \varnothing \}$.

$T{N}_{<l>}\left(x\right)=\{y\in X\mid N_{<l>}\left(x\right)\cap N_{<r>}\left(y\right)\ne \varnothing \}$.

$T{N}_{<i>}\left(x\right)=T{N}_{<r>}\left(x\right)\cap T{N}_{<l>}\left(x\right)$.

$T{N}_{<u>}\left(x\right)=T{N}_{<r>}\left(x\right)\cup T{N}_{<l>}\left(x\right)$.

Proposition 2. 

If $\mathcal{R}$ is a symmetric relation, then

$$T{N}_r\left(x\right)=T{N}_l\left(x\right)=T{N}_i\left(x\right)=T{N}_r\left(u\right)$$

$$T{N}_{<r>}\left(x\right)=T{N}_{<l>}\left(x\right)=T{N}_{<i>}\left(x\right)=T{N}_{<u>}\left(x\right)$$

Proof. 

This result follows from the fact that if $\mathcal{R}$ is symmetric, then $N_r\left(x\right)=N_l\left(x\right).$ □

Remark 2. 

If multiple binary relations are defined on a set X, we refer to their j-transitive neighborhoods by the relation’s name. For instance, let $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$ be two relations on X. Then, $T{N}_r^R\left(x\right)$ and $T{N}_r^{R^{\prime }}\left(x\right)$ are the r-transitive neighborhood sets of the point x with respect to the relations $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$, respectively.

Example 1. 

Let $X=\{a,b,c,d\}$ and $\mathcal{R}=\left\{\right(a,a),(c,c),(d,a),(a,c),(b,c),(c,b),(d,c),(d,b\left)\right\}$. The j-neighborhoods and j-transitive neighborhoods of the points of X are given in Table 1.

Table 1. The $N_j$ and $T{N}_j$ neighborhoods.

	a	b	c	d
$N_r$	$\{a,c\}$	$\left\{c\right\}$	$\{b,c\}$	$\{a,b,c\}$
$N_l$	$\{a,d\}$	$\{c,d\}$	X	∅
$N_i$	$\left\{a\right\}$	$\left\{c\right\}$	$\{b,c\}$	∅
$N_u$	$\{a,c,d\}$	$\{c,d\}$	X	$\{a,b,c\}$
$N_{<r>}$	$\{a,c\}$	$\{b,c\}$	$\left\{c\right\}$	∅
$N_{<l>}$	$\{a,d\}$	X	$\{c,d\}$	$\left\{d\right\}$
$N_{<i>}$	$\left\{a\right\}$	$\{b,c\}$	$\left\{c\right\}$	∅
$N_{<u>}$	$\{a,c,d\}$	X	$\{c,d\}$	$\left\{d\right\}$
$T{N}_r$	$\{a,b,c\}$	$\{b,c\}$	$\{b,c\}$	$\{a,b,c\}$
$T{N}_l$	$\{a,d\}$	X	X	∅
$T{N}_i$	$\left\{a\right\}$	$\{b,c\}$	$\{b,c\}$	∅
$T{N}_u$	X	X	X	$\{a,b,c\}$
$T{N}_{<r>}$	$\{a,b,c\}$	$\{b,c\}$	$\{b,c\}$	∅
$T{N}_{<l>}$	$\left\{a\right\}$	$\{a,b,c\}$	$\{a,b,c\}$	∅
$T{N}_{<i>}$	$\left\{a\right\}$	$\{b,c\}$	$\{b,c\}$	∅
$T{N}_{<u>}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	∅

The following example shows that the concepts j-neighborhood and j-transitive neighborhood are not comparable under the subset relation.

Example 2. 

Let $X=\{x,y\}$ and $\mathcal{R}=\left\{\right(x,y),(y,x\left)\right\}.$ Then, $x\in T{N}_r^{\mathcal{R}}\left(x\right)$, but $x\notin N_r^{\mathcal{R}}\left(x\right).$ Besides, $y\in T{N}_l^{\mathcal{R}}\left(y\right)$, but $y\notin N_l^{\mathcal{R}}\left(y\right).$ Let ${\mathcal{R}}^{\prime }=\left\{(x,y)\right\}.$ Then, $x\in N_l^{{\mathcal{R}}^{\prime }}\left(y\right)$ and $y\in N_r^{{\mathcal{R}}^{\prime }}\left(x\right).$ But $x\notin T{N}_r^{{\mathcal{R}}^{\prime }}\left(y\right)$ and $y\notin T{N}_r^{{\mathcal{R}}^{\prime }}\left(x\right).$

Proposition 3. 

Let $R\subseteq X\times X$ and $x,y\in X.$ Then, the following hold:

(i) 

$y\in T{N}_r\left(x\right)\iff x\in T{N}_l\left(y\right)$ and $y\in T{N}_{<r>}\left(x\right)\iff x\in T{N}_{<l>}\left(y\right)$.

(ii) 

$y\in T{N}_j\left(x\right)\iff x\in T{N}_j\left(y\right)$, $j\in \{i,u,<i>,<u>\}$.

Proof. 

(i) Straightforward.

(ii)

$y\in T{N}_i\left(x\right)\iff y\in T{N}_r\left(x\right)\cap T{N}_l\left(x\right)\iff x\in T{N}_l\left(y\right)\cap T{N}_r\left(y\right)\iff x\in T{N}_i\left(y\right).$ For $j\in \{u,<i>,<u>\}$, the results are obtained in a similar manner.

□

Lemma 1. 

Let ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ be two relations on X such that ${\mathcal{R}}_1\subseteq {\mathcal{R}}_2$. Then,

$$\begin{array}{c}\hfill T{N}_j^{{\mathcal{R}}_1}\left(x\right)\subseteq T{N}_j^{{\mathcal{R}}_2}\left(x\right)\end{array}$$

(3)

for $j\in \{r,l,i,u\}$.

Proof. 

Assume ${\mathcal{R}}_1\subseteq {\mathcal{R}}_2$ and let $y\in T{N}_r^{\mathcal{R}1}\left(x\right)$. This implies the existence of $z\in X$ such that $(x,z),(z,y)\in {\mathcal{R}}_1$. Consequently, $(x,z),(z,y)\in {\mathcal{R}}_2$, leading to $y\in T{N}_r^{{\mathcal{R}}_2}\left(x\right)$. Similarly, it can be observed that $T{N}_l^{{\mathcal{R}}_1}\left(x\right)\subseteq T{N}_l^{{\mathcal{R}}_2}\left(x\right)$. Moreover, since $T{N}_r^{{\mathcal{R}}_1}\left(x\right)\subseteq T{N}_r^{{\mathcal{R}}_2}\left(x\right)$ and $T{N}_l^{{\mathcal{R}}_1}\left(x\right)\subseteq T{N}_l^{{\mathcal{R}}_2}\left(x\right)$, it follows that

$$T{N}_i^{{\mathcal{R}}_1}\left(x\right)=T{N}_r^{{\mathcal{R}}_1}\left(x\right)\cap T{N}_l^{{\mathcal{R}}_1}\left(x\right)\subseteq T{N}_r^{{\mathcal{R}}_2}\left(x\right)\cap T{N}_l^{{\mathcal{R}}_2}\left(x\right)=T{N}_i^{{\mathcal{R}}_2}\left(x\right)$$

and

$$T{N}_u^{{\mathcal{R}}_1}\left(x\right)=T{N}_l^{{\mathcal{R}}_1}\left(x\right)\cup T{N}_l^{{\mathcal{R}}_1}\left(x\right)\subseteq T{N}_r^{{\mathcal{R}}_2}\left(x\right)\cup T{N}_l^{{\mathcal{R}}_2}\left(x\right)=T{N}_u^{{\mathcal{R}}_2}\left(x\right).$$

□

Example 3. 

Let $X=\{1,2,3,4,5,6,7\}$ and

$${\mathcal{R}}_1=\{(1,5),(1,6),(2,5),(2,6),(4,2),(5,3),(6,7),(7,3)\}$$

$${\mathcal{R}}_2=\{(1,5),(1,6),(2,5),(2,6),(3,3),(3,4),(4,2),(4,4),(5,3),(6,7),(7,3),(7,4)\}.$$

The j-transitive neighborhoods for $j\in J$ based on ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ are given in Table 2 and Table 3, respectively. As evident from the provided example, the assertion (3) presented in Lemma 1 holds. However, their conversations are typically not valid.

Table 2. The $T{N}_j^{{\mathcal{R}}_1}$ neighborhoods.

	1	2	3	4	5	6	7
$T{N}_r^{{\mathcal{R}}_1}$	$\{3,7\}$	$\{3,7\}$	∅	$\{5,6\}$	∅	{3}	∅
$T{N}_l^{{\mathcal{R}}_1}$	∅	∅	$\{1,2,6\}$	∅	$\left\{4\right\}$	$\left\{4\right\}$	$\{1,2\}$
$T{N}_i^{{\mathcal{R}}_1}$	∅	∅	∅	∅	∅	∅	∅
$T{N}_u^{{\mathcal{R}}_1}$	$\{3,7\}$	$\{3,7\}$	$\{1,2,6\}$	$\{5,6\}$	$\left\{4\right\}$	$\{3,4\}$	$\{1,2\}$
$T{N}_{<r>}^{{\mathcal{R}}_1}$	∅	$\{1,2\}$	∅	∅	$\{5,6,7\}$	$\{5,6,7\}$	$\{5,6,7\}$
$T{N}_{<l>}^{{\mathcal{R}}_1}$	$\left\{2\right\}$	$\left\{2\right\}$	∅	∅	$\{5,6,7\}$	$\{5,6\}$	$\{5,6,7\}$
$T{N}_{<i>}^{{\mathcal{R}}_1}$	∅	$\left\{2\right\}$	∅	∅	$\{5,6,7\}$	$\{5,6\}$	$\{5,6,7\}$
$T{N}_{<u>}^{{\mathcal{R}}_1}$	$\left\{2\right\}$	$\{1,2\}$	∅	∅	$\{5,6,7\}$	$\{5,6,7\}$	$\{5,6,7\}$

Table 3. The $T{N}_j^{{\mathcal{R}}_2}$ neighborhoods.

	1	2	3	4	5	6	7
$T{N}_r^{{\mathcal{R}}_2}$	$\{3,7\}$	$\{3,7\}$	$\{2,3,4\}$	$\{2,4,5,6\}$	$\{3,4\}$	$\{3,4\}$	$\{2,3,4\}$
$T{N}_l^{{\mathcal{R}}_2}$	∅	$\{3,4,7\}$	$\{1,2,3,5,6,7\}$	$\{3,4,5,6,7\}$	$\left\{4\right\}$	$\left\{4\right\}$	$\{1,2\}$
$T{N}_i^{{\mathcal{R}}_2}$	∅	$\{3,7\}$	$\{2,3,4\}$	$\{4,5,6\}$	$\left\{4\right\}$	$\left\{4\right\}$	$\left\{2\right\}$
$T{N}_u^{{\mathcal{R}}_2}$	$\{3,7\}$	$\{3,4,7\}$	X	$\{2,3,4,5,6,7\}$	$\{3,4\}$	$\{3,4\}$	$\{1,2,3,4\}$
$T{N}_{<r>}^{{\mathcal{R}}_2}$	∅	$\{1,2,4\}$	$\{3,5,7\}$	$\left\{4\right\}$	$\{5,6\}$	$\{5,6\}$	$\{5,7\}$
$T{N}_{<l>}^{{\mathcal{R}}_2}$	$\left\{2\right\}$	$\left\{2\right\}$	$\{3,7\}$	$\{2,4\}$	$\{3,5,6,7\}$	$\{5,6\}$	$\{3,7\}$
$T{N}_{<i>}^{{\mathcal{R}}_2}$	∅	$\left\{2\right\}$	$\{3,7\}$	$\left\{4\right\}$	$\{5,6\}$	$\{5,6\}$	$\left\{7\right\}$
$T{N}_{<u>}^{{\mathcal{R}}_2}$	$\left\{2\right\}$	$\{1,2,4\}$	$\{3,5,7\}$	$\{2,4\}$	$\{3,5,6,7\}$	$\{5,6\}$	$\{3,5,7\}$

Result 1. 

Example 3 illustrates that the converse of (3) in Lemma 1 does not hold, and it is invalid for $j\in \{<r>,<l>,<i>,<u>\}.$

Definition 9. 

Let $\mathcal{R}\subseteq X\times X$ and ${\mathrm{Y}}_j:X\to 2^X$ be a mapping such that ${\mathrm{Y}}_j\left(x\right)=T{N}_j\left(x\right)$ for all $j\in J$. Then $(X,\mathcal{R},{\mathrm{Y}}_j)$ is called a j-transitive neighborhood space, or briefly, a j-TN space.

Proposition 4. 

Let $(X,\mathcal{R},{\mathrm{Y}}_j)$ be a j-TN space. Then, $T{N}_i\left(x\right)\subseteq T{N}_r\left(x\right)$ and $T{N}_i\left(x\right)\subseteq T{N}_l\left(x\right)$ for all $x\in X.$

Proof. 

Let $y\in T{N}_i\left(x\right)$. This implies $y\in T{N}_l\left(x\right)\cap T{N}_r\left(x\right)$. Consequently, $N_r\left(y\right)\cap N_l\left(x\right)\ne \varnothing$ and $N_l\left(y\right)\cap N_r\left(x\right)\ne \varnothing$. Thus, there exist $a,b\in X$ such that $(y,a),(a,x),(x,b),(b,y)\in \mathcal{R}$. As a result, $y\in T{N}_r\left(x\right)$ and $y\in T{N}_l\left(x\right)$. Therefore, it follows that $T{N}_i\left(x\right)\subseteq T{N}_r\left(x\right)$ and $T{N}_i\left(x\right)\subseteq T{N}_l\left(x\right)$. □

Result 2. 

Let $(X,\mathcal{R},{\mathrm{Y}}_j)$ be a j-TN space. Then, the following hold:

(i) 

$T{N}_i\left(x\right)\subseteq T{N}_r\left(x\right)\subseteq T{N}_u\left(x\right)$

$T{N}_i\left(x\right)\subseteq T{N}_l\left(x\right)\subseteq T{N}_u\left(x\right)$.

(ii) 

$T{N}_{<i>}\left(x\right)\subseteq T{N}_{<r>}\left(x\right)\subseteq T{N}_{<u>}\left(x\right)$

$T{N}_{<i>}\left(x\right)\subseteq T{N}_{<l>}\left(x\right)\subseteq T{N}_{<u>}\left(x\right)$.

Proposition 5. 

Let $(X,\mathcal{R},{\mathrm{Y}}_j)$ be a j-TN space and $x,y,z\in X$. Then, the following hypotheses are held:

(i) 

If $\mathcal{R}$ is reflexive, then $x\in T{N}_j\left(x\right)$ for each $j\in J$.

(ii) 

If $\mathcal{R}$ is symmetric and $x\in N_l\left(y\right)$, then $y\in T{N}_j\left(y\right)$ and $x\in T{N}_j\left(x\right)$ for $j\in \{r,l\}$.

(iii) 

If $\mathcal{R}$ is transitive and $z\in T{N}_r\left(x\right)$, then $z\in N_r\left(x\right).$

(iv) 

If $\mathcal{R}$ is transitive, $z\in T{N}_r\left(x\right)$ and $y\in N_r\left(z\right)$, then $y\in T{N}_r\left(x\right).$

(v) 

If $\mathcal{R}$ is antisymmetric and $x\in T{N}_r\left(x\right)$, for all $x\in X$, then $\mathcal{R}$ is reflexive.

(vi) 

If $\mathcal{R}$ is an equivalence, then $T{N}_j\left(x\right)\ne \varnothing$, for all $j\in J$.

Proof. 

(i) It is straightforward.

(ii)

Since $x\in N_l\left(y\right)$, $(x,y)\in \mathcal{R}$. Given the symmetry of $\mathcal{R}$, it follows that $(y,x)\in \mathcal{R}$. Consequently, $y\in T{N}_l\left(y\right)$ and $x\in T{N}_r\left(x\right)$.

(iii)

Let $z\in T{N}_r\left(x\right)$. Then $\exists a\in X$, such that $(x,a),(a,z)\in \mathcal{R}.$ Since $\mathcal{R}$ is transitive, $(x,z)\in \mathcal{R}.$ Therefore, $z\in N_r\left(x\right).$

(iv)

From (iii), $(x,z)\in \mathcal{R}$. Since $(z,y)\in \mathcal{R}$, we obtain $y\in T{N}_r\left(x\right).$

(v)

Assume that $\mathcal{R}$ is not reflexive. Then, $\exists a\in X$ such that $(a,a)\notin \mathcal{R}.$ Since $a\in T{N}_r\left(a\right),$ then there exists $b\in X$ such that $(a,b),(b,a)\in \mathcal{R}.$ However, because $\mathcal{R}$ is antisymmetric, it follows that $a=b$ which contradicts the assumption that $(a,a)\notin \mathcal{R}.$ Therefore, $\mathcal{R}$ is reflexive.

(vi)

Let $\mathcal{R}$ be an equivalence relation. From (i), it follows that $x\in T{N}_j\left(x\right)$, for $j\in \{r,l,i,u\}$. Furthermore, since $x\in N_j\left(x\right)$, for $j\in \{<r>,<l>\}$, this implies $x\in T{N}_j\left(x\right)$, for $j\in \{<r>,<l>,<i>,<u>\}.$

□

Following, the correlation between the j-transitive neighborhoods based on the relation $\mathcal{R}$ and its inverse, denoted as ${\mathcal{R}}^{-1}$, is demonstrated.

Proposition 6. 

Let $(X,\mathcal{R},{\mathrm{Y}}_j)$ be a j-TN space. Then

(i) 

if $\mathcal{R}$ is symmetric, then $T{N}_j^{\mathcal{R}}\left(x\right)=T{N}_j^{{\mathcal{R}}^{-1}}\left(x\right)$

(ii) 

$y\in T{N}_r^{\mathcal{R}}\left(x\right)\iff x\in T{N}_l^{{\mathcal{R}}^{-1}}\left(y\right)$

Proof. 

(i) The desired result is obtained directly, as $\mathcal{R}={\mathcal{R}}^{-1}$ when $\mathcal{R}$ is symmetric.

(ii)

$$\begin{array}{ccc}y\in T{N}_r^{\mathcal{R}}\left(x\right)\hfill & \iff \hfill & (x,a),(a,y)\in \mathcal{R},\exists a\in X\hfill \\ \hfill & \iff \hfill & (y,a),(a,x)\in {\mathcal{R}}^{-1},\exists a\in X\hfill \\ \hfill & \iff \hfill & x\in T{N}_l^{{\mathcal{R}}^{-1}}\left(x\right).\hfill \end{array}$$

□

The following definition is introduced to characterize elements that facilitate the transition between two given elements in a j-TN space.

Definition 10. 

Let $\mathcal{R}$ be a binary relation on X. If $z\in N_r\left(x\right)\cap N_l\left(y\right)$, then z is called a transition element from x to y. The set of all transition elements from x to y is denoted by ${\mathcal{R}}^{x\to y}$. It follows that $y\in T{N}_r\left(x\right)\iff {\mathcal{R}}^{x\to y}\ne \varnothing$ and $y\in T{N}_l\left(x\right)\iff {\mathcal{R}}^{y\to x}\ne \varnothing$

Proposition 7. 

Let $R\subseteq X$ and $w,x,y,z\in X.$ Then

$${\mathcal{R}}^{w\to x}\cap {\mathcal{R}}^{y\to z}={\mathcal{R}}^{w\to z}\cap {\mathcal{R}}^{y\to x}.$$

Proof. 

Straightforward. □

Proposition 8. 

Let $(X,\mathcal{R},{\mathrm{Y}}_j)$ be a j-TN space.

(i) 

If $a\in {\mathcal{R}}^{x\to y}$, then $x\in T{N}_l\left(y\right)$ and $y\in T{N}_r\left(x\right).$

(ii) 

If $a\in {\mathcal{R}}^{x\to y}$ and $\mathcal{R}$ is symmetric, then $a\in {\mathcal{R}}^{y\to x}$.

(iii) 

If $\mathcal{R}$ is symmetric and $a\in {\mathcal{R}}^{x\to y}$, then $x\in T{N}_r\left(y\right).$

(iv) 

If $\mathcal{R}$ is reflexive and $a\in {\mathcal{R}}^{x\to y}$, then $a\in T{N}_r\left(x\right)$ and $a\in T{N}_l\left(y\right)$.

(v) 

If $x\in {\mathcal{R}}^{y\to y}$, then $x\in T{N}_i\left(x\right)$ and $x\in T{N}_u\left(x\right).$

(vi) 

If $a\in {\mathcal{R}}^{x\to y}\cap {\mathcal{R}}^{y\to x}$, then $x,y\in T{N}_i\left(a\right)$.

(vii) 

If $a\in {\mathcal{R}}^{x\to y},$ then $a\in {\left({\mathcal{R}}^{-1}\right)}^{y\to x}.$

(viii) 

If $\mathcal{R}$ is antisymmetric and $y\in {\mathcal{R}}^{x\to x}$, then $x=y.$

Proof. 

(i) It is clear from Definition 8 and Definition 10.

(ii)

Let $a\in {\mathcal{R}}^{x\to y}$ and $\mathcal{R}$ be symmetric. Then $(y,a),(a,x)\in \mathcal{R}.$ Therefore, $a\in {\mathcal{R}}^{y\to x}$.

(iii)

Since $a\in {\mathcal{R}}^{x\to y}$ and $\mathcal{R}$ is symmetric, then $(a,x),(y,a)\in \mathcal{R}$. Therefore, $x\in T{N}_r\left(y\right).$

(iv)

Since $\mathcal{R}$ is reflexive and $a\in {\mathcal{R}}^{x\to y}$, then $(x,x),(x,a),(a,y),(y,y)\in \mathcal{R}.$ Therefore, $a\in T{N}_r\left(x\right)$ and $a\in T{N}_l\left(y\right).$

(v)

Let $x\in {\mathcal{R}}^{y\to y}$, then $(y,x),(x,y)\in \mathcal{R}$. Therefore, $x\in T{N}_r\left(x\right)$ and $x\in T{N}_l\left(x\right)$. Hence, $x\in T{N}_i\left(x\right)$ and $x\in T{N}_u\left(x\right).$

(vi)

Let $a\in {\mathcal{R}}^{x\to y}\cap {\mathcal{R}}^{y\to x}$. Then $(x,a),(a,y),(y,a),(a,x)\in \mathcal{R}.$ Thus, $x,y\in T{N}_r\left(a\right)\cap T{N}_l\left(a\right)=T{N}_i\left(a\right)$.

(vii)

Let $a\in {\mathcal{R}}^{x\to y}.$ Then $(x,a),(a,y)\in \mathcal{R}.$ Therefore, $(a,x),(y,a)\in {\mathcal{R}}^{-1}.$ This implies that $a\in {\left({\mathcal{R}}^{-1}\right)}^{y\to x}.$

(viii)

Let $y\in {\mathcal{R}}^{x\to x}$. Then, $(x,y),(y,x)\in \mathcal{R}.$ Since $\mathcal{R}$ is antisymmetric, it follows that $x=y.$

□

4. Rough Approximation Based on j-Transitive Neighborhood

This section gives the concepts of j-transitive lower and j-transitive upper approximations utilizing transitive neighborhoods.

From this point onward, let $(X,\mathcal{R},{\mathrm{Y}}_j)$ be a j-TN space.

Definition 11. 

The j-transitive lower and j-transitive upper approximations with respect to ${\mathrm{Y}}_j$ of a subset $A\subseteq X$ are defined as, respectively,

$$\underline{{\mathcal{TN}}_j}\left(A\right)=\left(\bigcup _{T{N}_j\left(x\right)\subseteq A}T{N}_j\left(x\right)\right)$$

$$\overline{{\mathcal{TN}}_j}\left(A\right)=\left(\bigcap _{T{N}_j\left(x\right)\cap A\ne \varnothing }T{N}_j\left(x\right)\right)\cup A$$

for all $j\in J.$

Proposition 9. 

Let $(X,\mathcal{R},{\mathrm{Y}}_j)$ be a j-TN space, and $A,A_1,A_2\subseteq X$. The following conditions hold for all $j\in J$:

(i) 

$\underline{{\mathcal{TN}}_j}\left(A\right)\subseteq A\subseteq \overline{{\mathcal{TN}}_j}\left(A\right)$.

(ii) 

$\underline{{\mathcal{TN}}_j}(\varnothing )=\varnothing =\overline{{\mathcal{TN}}_j}(\varnothing )$ and $\underline{{\mathcal{TN}}_j}\left(X\right)\subset X=\overline{{\mathcal{TN}}_j}\left(X\right)$.

(iii) 

If $A_1\subseteq A_2$, then $\overline{{\mathcal{TN}}_j}\left(A_1\right)\subseteq \overline{{\mathcal{TN}}_j}\left(A_2\right)$ and $\underline{{\mathcal{TN}}_j}\left(A_1\right)\subseteq \underline{{\mathcal{TN}}_j}\left(A_2\right)$.

(iv) 

$\underline{{\mathcal{TN}}_j}\left(A_1\right)\cup \underline{{\mathcal{TN}}_j}\left(A_2\right)\subseteq \underline{{\mathcal{TN}}_j}(A_1\cup A_2)$

$\overline{{\mathcal{TN}}_j}(A_1\cup A_2)=\overline{{\mathcal{TN}}_j}\left(A_1\right)\cup \overline{{\mathcal{TN}}_j}\left(A_2\right)$.

(v) 

$\underline{{\mathcal{TN}}_j}(A_1\cap A_2)\subseteq \underline{{\mathcal{TN}}_j}\left(A_1\right)\cap \underline{{\mathcal{TN}}_j}\left(A_2\right)$

$\overline{{\mathcal{TN}}_j}(A_1\cap A_2)\subseteq \overline{{\mathcal{TN}}_j}\left(A_1\right)\cap \overline{{\mathcal{TN}}_j}\left(A_2\right)$.

(vi) 

$\underline{{\mathcal{TN}}_j}\left(A^c\right)={\left(\overline{{\mathcal{TN}}_j}\left(A\right)\right)}^c$

$\overline{{\mathcal{TN}}_j}\left(A^c\right)={\left(\underline{{\mathcal{TN}}_j}\left(A\right)\right)}^c$.

(vii) 

$\underline{{\mathcal{TN}}_j}\left(\underline{{\mathcal{TN}}_j}\left(A\right)\right)=\underline{{\mathcal{TN}}_j}\left(A\right)$

$\overline{{\mathcal{TN}}_j}\left(\overline{{\mathcal{TN}}_j}\left(A\right)\right)=\overline{{\mathcal{TN}}_j}\left(A\right)$.

Proof. 

Statements (i) and (ii) are obvious.

(iii)

Let $y\in \overline{{\mathcal{TN}}_j}\left(A_1\right)$. This implies that $y\in A_1$ or $y\in T{N}_j\left(x\right)$ for some $x\in X$ such that $T{N}_j\left(x\right)\cap A_1\ne \varnothing$. Since $A_1\subseteq A_2$, we have $y\in A_2$ or $T{N}_j\left(x\right)\cap A_2\ne \varnothing$, which implies $y\in \overline{{\mathcal{TN}}_j}\left(A_2\right)$.

Let $y\in \underline{{\mathcal{TN}}_j}\left(A_1\right).$ Since $A_1\subseteq A_2$, we have $y\in T{N}_j\left(x\right)\subseteq A_1\subseteq A_2,$ for all $x\in X$. Therefore, $y\in \underline{{\mathcal{TN}}_j}\left(A_2\right).$

(iv)

Let $y\in \underline{{\mathcal{TN}}_j}\left(A_1\right)\cup \underline{{\mathcal{TN}}_j}\left(A_2\right)$. Then $y\in \underline{{\mathcal{TN}}_j}\left(A_1\right)$ or $y\in \underline{{\mathcal{TN}}_j}\left(A_2\right)$. This implies that $y\in T{N}_j\left(x\right)\subseteq A_1$ or $y\in T{N}_j\left(x\right)\subseteq A_2$, for some $x\in X$. Thus, we have $y\in T{N}_j\left(x\right)\subseteq A_1\cup A_2$. As a result, $y\in \underline{{\mathcal{TN}}_j}(A_1\cup A_2)$.

It can be derived from (iii) that $\overline{{\mathcal{TN}}_j}\left(A_1\right)\cup \overline{{\mathcal{TN}}_j}\left(A_2\right)\subseteq \overline{{\mathcal{TN}}_j}(A_1\cup A_2).$ Let $y\in \overline{{\mathcal{TN}}_j}(A_1\cup A_2).$ Then, $y\in A_1\cup A_2$ or $y\in T{N}_j\left(x\right)$ for all $x\in X$ such that $T{N}_j\left(x\right)\cap (A_1\cup A_2)\ne \varnothing$. This implies that $[y\in A_1$ or $y\in T{N}_j\left(x\right)$ such that $T{N}_j\left(x\right)\cap A_1\ne \varnothing ]$ or [$y\in A_2$ or $y\in T{N}_j\left(x\right)$ such that $T{N}_j\left(x\right)\cap A_2\ne \varnothing$]. Thus, $y\in \overline{{\mathcal{TN}}_j}\left(A_1\right)$ or $y\in \overline{{\mathcal{TN}}_j}\left(A_2\right)$. Consequently, it follows that $y\in \overline{{\mathcal{TN}}_j}\left(A_1\right)\cup \overline{{\mathcal{TN}}_j}\left(A_2\right).$

(v)

As established in (iii), we have $\underline{{\mathcal{TN}}_j}(A_1\cap A_2)\subseteq \underline{{\mathcal{TN}}_j}\left(A_1\right)\cap \underline{{\mathcal{TN}}_j}\left(A_2\right)$ and $\overline{T{N}_j}(A_1\cap A_2)\subseteq \overline{T{N}_j}\left(A_1\right)\cap \overline{T{N}_j}\left(A_2\right).$

(vi)

$$\begin{array}{ccc}\hfill {\left(\overline{{\mathcal{TN}}_j}\left(A\right)\right)}^c& =& {\left[\left(\bigcap _{T{N}_j\left(x\right)\cap A\ne \varnothing }T{N}_j\left(x\right)\right)\cup A\right]}^c\hfill \\ & =& \left(\bigcup _{T{N}_j\left(x\right)\cap A=\varnothing }T{N}_j\left(x\right)\right)\cap A^c\hfill \\ & =& \left(\bigcup _{T{N}_j\left(x\right)\subseteq A^c}T{N}_j\left(x\right)\right)\hfill \\ & =& \underline{{\mathcal{TN}}_j}\left(A^c\right)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\left(\underline{{\mathcal{TN}}_j}\left(A\right)\right)}^c& =& {\left(\bigcup _{T{N}_j\left(x\right)\subseteq A}T{N}_j\left(x\right)\right)}^c\hfill \\ & =& {\left[\left(\bigcup _{T{N}_j\left(x\right)\subseteq A}T{N}_j\left(x\right)\right)\cap A\right]}^c\hfill \\ & =& \left(\bigcap _{T{N}_j\left(x\right)\cap A^c\ne \varnothing }T{N}_j\left(x\right)\right)\cup A^c\hfill \\ & =& \overline{{\mathcal{TN}}_j}\left(A^c\right)\hfill \end{array}$$

(vii)

From statement (i), we derive that $\underline{{\mathcal{TN}}_j}\left(\underline{{\mathcal{TN}}_j}\left(A\right)\right)\subseteq \underline{{\mathcal{TN}}_j}\left(A\right)$. For the sake of contradiction, assume that $\underline{{\mathcal{TN}}_j}\left(A\right)⊈\underline{{\mathcal{TN}}_j}\left(\underline{{\mathcal{TN}}_j}\left(A\right)\right)$. Then, there exists some $z\in \underline{{\mathcal{TN}}_j}\left(A\right)$ such that $z\notin \underline{{\mathcal{TN}}_j}\left(\underline{{\mathcal{TN}}_j}\left(A\right)\right)$. This implies that $z\notin T{N}_j\left(w\right)$ for all $w\in X$ that satisfy $T{N}_j\left(w\right)\subseteq \underline{{\mathcal{TN}}_j}\left(A\right)$. However, since $\underline{{\mathcal{TN}}_j}\left(A\right)\subseteq A$, we have $z\notin \underline{{\mathcal{TN}}_j}\left(A\right)$, which is a contradiction. Therefore, it must be the case that $\underline{{\mathcal{TN}}_j}\left(\underline{{\mathcal{TN}}_j}\left(A\right)\right)=\underline{{\mathcal{TN}}_j}\left(A\right)$.

Similarly, the statement $\overline{{\mathcal{TN}}_j}\left(\overline{{\mathcal{TN}}_j}\left(A\right)\right)=\overline{{\mathcal{TN}}_j}\left(A\right)$ can be derived.

□

The presented example illustrates that the equality of statement (ii) in Proposition 9 does not hold.

Example 4. 

Let $X=\{a,b,c,d\}$ and $\mathcal{R}=\left\{\right(a,b\left)\right\}$. Then, $\underline{{\mathcal{TN}}_j}\left(X\right)=\varnothing$, for all $j\in J$.

The following examples demonstrate that statements (iv) and (v) in Proposition 9 do not have equality.

Example 5. 

Let $A=\left\{x\right\}$ and $B=\left\{y\right\}$ be subsets of the j-TN space $(X,\mathcal{R},{\mathrm{Y}}_j)$ where $X=\{x,y,z\}$, $\mathcal{R}=\left\{\right(x,x),(x,y\left)\right\}$. Then, $\underline{{\mathcal{TN}}_r}(A\cup B)=\{x,y\}$, while $\underline{{\mathcal{TN}}_r}\left(A\right)\cup \underline{{\mathcal{TN}}_r}\left(B\right)=\varnothing$. Moreover, $\overline{{\mathcal{TN}}_r}(A\cap B)=\varnothing$, but $\overline{{\mathcal{TN}}_r}\left(A\right)\cap \overline{{\mathcal{TN}}_r}\left(B\right)=\{x,y\}.$

Example 6. 

Let $A=\{x,z,w\}$ and $B=\{x,w,t\}$ be subsets of the j-TN space $(X,\mathcal{R},{\mathrm{Y}}_j)$ where $X=\{x,y,z,w,t\}$ and $\mathcal{R}=\left\{\right(x,x),(x,w),(w,x),(w,z),(z,t\left)\right\}.$ Then $\underline{{\mathcal{TN}}_r}\left(A\right)\cap \underline{{\mathcal{TN}}_r}\left(B\right)=\{x,z,w\}\cap \{x,w,t\}=\{x,w\}.$ However, $\underline{{\mathcal{TN}}_r}(A\cap B)=\varnothing$.

Example 7. 

Table 4 and Table 5 give all j-transitive neighborhoods of subsets of X in Example 1.

Table 4. The j-transitive neighborhoods of all subsets when $j\in \{r,l,i,u\}$.

A	${\underline{\mathcal{TN}}}_{\mathit{r}}\left(\mathit{A}\right)$	${\overline{\mathcal{TN}}}_{\mathit{r}}\left(\mathit{A}\right)$	${\underline{\mathcal{TN}}}_{\mathit{l}}\left(\mathit{A}\right)$	${\overline{\mathcal{TN}}}_{\mathit{l}}\left(\mathit{A}\right)$	${\underline{\mathcal{TN}}}_{\mathit{i}}\left(\mathit{A}\right)$	${\overline{\mathcal{TN}}}_{\mathit{i}}\left(\mathit{A}\right)$	${\underline{\mathcal{TN}}}_{\mathit{u}}\left(\mathit{A}\right)$	${\overline{\mathcal{TN}}}_{\mathit{u}}\left(\mathit{A}\right)$
$\left\{a\right\}$	∅	$\{a,b,c\}$	∅	$\{a,d\}$	$\left\{a\right\}$	$\left\{a\right\}$	∅	$\{a,b,c\}$
$\left\{b\right\}$	∅	$\{b,c\}$	∅	X	∅	$\{b,c\}$	∅	$\{a,b,c\}$
$\left\{c\right\}$	∅	$\{b,c\}$	∅	$\{a,c\}$	∅	$\left\{c\right\}$	∅	$\{a,b,c\}$
$\left\{d\right\}$	∅	$\left\{d\right\}$	∅	$\left\{d\right\}$	∅	$\left\{d\right\}$	∅	$\left\{d\right\}$
$\{a,b\}$	∅	$\{a,b,c\}$	∅	$\{a,b,c\}$	∅	$\{a,b,c\}$	∅	$\{a,b,c\}$
$\{a,c\}$	∅	$\{a,b,c\}$	∅	$\{a,c,d\}$	$\left\{a\right\}$	$\{a,c\}$	∅	$\{a,b,c\}$
$\{a,d\}$	∅	X	$\{a,d\}$	$\{a,d\}$	$\left\{a\right\}$	$\{a,d\}$	∅	X
$\{b,c\}$	$\{b,c\}$	$\{b,c\}$	∅	X	$\{b,c\}$	$\{b,c\}$	∅	$\{a,b,c\}$
$\{b,d\}$	∅	$\{b,c,d\}$	∅	$\{a,b,d\}$	∅	$\{b,c,d\}$	∅	X
$\{c,d\}$	∅	$\{b,c,d\}$	∅	$\{a,c,d\}$	∅	$\{b,c,d\}$	∅	X
$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	∅	X	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$
$\{a,b,d\}$	∅	X	$\{a,d\}$	$\{a,b,d\}$	$\left\{a\right\}$	$\{a,b,d\}$	∅	X
$\{a,c,d\}$	∅	X	$\{a,d\}$	X	$\left\{a\right\}$	$\{a,c,d\}$	∅	X
$\{b,c,d\}$	$\{b,c\}$	$\{b,c,d\}$	∅	X	$\{b,c\}$	$\{b,c,d\}$	∅	X
$\{a,b,c,d\}$	$\{a,b,c\}$	X	X	X	$\{a,b,c\}$	X	X	X

Table 5. The j-transitive neighborhoods of all subsets when $j\in \{<r>,<l>,<i>,<u>\}$.

A	${\underline{\mathcal{TN}}}_{<\mathit{r}>}\left(\mathit{A}\right)$	${\overline{\mathcal{TN}}}_{<\mathit{r}>}\left(\mathit{A}\right)$	${\underline{\mathcal{TN}}}_{<\mathit{l}>}\left(\mathit{A}\right)$	${\overline{\mathcal{TN}}}_{<\mathit{l}>}\left(\mathit{A}\right)$	${\underline{\mathcal{TN}}}_{<\mathit{i}>}\left(\mathit{A}\right)$	${\overline{\mathcal{TN}}}_{<\mathit{i}>}\left(\mathit{A}\right)$	${\underline{\mathcal{TN}}}_{<\mathit{u}>}\left(\mathit{A}\right)$	${\overline{\mathcal{TN}}}_{<\mathit{u}>}\left(\mathit{A}\right)$
$\left\{a\right\}$	∅	$\{a,b,c\}$	$\left\{a\right\}$	$\left\{a\right\}$	∅	$\{a,c\}$	∅	$\{a,b,c\}$
$\left\{b\right\}$	∅	$\{b,c\}$	∅	$\{a,b,c\}$	∅	$\{b,c\}$	∅	$\{a,b,c\}$
$\left\{c\right\}$	∅	$\{b,c\}$	∅	$\{a,b,c\}$	∅	$\{b,c\}$	∅	$\{a,b,c\}$
$\left\{d\right\}$	∅	$\left\{d\right\}$	∅	$\left\{d\right\}$	∅	$\left\{d\right\}$	∅	∅
$\{a,b\}$	∅	$\{a,b,c\}$	$\left\{a\right\}$	$\{a,b\}$	$\left\{a\right\}$	$\{a,b\}$	∅	$\{a,b,c\}$
$\{a,c\}$	∅	$\{a,b,c\}$	$\left\{a\right\}$	$\{a,c\}$	$\{a,c\}$	$\{a,c\}$	∅	$\{a,b,c\}$
$\{a,d\}$	∅	X	$\left\{a\right\}$	$\{a,d\}$	∅	$\{c,d\}$	∅	X
$\{b,c\}$	$\{b,c\}$	$\{b,c\}$	∅	$\{a,b,c\}$	$\{b,c\}$	$\{b,c\}$	∅	$\{a,b,c\}$
$\{b,d\}$	∅	$\{b,c,d\}$	∅	X	∅	$\{b,c,d\}$	∅	X
$\{c,d\}$	∅	$\{b,c,d\}$	∅	X	∅	$\{b,c,d\}$	∅	X
$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$
$\{a,b,d\}$	∅	X	$\left\{a\right\}$	$\{a,b,d\}$	$\left\{a\right\}$	$\{a,b,d\}$	∅	X
$\{a,c,d\}$	∅	X	$\{a,d\}$	X	$\left\{a\right\}$	$\{a,c,d\}$	∅	X
$\{b,c,d\}$	$\{b,c\}$	$\{b,c,d\}$	∅	X	$\{b,c\}$	$\{b,c,d\}$	∅	X
$\{a,b,c,d\}$	$\{a,b,c\}$	X	$\{a,b,c\}$	X	$\{a,b,c\}$	X	X	X

Definition 12. 

The $T{N}_j$-boundary, $T{N}_j$-positive, and $T{N}_j$-negative regions of a subset $A\subseteq X$ are defined as, respectively,

$${\mathcal{B}}_{{\mathcal{TN}}_j}\left(A\right)=\overline{{\mathcal{TN}}_j}\left(A\right)-\underline{{\mathcal{TN}}_j}\left(A\right)$$

$${\mathcal{POS}}_{{\mathcal{TN}}_j}\left(A\right)=\underline{{\mathcal{TN}}_j}\left(A\right)$$

$$\mathcal{N}E{G}_{{\mathcal{TN}}_j}\left(A\right)=X-\overline{{\mathcal{TN}}_j}\left(A\right)$$

Definition 13. 

The $T{N}_j$ accuracy of $\varnothing \ne A\subseteq X$ is defined as

$${\mathcal{A}}_{{\mathcal{TN}}_j}\left(A\right)={\displaystyle \frac{|\underline{{\mathcal{TN}}_j}\left(A\right)|}{|\overline{{\mathcal{TN}}_j}\left(A\right)|}}$$

where $|\overline{{\mathcal{TN}}_j}\left(A\right)|\ne \varnothing .$ If ${\mathcal{A}}_{{\mathcal{TN}}_j}\left(A\right)=1$, then A is called $j-TN$ exact set; otherwise, it is called $j-TN$ rough.

Definition 14. 

Table 6 and Table 7 give the j-accuracy and $T{N}_j$-accuracy of all subsets of X defined in Example 1 for each $j\in J$.

Table 6. The j-accuracy and $T{N}_j$-accuracy of all subsets for each $j\in \{r,l,i,u\}$.

A	${\mathcal{A}}_{{\mathcal{N}}_{\mathit{r}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{TN}}_{\mathit{r}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{N}}_{\mathit{l}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{TN}}_{\mathit{l}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{N}}_{\mathit{i}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{TN}}_{\mathit{i}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{N}}_{\mathit{u}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{TN}}_{\mathit{u}}}\left(\mathit{A}\right)$
$\left\{a\right\}$	0	0	0	0	1	1	0	0
$\left\{b\right\}$	0	0	0	0	0	0	0	0
$\left\{c\right\}$	0	0	0	0	0	0	0	0
$\left\{d\right\}$	0	0	${\displaystyle \frac{1}{4}}$	0	1	0	0	0
$\{a,b\}$	0	0	0	0	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{1}{2}}$	0	0
$\{a,c\}$	${\displaystyle \frac{1}{4}}$	0	0	0	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{1}{2}}$	0	0
$\{a,d\}$	0	0	${\displaystyle \frac{1}{2}}$	1	1	1	0	0
$\{b,c\}$	${\displaystyle \frac{1}{2}}$	1	0	0	1	1	0	0
$\{b,d\}$	0	0	${\displaystyle \frac{1}{4}}$	0	${\displaystyle \frac{1}{3}}$	0	0	0
$\{c,d\}$	0	0	${\displaystyle \frac{1}{4}}$	0	${\displaystyle \frac{1}{3}}$	0	0	0
$\{a,b,c\}$	${\displaystyle \frac{3}{4}}$	1	0	0	1	1	0	1
$\{a,b,d\}$	0	0	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{2}{3}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{3}}$	0	0
$\{a,c,d\}$	${\displaystyle \frac{1}{3}}$	0	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{1}{4}}$	0
$\{b,c,d\}$	${\displaystyle \frac{2}{3}}$	${\displaystyle \frac{2}{3}}$	${\displaystyle \frac{1}{2}}$	0	1	${\displaystyle \frac{2}{3}}$	${\displaystyle \frac{1}{4}}$	0
$\{a,b,c,d\}$	1	${\displaystyle \frac{3}{4}}$	1	1	1	${\displaystyle \frac{3}{4}}$	1	1

Table 7. The j-accuracy and the $T{N}_j$-accuracy of all subsets for each $j\in \{<r>,<l>,<i>,<u>\}$.

A	${\mathcal{A}}_{{\mathcal{N}}_{<\mathit{r}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{TN}}_{<\mathit{r}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{N}}_{<\mathit{l}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{TN}}_{<\mathit{l}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{N}}_{<\mathit{i}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{TN}}_{<\mathit{i}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{N}}_{<\mathit{u}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathcal{TN}}_{<\mathit{u}>}}\left(\mathit{A}\right)$
$\left\{a\right\}$	0	0	0	1	1	0	${\displaystyle \frac{1}{2}}$	0
$\left\{b\right\}$	0	0	0	0	0	0	1	0
$\left\{c\right\}$	${\displaystyle \frac{1}{3}}$	0	0	0	${\displaystyle \frac{1}{2}}$	0	${\displaystyle \frac{1}{3}}$	0
$\left\{d\right\}$	1	0	${\displaystyle \frac{1}{4}}$	0	1	0	${\displaystyle \frac{1}{4}}$	0
$\{a,b\}$	0	0	0	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	1	0
$\{a,c\}$	${\displaystyle \frac{2}{3}}$	0	0	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{2}{3}}$	1	${\displaystyle \frac{1}{2}}$	0
$\{a,d\}$	${\displaystyle \frac{1}{2}}$	0	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{1}{2}}$	1	0	${\displaystyle \frac{1}{4}}$	0
$\{b,c\}$	${\displaystyle \frac{2}{3}}$	1	0	0	1	1	${\displaystyle \frac{2}{3}}$	0
$\{b,d\}$	${\displaystyle \frac{1}{2}}$	0	${\displaystyle \frac{1}{4}}$	0	${\displaystyle \frac{1}{2}}$	0	${\displaystyle \frac{1}{4}}$	0
$\{c,d\}$	${\displaystyle \frac{1}{2}}$	0	${\displaystyle \frac{1}{2}}$	0	${\displaystyle \frac{1}{3}}$	0	${\displaystyle \frac{1}{2}}$	0
$\{a,b,c\}$	1	1	0	1	1	1	1	1
$\{a,b,d\}$	${\displaystyle \frac{1}{3}}$	0	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{2}{3}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{1}{4}}$	0
$\{a,c,d\}$	${\displaystyle \frac{3}{4}}$	0	${\displaystyle \frac{3}{4}}$	1	${\displaystyle \frac{3}{4}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{3}{4}}$	0
$\{b,c,d\}$	${\displaystyle \frac{3}{4}}$	${\displaystyle \frac{2}{3}}$	${\displaystyle \frac{1}{2}}$	0	1	${\displaystyle \frac{2}{3}}$	${\displaystyle \frac{1}{2}}$	0
$\{a,b,c,d\}$	1	${\displaystyle \frac{3}{4}}$	1	${\displaystyle \frac{3}{4}}$	1	${\displaystyle \frac{3}{4}}$	1	1

5. Topological Structure and Approximations Derived from j-Transitive Neighborhood

This section defines the topology ${\tau }_j$, referred to as $T{N}_j$-topology, on a j-TN space using j-transitive neighborhood relationships and subsequently investigates various associated characteristics. In addition, approximations within $T{N}_j$-topological spaces are established and analyzed.

Theorem 1. 

Let $(X,\mathcal{R},{\mathrm{Y}}_j)$ be a j-TN space. Then, the set

$${\tau }_j=\{U\subseteq X\mid {\mathrm{Y}}_j\left(x\right)\subseteq U,\mathit{for}\mathit{all}x\in U\}$$

is a topology on X, for all $j\in J$. $(X,\mathcal{R},{\tau }_j)$ is called as j-TN topological space.

Proof. 

It is evident that $\varnothing ,X\in {\tau }_j.$ Let $U,V\in {\tau }_j$ and $x\in U\cap V$. For all $j\in J$, it follows that ${\mathrm{Y}}_j\left(x\right)\subseteq U$ and ${\mathrm{Y}}_j\left(x\right)\subseteq V.$ Hence, ${\mathrm{Y}}_j\left(x\right)\subseteq U\cap V$, implying $U\cap V\in {\tau }_j.$ Now, consider $U_{\alpha }\in {\tau }_j$ for $\alpha \in \Delta$ where $\Delta$ is an index set. Let $x\in \bigcup U_{\alpha }.$ There exists ${\alpha }^{\prime }\in \Delta$ such that $x\in U_{{\alpha }^{\prime }}.$ Consequently, ${\mathrm{Y}}_j\left(x\right)\subseteq U_{{\alpha }^{\prime }}$, for all $j\in J$. This implies that ${\mathrm{Y}}_j\left(x\right)\subseteq \bigcup U_{\alpha }$ establishes that $\bigcup U_{\alpha }\in {\tau }_j.$

□

Example 8. 

In the set X provided in Example 1, the topologies $T_j$ and ${\tau }_j$ with respect to j-neighborhoods and j-transitive neighborhoods become as follows:

$T_r=\{\varnothing ,X,\{a,b,c\}\}$

$T_l=\{\varnothing ,X,\{a,d\},\left\{d\right\}\}$

$T_i=\{\varnothing ,X\}$

$T_u=\{\varnothing ,X\}$

$T_{<r>}=\{\varnothing ,X,\{a,c\},\left\{c\right\},\left\{d\right\},\{c,d\},\{a,b,c\},\{a,c,d\},\{b,c\},\{b,c,d\}\}$

$T_{<l>}=\{\varnothing ,X,\{a,d\},\{c,d\},\{a,c,d\},\left\{d\right\}\}$

$T_{<i>}=\{\varnothing ,X,\left\{a\right\},\left\{c\right\},\left\{d\right\},\{b,c\},\{a,c\},\{a,d\},\{c,d\},\{a,b,c\},\{b,c,d\},\{a,c,d\}\}$

$T_{<u>}=\{\varnothing ,X,\left\{d\right\},\{c,d\},\{a,c,d\}\}$

${\tau }_r=\{\varnothing ,X,\{a,b,c\},\{b,c\}\}$

${\tau }_l=\{\varnothing ,X,\{a,d\},\left\{d\right\}\}$

${\tau }_i=\{\varnothing ,X,\left\{a\right\},\{b,c\},\left\{d\right\},\{a,b,c\},\{b,c,d\},\{a,d\}\}$

${\tau }_u=\{\varnothing ,X\}$

${\tau }_{<r>}=\{\varnothing ,X,\left\{d\right\},\{a,b,c\},\{b,c\},\{b,c,d\}\}$

${\tau }_{<l>}=\{\varnothing ,X,\left\{a\right\},\left\{d\right\},\{a,d\},\{a,b,c\}\}$

${\tau }_{<i>}=\{\varnothing ,X,\left\{a\right\},\{b,c\},\left\{d\right\},\{a,b,c\},\{b,c,d\},\{a,d\}\}$

${\tau }_{<u>}=\{\varnothing ,X,\{a,b,c\}\}$.

The following theorem is derived from properties (i) and (ii) presented in Result 2.

Theorem 2. 

$(X,\mathcal{R},{\tau }_j)$ be a j-TN topological space. Then, the following hold:

(i) 

${\tau }_u\subseteq {\tau }_r\subseteq {\tau }_i$.

(ii) 

${\tau }_u\subseteq {\tau }_l\subseteq {\tau }_i$.

(iii) 

${\tau }_{<u>}\subseteq {\tau }_{<r>}\subseteq {\tau }_{<i>}$.

(iv) 

${\tau }_{<u>}\subseteq {\tau }_{<l>}\subseteq {\tau }_{<i>}$.

Example 8 illustrates that the converse of relations given in Theorem 2 is invalid.

Theorem 3. 

Let $(X,\mathcal{R},{\tau }_j)$ be a j-TN topological space. Then ${\tau }_u={\tau }_r\cap {\tau }_l$ and ${\tau }_{<u>}={\tau }_{<r>}\cap {\tau }_{<l>}.$

Proof. 

Suppose $U\in {\tau }_u$. This implies that for all $x\in U$, $T{N}_u\left(x\right)\subseteq U$. Consequently, we have $T{N}_r\left(x\right)\cup T{N}_l\left(x\right)\subseteq U$. Therefore, for all $x\in U$, it follows that $T{N}_r\left(x\right)\subseteq U$ and $T{N}_l\left(x\right)\subseteq U$. Hence, $U\in {\tau }_r\cap {\tau }_l$. In contrast, we can demonstrate that ${\tau }_r\cap {\tau }_l\subseteq {\tau }_u$, which completes the proof.

Similarly, it can be shown that ${\tau }_{<u>}={\tau }_{<r>}\cap {\tau }_{<l>}.$ □

Theorem 4. 

$(X,\mathcal{R},{\tau }_j)$ be a j-TN topological space. Then, ${\tau }_l$ is the dual topology of ${\tau }_r.$

Proof. 

Let $U\in {\tau }_l$ and $y\notin U.$ Let us assume that $T{N}_r\left(y\right)\cap U\ne \varnothing .$ Then, there exists $z\in T{N}_r\left(y\right)\cap U.$ Therefore, by Proposition 3, $y\in T{N}_l\left(z\right).$ Since $U\in {\tau }_l$ and $z\in U$, then $T{N}_l\left(z\right)\subseteq U.$ This means $y\in U$, which is a contradiction. Consequently, $T{N}_r\left(y\right)\cap U=\varnothing \Rightarrow T{N}_r\left(y\right)\subseteq U^c$. □

Theorem 5. 

Let $(X,\mathcal{R},T_j)$ be a j-topological space and $(X,{\mathcal{R}}^{\prime },{\tau }_j)$ be a j-TN topological space. Then, we have $T_j\subseteq {\tau }_j$, for $j\in \{r,l,u\}$.

Proof. 

($j=r$) Let $A\in T_r$, and let $x\in A$. Consider $y\in T{N}_r\left(x\right)$. It follows that $N_r\left(x\right)\cap N_l\left(y\right)\ne \varnothing$, which implies the existence of $z\in N_r\left(x\right)\cap N_l\left(y\right)$. Consequently, $z\in N_r\left(x\right)$ and $y\in N_r\left(z\right)$. Given that $A\in T_r$ and $x\in A$, we can deduce that $N_r\left(x\right)\subseteq A$, thus establishing that $z\in A$. Furthermore, since $y\in N_r\left(z\right)$ and $z\in A$, we can conclude that $y\in A$. As a result, $T{N}_r\left(x\right)\subseteq A$, which establishes $A\in {\tau }_r$.

($j=l$) Analogously, we have $T_l\subseteq {\tau }_l$.

($j=u$) Let $A\in T_u$, and $x\in A$. We observe that $N_r\left(x\right)\cup N_l\left(x\right)\subseteq A$. Consequently, $N_r\left(x\right)\subseteq A$ and $N_l\left(x\right)\subseteq A$. For any $y\in T{N}_u\left(x\right)$, it follows that $y\in T{N}_l\left(x\right)\cup T{N}_r\left(x\right)$. This implies either $N_r\left(x\right)\cap N_l\left(y\right)\ne \varnothing$ or $N_l\left(x\right)\cap N_r\left(y\right)\ne \varnothing$. In turn, this implies the existence of $z\in N_r\left(x\right)\cap N_l\left(y\right)$ or $z^{\prime }\in N_l\left(x\right)\cap N_r\left(y\right)$. Thus, $z\in N_l\left(x\right)$ with $y\in N_l\left(z\right)$ or $z^{\prime }\in N_r\left(x\right)$ with $y\in N_r\left(z^{\prime }\right)$. In both cases, we have $z\in A$ and $N_l\left(z\right)\subseteq A$ or $z^{\prime }\in A$ with $N_r\left(z^{\prime }\right)\subseteq A$. Therefore, we conclude that $y\in A$, establishing that $A\in {\tau }_u$. This completes the proof.

□

The following examples illustrate that the converse of Theorem 5 is not generally true and is not valid for $j=i.$

Example 9. 

Consider the set $X=\{a,b,c\}$ with the relation $R=\left\{\right(a,b\left)\right\}$. In this context, we observe that ${\tau }_r={\tau }_l=2^X$, while $T_r=\{\varnothing ,\left\{b\right\},\left\{c\right\},\{b,c\},\{a,b\},X\}$ and $T_l=\{\varnothing ,\left\{a\right\},\left\{c\right\},\{a,c\},\{a,b\},X\}$.

Example 10. 

Let $X=\{a,b,c\}$ and $R=\left\{\right(a,b),(b,c),(b,a\left)\right\}$. Then, $T_r=\{\varnothing ,X,\left\{c\right\}\},$$T_l=\{\varnothing ,X,\{a,b\}\}$, $T_u=\{\varnothing ,X\}$, while ${\tau }_r=\{\varnothing ,X,\left\{b\right\},\left\{c\right\},\{b,c\},\{a,c\}\}$, ${\tau }_l=\{\varnothing ,X,\left\{a\right\},\left\{b\right\},\{a,b\},\{a,c\}\}$, and ${\tau }_u=\{\varnothing ,X,\left\{b\right\},\{a,c\}\}.$

Example 11. 

Let $X=\{a,b,c,d,e\}$ and $\mathcal{R}=\left\{\right(a,b),(a,d),(a,e),(c,d),(d,a),(e,c\left)\right\}$. Then, $N_i\left(a\right)=\left\{d\right\},N_i\left(b\right)=N_i\left(c\right)=N_i\left(e\right)=\{\varnothing \},N_i\left(d\right)=\left\{a\right\}$, and $T{N}_i\left(a\right)=\{a,c\},T{N}_i\left(b\right)=\varnothing ,T{N}_i\left(c\right)=\left\{a\right\},T{N}_i\left(d\right)=\{d,e\},T{N}_i\left(e\right)=\left\{d\right\}$. Therefore, $\{a,d\}\in T_i$, but $\{a,d\}\notin {\tau }_i.$

Remark 3. 

From Example 8, it can be inferred that $T_{<j>}⊈{\tau }_{<j>}$, and ${\tau }_{<j>}⊈T_{<j>}$ for $j\in \{r,l\}$.

Result 3. 

The following inclusions hold:

(i) 

$T_u\subseteq {\tau }_r\cup {\tau }_i$.

(ii) 

$T_u\subseteq {\tau }_l\cup {\tau }_i$.

Below, the approximation concepts relevant to the topology derived from j-transitive neighborhoods are defined.

Definition 15. 

The j-lower and j-upper approximations of a subset A of a j-TN topological space $(X,\mathcal{R},{\tau }_j)$ are defined as

$${\underline{\tau }}_j\left(A\right)=\bigcup \{U\in {\tau }_j\mid U\subseteq A\}$$

$${\overline{\tau }}_j\left(A\right)=\bigcap \{F\mid F^c\in {\tau }_j\mathrm{and}A\subseteq F\}$$

Proposition 10. 

Let $(X,\mathcal{R},{\tau }_j)$ be a j-TN topological space and $A,B\subseteq X$. Then, the subsequent properties hold:

(i) 

${\underline{\tau }}_j(\varnothing )={\overline{\tau }}_j(\varnothing )=\varnothing$ and ${\underline{\tau }}_j\left(X\right)={\overline{\tau }}_j\left(X\right)=X$.

(ii) 

${\underline{\tau }}_j\left(A\right)\subseteq A\subseteq {\overline{\tau }}_j\left(A\right)$.

(iii) 

If $A\subseteq B$, then ${\underline{\tau }}_j\left(A\right)\subseteq {\underline{\tau }}_j\left(B\right)$ and ${\overline{\tau }}_j\left(A\right)\subseteq {\overline{\tau }}_j\left(B\right)$.

(iv) 

${\underline{\tau }}_j(A\cap B)={\underline{\tau }}_j\left(A\right)\cap {\underline{\tau }}_j\left(B\right)$ and ${\overline{\tau }}_j(A\cap B)\subseteq {\overline{\tau }}_j\left(A\right)\cap {\overline{\tau }}_j\left(B\right)$.

(v) 

${\underline{\tau }}_j\left(A\right)\cup {\underline{\tau }}_j\left(B\right)\subseteq {\underline{\tau }}_j(A\cup B)$ and ${\overline{\tau }}_j(A\cup B)={\overline{\tau }}_j\left(A\right)\cup {\overline{\tau }}_j\left(B\right)$.

(vi) 

${\underline{\tau }}_j\left({\underline{\tau }}_j\left(A\right)\right)={\underline{\tau }}_j\left(A\right)$ and ${\overline{\tau }}_j\left({\overline{\tau }}_j\left(A\right)\right)=({\overline{\tau }}_j\left(A\right)$.

Proof. 

The proofs of properties $\left(i\right)$, $\left(ii\right)$, and $\left(iii\right)$ are derived from Definition 15.

(iv)

From $\left(iii\right)$, we have ${\underline{\tau }}_j(A\cap B)\subseteq {\underline{\tau }}_j\left(A\right)\cap {\underline{\tau }}_j\left(B\right)$. To show the reverse inclusion, let $w\in {\underline{\tau }}_j\left(A\right)\cap {\underline{\tau }}_j\left(B\right)$. Then, there exist $U,V\in {\tau }_j$ such that $w\in U\subseteq A$ and $w\in V\subseteq B$. Hence, $w\in U\cap V\subseteq A\cap B.$ Since $U\cap V\in {\tau }_j$, it follows that $w\in {\underline{\tau }}_j(A\cap B).$

Similarly, we can prove that ${\overline{\tau }}_j(A\cap B)\subseteq {\overline{\tau }}_j\left(A\right)\cap {\overline{\tau }}_j\left(B\right)$.

(v)

Since ${\underline{\tau }}_j\left(A\right)\subseteq {\underline{\tau }}_j(A\cup B)$ and ${\underline{\tau }}_j\left(B\right)\subseteq {\underline{\tau }}_j(A\cup B)$, we have ${\underline{\tau }}_j\left(A\right)\cup {\underline{\tau }}_j\left(B\right)\subseteq {\underline{\tau }}_j(A\cup B)$. Similarly, ${\overline{\tau }}_j\left(A\right)\cup {\overline{\tau }}_j\left(B\right)\subseteq {\overline{\tau }}_j(A\cup B)$. Conversely, for any $w\in {\overline{\tau }}_j(A\cup B)$, there exists $F^c\in {\tau }_j$ such that $w\in F$ and $A\cup B\subseteq F$. This implies that $A\subseteq F$ and $B\subseteq F$. Thus, $w\in {\overline{\tau }}_j\left(A\right)\cup {\overline{\tau }}_j\left(B\right).$ Therefore, ${\overline{\tau }}_j(A\cup B)={\overline{\tau }}_j\left(A\right)\cup {\overline{\tau }}_j\left(B\right)$.

(vi)

Since ${\underline{\tau }}_j\left(A\right)\subseteq A$, it follows that ${\underline{\tau }}_j\left({\underline{\tau }}_j\left(A\right)\right)\subseteq {\underline{\tau }}_j\left(A\right))$. To show the converse, assume $w\in {\underline{\tau }}_j\left(A\right).$ Then, $w\in U\subseteq A$ for some $U\in {\tau }_j.$ Given that $U\in {\tau }_j$, we have ${\underline{\tau }}_j\left(U\right)=U$. Thus, $w\in U\subseteq {\underline{\tau }}_j\left({\underline{\tau }}_j\left(A\right)\right)$.

Similarly, we can prove that ${\overline{\tau }}_j\left({\overline{\tau }}_j\left(A\right)\right)=\left({\overline{\tau }}_j\left(A\right)\right)$.

□

Example 12. 

Table 8 and Table 9 give the j-approximations of all subsets of the topological space $(X,\mathcal{R},{\tau }_j)$ given in Example 1.

Table 8. The j-approximations of all subsets when $j\in \{r,l,i,u\}$.

A	${\underline{\mathit{\tau }}}_{\mathit{r}}\left(\mathit{A}\right)$	${\overline{\mathit{\tau }}}_{\mathit{r}}\left(\mathit{A}\right)$	${\underline{\mathit{\tau }}}_{\mathit{l}}\left(\mathit{A}\right)$	${\overline{\mathit{\tau }}}_{\mathit{l}}\left(\mathit{A}\right)$	${\underline{\mathit{\tau }}}_{\mathit{i}}\left(\mathit{A}\right)$	${\overline{\mathit{\tau }}}_{\mathit{i}}\left(\mathit{A}\right)$	${\underline{\mathit{\tau }}}_{\mathit{u}}\left(\mathit{A}\right)$	${\overline{\mathit{\tau }}}_{\mathit{u}}\left(\mathit{A}\right)$
$\left\{a\right\}$	∅	$\{a,d\}$	∅	$\{a,b,c\}$	$\left\{a\right\}$	$\left\{a\right\}$	∅	X
$\left\{b\right\}$	∅	X	∅	$\{b,c\}$	∅	$\{b,c\}$	∅	X
$\left\{c\right\}$	∅	X	∅	$\{b,c\}$	∅	$\{b,c\}$	∅	X
$\left\{d\right\}$	∅	$\left\{d\right\}$	$\left\{d\right\}$	X	$\left\{d\right\}$	$\left\{d\right\}$	∅	X
$\{a,b\}$	∅	X	∅	$\{a,b,c\}$	$\left\{a\right\}$	$\{a,b,c\}$	∅	X
$\{a,c\}$	∅	X	∅	$\{a,b,c\}$	$\left\{a\right\}$	$\{a,b,c\}$	∅	X
$\{a,d\}$	∅	$\{a,d\}$	$\{a,d\}$	X	$\{a,d\}$	$\{a,d\}$	∅	X
$\{b,c\}$	$\{b,c\}$	X	∅	$\{b,c\}$	$\{b,c\}$	$\{b,c\}$	∅	X
$\{b,d\}$	∅	X	$\left\{d\right\}$	X	$\left\{d\right\}$	$\{b,c,d\}$	∅	X
$\{c,d\}$	∅	X	$\left\{d\right\}$	X	$\left\{d\right\}$	$\{b,c,d\}$	∅	X
$\{a,b,c\}$	$\{a,b,c\}$	X	∅	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	∅	X
$\{a,b,d\}$	∅	X	$\{a,d\}$	X	$\{a,d\}$	X	∅	X
$\{a,c,d\}$	∅	X	$\{a,d\}$	X	$\{a,d\}$	X	∅	X
$\{b,c,d\}$	$\{b,c\}$	X	$\left\{d\right\}$	X	$\{b,c,d\}$	$\{b,c,d\}$	∅	X

Table 9. The j-approximations of all subsets when $j\in \{<r>,<l>,<i>,<u>\}$.

A	${\underline{\mathit{\tau }}}_{<\mathit{r}>}\left(\mathit{A}\right)$	${\overline{\mathit{\tau }}}_{<\mathit{r}>}\left(\mathit{A}\right)$	${\underline{\mathit{\tau }}}_{<\mathit{l}>}\left(\mathit{A}\right)$	${\overline{\mathit{\tau }}}_{<\mathit{l}>}\left(\mathit{A}\right)$	${\underline{\mathit{\tau }}}_{<\mathit{i}>}\left(\mathit{A}\right)$	${\overline{\mathit{\tau }}}_{<\mathit{i}>}\left(\mathit{A}\right)$	${\underline{\mathit{\tau }}}_{<\mathit{u}>}\left(\mathit{A}\right)$	${\overline{\mathit{\tau }}}_{<\mathit{u}>}\left(\mathit{A}\right)$
$\left\{a\right\}$	∅	$\left\{a\right\}$	$\left\{a\right\}$	$\{a,b,c\}$	$\left\{a\right\}$	$\left\{a\right\}$	∅	X
$\left\{b\right\}$	∅	$\{a,b,c\}$	∅	$\{b,c\}$	∅	$\{b,c\}$	∅	X
$\left\{c\right\}$	∅	$\{a,b,c\}$	∅	$\{b,c\}$	∅	$\{b,c\}$	∅	X
$\left\{d\right\}$	$\left\{d\right\}$	$\left\{d\right\}$	$\left\{d\right\}$	$\left\{d\right\}$	$\left\{d\right\}$	$\left\{d\right\}$	∅	$\left\{d\right\}$
$\{a,b\}$	∅	$\{a,b,c\}$	$\left\{a\right\}$	$\{a,b,c\}$	$\left\{a\right\}$	$\{a,b,c\}$	∅	X
$\{a,c\}$	∅	$\{a,b,c\}$	$\left\{a\right\}$	$\{a,b,c\}$	$\left\{a\right\}$	$\{a,b,c\}$	∅	X
$\{a,d\}$	$\left\{d\right\}$	$\{a,d\}$	$\{a,d\}$	X	$\{a,d\}$	$\{a,d\}$	∅	X
$\{b,c\}$	$\{b,c\}$	$\{a,b,c\}$	∅	$\{b,c\}$	$\{b,c\}$	$\{b,c\}$	∅	X
$\{b,d\}$	$\left\{d\right\}$	X	$\left\{d\right\}$	$\{b,c,d\}$	$\left\{d\right\}$	$\{b,c,d\}$	∅	X
$\{c,d\}$	$\left\{d\right\}$	X	$\left\{d\right\}$	$\{b,c,d\}$	$\left\{d\right\}$	$\{b,c,d\}$	∅	X
$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	$\{a,b,c\}$	X
$\{a,b,d\}$	$\left\{d\right\}$	X	$\{a,d\}$	X	$\{a,d\}$	X	∅	X
$\{a,c,d\}$	$\left\{d\right\}$	X	$\{a,d\}$	X	$\{a,d\}$	X	∅	X
$\{b,c,d\}$	$\{b,c,d\}$	X	$\left\{d\right\}$	$\{b,c,d\}$	$\{b,c,d\}$	$\{b,c,d\}$	∅	X

Definition 16. 

Let $(X,\mathcal{R},{\tau }_j)$ be a j-TN-topological space. The regions and measures of subsets are derived using ${\underline{\tau }}_j\left(A\right)$ and ${\overline{\tau }}_j\left(A\right)$, analogous to the corresponding Definition 13.

Example 13. 

The j-accuracy of all subsets of the topological space $(X,\mathcal{R},{\tau }_j)$ and the topological space $(X,\mathcal{R},T_j)$, as presented in Example 1, are provided in Table 10 and Table 11, respectively.

Table 10. The j-accuracy of all subsets of $(X,\mathcal{R},T_j)$ for each $j\in J$.

A	${\mathcal{A}}_{{\mathit{T}}_{\mathit{r}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{T}}_{\mathit{l}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{T}}_{\mathit{i}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{T}}_{\mathit{u}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{T}}_{<\mathit{r}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{T}}_{<\mathit{l}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{T}}_{<\mathit{i}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{T}}_{<\mathit{u}>}}\left(\mathit{A}\right)$
$\left\{a\right\}$	0	0	0	0	0	0	1	0
$\left\{b\right\}$	0	0	0	0	0	0	0	0
$\left\{c\right\}$	0	0	0	0	${\displaystyle \frac{1}{3}}$	0	${\displaystyle \frac{1}{2}}$	0
$\left\{d\right\}$	0	${\displaystyle \frac{1}{4}}$	0	0	1	${\displaystyle \frac{1}{4}}$	1	${\displaystyle \frac{1}{4}}$
$\{a,b\}$	0	0	0	0	0	0	${\displaystyle \frac{1}{2}}$	0
$\{a,c\}$	0	0	0	0	${\displaystyle \frac{2}{3}}$	0	${\displaystyle \frac{2}{3}}$	0
$\{a,d\}$	0	${\displaystyle \frac{1}{2}}$	0	0	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	1	${\displaystyle \frac{1}{4}}$
$\{b,c\}$	0	0	0	0	${\displaystyle \frac{2}{3}}$	0	1	0
$\{b,d\}$	0	${\displaystyle \frac{1}{4}}$	0	0	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{4}}$
$\{c,d\}$	0	${\displaystyle \frac{1}{4}}$	0	0	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{2}{3}}$	${\displaystyle \frac{1}{2}}$
$\{a,b,c\}$	${\displaystyle \frac{3}{4}}$	0	0	0	1	0	1	0
$\{a,b,d\}$	0	${\displaystyle \frac{1}{2}}$	0	0	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{2}{3}}$	${\displaystyle \frac{1}{4}}$
$\{a,c,d\}$	0	${\displaystyle \frac{1}{2}}$	0	0	${\displaystyle \frac{3}{4}}$	${\displaystyle \frac{3}{4}}$	${\displaystyle \frac{3}{4}}$	${\displaystyle \frac{3}{4}}$
$\{b,c,d\}$	0	${\displaystyle \frac{1}{4}}$	0	0	${\displaystyle \frac{3}{4}}$	${\displaystyle \frac{1}{2}}$	1	${\displaystyle \frac{1}{2}}$

Table 11. The j-accuracy of all subsets of $(X,\mathcal{R},{\tau }_j)$ for each $j\in J$.

A	${\mathcal{A}}_{{\mathit{\tau }}_{\mathit{r}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{\tau }}_{\mathit{l}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{\tau }}_{\mathit{i}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{\tau }}_{\mathit{u}}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{\tau }}_{<\mathit{r}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{\tau }}_{<\mathit{l}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{\tau }}_{<\mathit{i}>}}\left(\mathit{A}\right)$	${\mathcal{A}}_{{\mathit{\tau }}_{<\mathit{u}>}}\left(\mathit{A}\right)$
$\left\{a\right\}$	0	0	1	0	0	${\displaystyle \frac{1}{3}}$	1	0
$\left\{b\right\}$	0	0	0	0	0	0	0	0
$\left\{c\right\}$	0	0	0	0	0	0	0	0
$\left\{d\right\}$	0	${\displaystyle \frac{1}{4}}$	1	0	1	1	1	0
$\{a,b\}$	0	0	${\displaystyle \frac{1}{3}}$	0	0	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{1}{3}}$	0
$\{a,c\}$	0	0	${\displaystyle \frac{1}{3}}$	0	0	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{1}{3}}$	0
$\{a,d\}$	0	${\displaystyle \frac{1}{2}}$	1	0	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	1	0
$\{b,c\}$	${\displaystyle \frac{1}{2}}$	0	1	0	${\displaystyle \frac{2}{3}}$	0	${\displaystyle \frac{2}{3}}$	0
$\{b,d\}$	0	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{1}{3}}$	0	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{1}{3}}$	0
$\{c,d\}$	0	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{1}{3}}$	0	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{1}{3}}$	${\displaystyle \frac{1}{3}}$	0
$\{a,b,c\}$	${\displaystyle \frac{3}{4}}$	0	1	0	1	1	1	${\displaystyle \frac{3}{4}}$
$\{a,b,d\}$	0	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	0	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	0
$\{a,c,d\}$	0	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	0	${\displaystyle \frac{1}{4}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{2}}$	0
$\{b,c,d\}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{4}}$	1	0	${\displaystyle \frac{3}{4}}$	${\displaystyle \frac{1}{3}}$	1	0

6. A Medical Application

Both direct and indirect contact play a major role in the spreading of infectious diseases throughout a community. Direct contact refers to the physical contact between an infected individual and another person that may lead to the fast-spreading of diseases. For instance, physical contact such as handshaking or kissing is effective in transferring viruses and bacteria. Indirect contact is the transmission of disease through an intermediate agent. One may not have contact with the infected person directly but nevertheless be exposed to a risk of the disease through the third party that has been in contact with an infected person. For instance, in a case where individual A is infected and has direct contact with individual B, and later on, individual C comes into contact with individual B, individual C is considered to have indirect contact with the disease of individual A. Differentiating the modes of transmission, especially direct and indirect contact, is thus paramount in infectious disease control.

The j-neighborhood concept has been applied to various decision-making tools in studies that aim to identify and isolate contacts to control infectious disease spread and break the transmission chain [29]. These studies have mainly focused on direct contacts. However, determining indirect contacts becomes especially essential in those illnesses with a long incubation period where indirect contact remains asymptomatic for a longer period of time, so preventive measures may be taken more effectively.

A scenario is considered below where an epidemic is present. The concept of j-transitive neighborhoods (j-TN) is adapted to identify indirect contacts and implement necessary measures to mitigate the spread of the epidemic.

Consider the set X, which comprises individuals working in an institution and their families (or people they are in close contact with). When an infectious disease starts spreading among employees, those who test positive for the disease are immediately placed in quarantine. Let $E\subseteq X$ represent the set of infected people. To mitigate the risk of further transmission, people who have had direct contact with infected patients are also placed under observation. Let $D\subseteq X-E$ denote the set of these direct contacts.

Now, our objective is to investigate the risk of infection among those who are not infected or who have not had direct contact with infected individuals. This subset, represented as $T=X-(E\cup D)$, includes employees and their families who have not directly interacted with infected patients.

We define a relation on X such that $(x,y)\in \mathcal{R}\iff$“$x$ is in direct contact with y and $x\ne y$”. Since $\mathcal{R}$ is symmetric, we have only two types of j-transitive neighborhoods by Proposition 2. Therefore, the set of indirect contacts of x is $T{N}_l\left(x\right)(T{N}_r\left(x\right)=T{N}_i\left(x\right)=T{N}_u\left(x\right))$.

To classify each $x\in T$ according to their risk of infection, we define the following risk levels (these numbers can be changed with an expert opinion) based on the number of contacts.

Low Risk: The number of contacts is between 0 and 3 (inclusive).

Moderate Risk: The number of contacts is between 4 and 7 (inclusive).

High Risk: The number of contacts exceeds 7.

• Afterward, precautions can be taken to the extent that the available resources allow.

To better understand this scenario, we examine the following example. Let $X=\{x_1,x_2,\dots ,x_{22}\}$ and $E=\{x_1,x_2,\dots ,x_8\}$. In Table 12, the symbol “✓” is used to indicate direct contact within the set X.

Table 12. Direct Contacts.

	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$	${\mathit{x}}_9$	${\mathit{x}}_{10}$	${\mathit{x}}_{11}$	${\mathit{x}}_{12}$	${\mathit{x}}_{13}$	${\mathit{x}}_{14}$	${\mathit{x}}_{15}$	${\mathit{x}}_{16}$	${\mathit{x}}_{17}$	${\mathit{x}}_{18}$	${\mathit{x}}_{19}$	${\mathit{x}}_{20}$	${\mathit{x}}_{21}$	${\mathit{x}}_{22}$
$x_1$									✓		✓	✓		✓	✓
$x_2$									✓	✓	✓
$x_3$									✓	✓	✓	✓	✓
$x_4$									✓		✓				✓
$x_5$											✓	✓	✓
$x_6$												✓		✓	✓
$x_7$												✓	✓	✓
$x_8$												✓		✓	✓
$x_9$	✓	✓	✓	✓												✓
$x_{10}$		✓	✓														✓
$x_{11}$	✓	✓	✓	✓	✓												✓					✓
$x_{12}$	✓		✓		✓	✓	✓	✓								✓			✓
$x_{13}$			✓		✓		✓											✓	✓
$x_{14}$	✓					✓	✓	✓												✓	✓
$x_{15}$	✓			✓		✓		✓											✓	✓
$x_{16}$									✓			✓
$x_{17}$										✓	✓
$x_{18}$													✓
$x_{19}$												✓	✓		✓
$x_{20}$														✓	✓
$x_{21}$														✓
$x_{22}$											✓

According to the Table 12, the set of direct contacts is $D=\{x_9,x_{10},x_{11},x_{12},x_{13},x_{14},x_{15}\}$, and the set of indirect contacts is $T=\{x_{16},x_{17},x_{18},x_{19},x_{20},x_{21},x_{22}\}$. The l-transitive neighborhoods of the individuals in T are given below:

$T{N}_l\left(x_{16}\right)=\{x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8\}$.

$T{N}_l\left(x_{17}\right)=\{x_1,x_2,x_3,x_4,x_5\}$.

$T{N}_l\left(x_{18}\right)=\{x_1,x_3,x_5\}$.

$T{N}_l\left(x_{19}\right)=\{x_1,x_3,x_4,x_5,x_6,x_7,x_8\}$.

$T{N}_l\left(x_{20}\right)=\{x_1,x_4,x_6,x_7,x_8\}$.

$T{N}_l\left(x_{21}\right)=\{x_1,x_6,x_7,x_8\}$.

$T{N}_l\left(x_{22}\right)=\{x_1,x_2,x_3,x_4,x_5\}$.

• Since T is the set of indirect contacts of E, no individual in T has an l-neighborhood in the set E. Thus, the classifications derived using the concepts of l-transitive neighborhood and l-neighborhood, and the given risk levels above are presented in Table 13.
  Table 13. Risk Classification.

  	l-Transitive Neighborhood	l-Neighborhood
  Low Risk	$\{x_{18},x_{23}\}$	T
  Moderate Risk	$\{x_{17},x_{19},x_{20},x_{21},x_{22}\}$	∅
  High Risk	$\left\{x_{16}\right\}$	∅

Table 13 reveals that individuals who should be categorized as high-risk are instead classified as low-risk when the j-neighborhood concept is applied.

7. Conclusions and Future Work

In order to cope with the cases where it is not clear whether an element belongs to a set, equivalence classes are examined and three different cases are determined: the element is not a member of the set, it is definitely a member, or it is probably a member. By introducing the concepts of right and left neighborhoods, the requirement for an equivalence relation is eliminated, allowing for a reduction in uncertainty in more complex or ambiguous situations. For this purpose, the neighborhood concept has been generalized in many studies, and the rough set concept has been reconsidered from different perspectives [30,31]. In almost all neighborhood concepts defined previously in the literature, other elements directly related to an element have been examined. The j-transitive neighborhood (j-TN) concept defined in this study focuses on resolving uncertainties where indirect relationships are involved. The main results obtained within the scope of the study are summarized below:

* As demonstrated in Example 1, the concepts of j-neighborhood and j-transitive neighborhood are not directly related.

* Theorem 9 shows that, unlike certain generalized approximations present in the literature [9,11], the lower–upper approximation concepts derived using j-TN satisfy the properties of the Pawlak approximation, as described in Proposition 1, except the equality in (P5).

* Table 6 and Table 7 show that a clear comparison cannot be made between the j-accuracy and $T{N}_j$ accuracy values. However, in terms of the topological accuracy values examined in Table 11, a higher accuracy value is obtained in the case of $j=i$.

* The concept of j-transitive neighborhoods (j-TN) was examined in the context of preventing the spread of infectious diseases, where indirect contact plays a critical role. Indirect contacts were grouped according to their risk of transmitting the disease. The classification revealed that individuals categorized as high-risk under the j-transitive neighborhood were classified as low-risk when evaluated using the j-neighborhood (see Table 13). This observation indicates that the defined neighborhood concept contributes to a more accurate categorization of at-risk individuals.

In a future study, more comprehensive analyses of the approximations obtained with j-TN will be carried out, and these approximations will be compared with the approximation concepts defined previously in the literature. In addition, some topological concepts such as generalized open sets, supratopology, and infratopology are planned to be adapted to the j-TN concept. Since the hybridization of rough neighborhoods with ideal improves the properties of approximation operators and increases the accuracy value, one can examine the current concepts from this perspective.