Idealizing Rough Topological Structures Generated by Several Types of Maximal Neighborhoods and Exploring Their Applications

Abstract

Several different topologies utilizing ideals are created and compared with previous topologies. The results show that the previous ones are weaker than the current ones and that the current ones are stronger. The merits of these topologies are proposed, and the smallest and largest among them are identified; this merit distinguishes the present study from previous ones. Afterwards, these topologies are employed to conduct more in-depth investigations on broadened rough sets. The proposed approximate models are particularly significant as applied to rough sets because they diminish vagueness and uncertainty compared to prior models. Moreover, the proposed models stand out from their predecessors because they can compare all types of approximations, display all the features described by Pawlak, and possess the property of monotonicity across any relations. Furthermore, a medical application is showcased to emphasize the significance of the current findings. Additionally, the advantages of the adopted approach are examined, alongside an evaluation of its limitations. The paper wraps up with the essential features of the proposed manner and recommend avenues for future research.

1. Introduction

Rough set theory, developed by Pawlak in 1980s [1,2], was introduced to handle uncertainty in data analysis. It is particularly useful for managing and interpreting imperfect or incomplete information by categorizing data into equivalence classes. Each class represents a collection of objects that are indistinguishable from each other given the information at hand. The theory uses two key approximations to address uncertainty. This distinction helps in managing and analyzing data where exact boundaries are not clear. It provides a formal method for dealing with incomplete or vague information by approximating sets that cannot be precisely defined.

The generalization of rough set theory extends the original framework to address a broader range of uncertainties and complexities in data analysis (see [3,4,5,6]). While traditional rough set theory focuses on equivalence relation, its generalizations aim to refine and expand this approach to handle more nuanced types of information. These generalizations enhance the ability of rough set theory to manage diverse and intricate datasets, making it applicable to a wider range of problems in fields such as data mining, artificial intelligence, and decision support systems. By broadening the scope of rough set theory, researchers and practitioners can achieve more accurate and meaningful insights from data with varying degrees of uncertainty.

One of these generalizations was achieved through topology. The link between topology and rough sets was first recognized by [7,8]. In this context, the lower approximation is associated with the topological notion of the interior, whereas the upper approximation relates to the closure (see [9,10,11,12,13,14]). An ideal is a nonempty collection of sets that is closed under a hereditary property and finite additivity [15]. Ideals in topological spaces have been studied since 1930, with significant contributions from Vaidyanathaswamy [15]. In 1990, Jankovic and Hamlett [16] expanded on this concept by introducing ideal topological spaces and exploring their applications in various fields. Since then, ideals have been recognized as a key topological concept for investigating rough sets. The primary advantage of incorporating ideals into this theory is their ability to diminish ambiguity by refining conceptual boundaries. This increased precision enhances the certainty of knowledge and improves the reliability of decision-making processes (see [17,18,19,20,21]).

Another important topological concept crucial for grasping and examining set approximations is neighborhoods. In [22,23], approximation spaces were created by neighborhoods and any relation. In 2020, Hosny [17] was the first to combine neighborhoods and ideals to create different topologies as a generalization of the previous ones [24,25]. She employed these topologies to develop approximations through ideals as an extension of the earlier work [24,25]. Subsequently, many researchers applied the same idea of ideals in various types of neighborhoods (see [26,27,28]) to create different topologies and then used them to define approximation operators. A limitation of these approximations is that they sacrifice some of Pawlak’s properties and do not possess the property of monotonicity. The concept of maximal neighborhoods was proposed in [29]. This concept was applied to propose three sorts of approximations by similarity relations. Later, the other types of maximal neighborhoods were described by Al-shami [30]. He employed these neighborhoods to suggest one type of approximation as an extension of [29], and the other types of approximation was studied in [31]. These approximations [29,30,31] formed directly by various kinds of maximal neighborhoods have been generalized using ideals in [18,32,33,34]. Later, Taher et al. [35] generated topology using maximal right neighborhoods. More recently, Hosny [36] generalized Taher et al.’s topology [35] by presenting topologies induced by the remaining types of maximal neighborhoods. Additionally, she [36] utilized these topologies to evolve new approximations.

A particular extension employing ideals is investigated in this study. Recognizing the pivotal role of ideals in affecting the topological rough set problems, this work concentrates on creating various topologies using ideals and emphasizes the interconnections between these topologies and rough sets. This article is organized into eight sections, presented in the following sequence: Section 2 offers a summary of key definitions. Section 3 explores the different topologies formed by ideals. It introduces comparisons among these topologies and identifies the smallest and largest ones, unlike existing approaches [17,26,27,28] that only compare topologies within distinct sets. These topologies are finer and more desirable than those in [35,36] and when the ideal is the empty set, they align with Hosny’s topologies [36] and Taher et al.’s topology [35], thus making the prior topologies [35,36] a specific instance of the current ones. The section concludes by establishing conditions for deriving equivalences among these topologies. New approximations are inspected by employing the suggested topologies and their features are detailed in Section 4. They fulfill all of Pawlak’s properties without restrictions and possess the property of monotonicity, in contrast to the earlier ones [26,27,28]. This property ensures that the approximations either become more precise or stay the same as additional information is added, never becoming coarser. This feature is vital for preserving the reliability of rough set analysis, providing a solid framework for data interpretation and understanding. Section 5 presents the relationships among all types of the suggested approximations, which are not available in [17,26,27,28]. These approximations extend those in [36] to any relation, offering reduced boundary effects and improved accuracy. Additionally, a medical application is proposed in Section 6. It showcases the practical utility and efficacy of the proposed models, emphasizing the vital role of ideals in decision-making. As a result, these methods allow doctors to diagnose dengue fever both easily and with great precision. Section 7 contains the discussion, and this paper comes to a close in the conclusion section.

2. Preliminaries

This section outlines the key concepts essential for comprehending this manuscript.

Definition 1. 

Take ⨿ as an arbitrary binary relation on a finite set $\mathcal{Q}\ne \varnothing$, and suppose that ${♭}_5\in \mathcal{Q}.$ Then, the following terms are defined:

(i) 

${\mathcal{N}}_r\left({♭}_5\right)=\{{♭}_4\in \mathcal{Q}:({♭}_5,{♭}_4)\in ⨿\}$ [37];

(ii) 

${\mathcal{N}}_l\left({♭}_5\right)=\{{♭}_4\in \mathcal{Q}:({♭}_4,{♭}_5)\in ⨿\}$ [37];

(iii) 

${\mathcal{N}}_i\left({♭}_5\right)={\mathcal{N}}_r\left({♭}_5\right)\cap {\mathcal{N}}_l\left({♭}_5\right)$ [37];

(iv) 

${\mathcal{N}}_u\left({♭}_5\right)={\mathcal{N}}_r\left({♭}_5\right)\cup {\mathcal{N}}_l\left({♭}_5\right)$ [37];

(v) 

${\mathcal{N}}_{\langle r\rangle }\left({♭}_5\right)={\bigcap }_{{♭}_5\in {\mathcal{N}}_r\left({♭}_4\right)}{\mathcal{N}}_r\left({♭}_4\right)$ [38];

(vi) 

${\mathcal{N}}_{\langle \mathfrak{l}\rangle }\left({♭}_5\right)={\bigcap }_{{♭}_5\in {\mathcal{N}}_l\left({♭}_4\right)}{\mathcal{N}}_l\left({♭}_4\right)$ [38];

(vii) 

${\mathcal{N}}_{\langle i\rangle }\left({♭}_5\right)={\mathcal{N}}_{\langle r\rangle }\left({♭}_5\right)\cap {\mathcal{N}}_{\langle l\rangle }\left({♭}_5\right)$ [39];

(viii) 

${\mathcal{N}}_{\langle u\rangle }\left({♭}_5\right)={\mathcal{N}}_{\langle r\rangle }\left({♭}_5\right)\cup {\mathcal{N}}_{\langle l\rangle }\left({♭}_5\right),$ [25];

(ix) 

The triple $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ is known as a ≀-neighborhood space (abbreviated as ≀-NS), $\wr \in \{r,\mathfrak{l},i,u,\langle r\rangle ,\langle \mathfrak{l}\rangle ,\langle i\rangle ,\langle u\rangle \},$ and ${\varsigma }_{\wr }$ is a function from $\mathcal{Q}$ to $P\left(\mathcal{Q}\right)$ that connects each ${♭}_5\in \mathcal{Q}$ with a ≀-neighborhood [25].

Theorem 1 

([17]). Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ Then,

$${\mathfrak{J}}_{{\mathcal{N}}_{\wr }}^{\pounds }=\{V\subseteq \mathcal{Q}:{\mathcal{N}}_{\wr }\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\mathit{for}\mathit{all}{♭}_5\in V\}$$

represents an ${\mathcal{N}}_{\wr }^{\pounds }$-topology on $\mathcal{Q}$ regarding the ideal $\pounds ,$ in which $V^{\prime }$ signifies the complementary set of $V.$

Definition 2 

([17]). Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ The £-${\mathcal{N}}_{\wr }$-lower approximation ${\underline{⨿}}_{{\mathcal{N}}_{\wr }}^{\pounds }$ and £-${\mathcal{N}}_{\wr }$-upper approximation ${\overline{⨿}}_{{\mathcal{N}}_{\wr }}^{\pounds }$ of $V\subseteq \mathcal{Q}$ are

$${\underline{⨿}}_{{\mathcal{N}}_{\wr }}^{\pounds }\left(V\right)=\cup \{S\in {\mathfrak{J}}_{{\mathcal{N}}_{\wr }}^{\pounds }:S\subseteq V\}=In{t}_{{\mathcal{N}}_{\wr }}^{\pounds }\left(V\right),$$

$${\overline{⨿}}_{{\mathcal{N}}_{\wr }}^{\pounds }\left(V\right)=\cap \{X:X^{\prime }\in {\mathfrak{J}}_{{\mathcal{N}}_{\wr }}^{\pounds }:V\subseteq X\}=C{l}_{{\mathcal{N}}_{\wr }}^{\pounds }\left(V\right).$$

Definition 3 

([29,30]). Take ⨿ as an arbitrary binary relation on $\mathcal{Q}\ne \varnothing .$ Then, the maximal neighborhoods of ${♭}_5\in \mathcal{Q}$ are as follows:

(i) 

${\mathcal{M}}_r\left({♭}_5\right)={\bigcup }_{{♭}_5\in {\mathcal{N}}_r\left({♭}_4\right)}{\mathcal{N}}_r\left({♭}_4\right);$

(ii) 

${\mathcal{M}}_{\mathfrak{l}}\left({♭}_5\right)={\bigcup }_{{♭}_5\in {\mathcal{N}}_{\mathfrak{l}}\left({♭}_4\right)}{\mathcal{N}}_{\mathfrak{l}}\left({♭}_4\right);$

(iii) 

${\mathcal{M}}_i\left({♭}_5\right)={\mathcal{M}}_r\left({♭}_5\right)\cap {\mathcal{M}}_l\left({♭}_5\right);$

(iv) 

${\mathcal{M}}_u\left({♭}_5\right)={\mathcal{M}}_r\left({♭}_5\right)\cup {\mathcal{M}}_l\left({♭}_5\right);$

(v) 

${\mathcal{M}}_{\langle r\rangle }\left({♭}_5\right)={\bigcap }_{{♭}_5\in {\mathcal{M}}_r\left(\mathcal{M}\right)}{\mathcal{M}}_r\left({♭}_4\right);$

(vi) 

${\mathcal{M}}_{\langle \mathfrak{l}\rangle }\left({♭}_5\right)={\bigcap }_{{♭}_5\in {\mathcal{M}}_{\mathfrak{l}}\left({♭}_4\right)}{\mathcal{M}}_{\mathfrak{l}}\left({♭}_4\right);$

(vii) 

${\mathcal{M}}_{\langle i\rangle }\left({♭}_5\right)={\mathcal{M}}_{\langle r\rangle }\left({♭}_5\right)\cap {\mathcal{M}}_{\langle \mathfrak{l}\rangle }\left({♭}_5\right);$

(viii) 

${\mathcal{M}}_{\langle u\rangle }\left({♭}_5\right)={\mathcal{M}}_{\langle r\rangle }\left({♭}_5\right)\cup {\mathcal{M}}_{\langle \mathfrak{l}\rangle }\left({♭}_5\right).$

Theorem 2 

([30]). Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and ${♭}_5\in \mathcal{Q}$. Then, the following statements are true:

(i) 

${\mathcal{M}}_{\langle \wr \rangle }\left({♭}_5\right)\subseteq {\mathcal{M}}_{\wr }\left({♭}_5\right),\wr \in \{r,\mathfrak{l},i,u\};$

(ii) 

${\mathcal{M}}_r\left({♭}_5\right)={\mathcal{M}}_l\left({♭}_5\right)={\mathcal{M}}_i\left({♭}_5\right)={\mathcal{M}}_u\left({♭}_5\right)$ and ${\mathcal{M}}_{\langle r\rangle }\left({♭}_5\right)={\mathcal{M}}_{\langle l\rangle }\left({♭}_5\right)={\mathcal{M}}_{\langle i\rangle }\left({♭}_5\right)={\mathcal{M}}_{\langle u\rangle }\left({♭}_5\right)$ when ⨿ is symmetric;

(iii) 

${\mathcal{M}}_{\wr }\left({♭}_5\right)={\mathcal{M}}_{\langle \wr \rangle }\left({♭}_5\right),\forall \wr \in \{r,l,i,u\}$ when ⨿ is symmetric and transitive;

(iv) 

All types of ${\mathcal{M}}_{\wr }\left({♭}_5\right)$ are equal when ⨿ is equivalence.

Proposition 1 

([30]). Let $(\mathcal{Q},{⨿}_1,{\varsigma }_{1\wr })$ and $(\mathcal{Q},{⨿}_2,{\varsigma }_{2\wr })$ be two ≀-NSs. If ${⨿}_1\subseteq {⨿}_2,$ then ${\mathcal{M}}_{1\wr }\left({♭}_5\right)\subseteq {\mathcal{M}}_{2\wr }\left({♭}_5\right),\forall {♭}_5\in \mathcal{Q}$, and $\wr \in \{r,\mathfrak{l},i,u\}.$

Definition 4 

([18,32,33,34]). Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ The ${{\mathcal{M}}_{\wr }}^{{\pounds }^{*}}$-lower and ${{\mathcal{M}}_{\wr }}^{{\pounds }^{*}}$-upper approximations of a set V are as follows

$${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{*}}\left(V\right)=\{{♭}_5\in \mathcal{Q}:{\mathcal{M}}_{\wr }\left({♭}_5\right)\cap V^{\prime }\in \pounds \},$$

$${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{*}}\left(V\right)=\{{♭}_5\in \mathcal{Q}:{\mathcal{M}}_{\wr }\left({♭}_5\right)\cap V\notin \pounds \}.$$

Definition 5 

([18,32,33,34]). Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ The ${{\mathcal{M}}_{\wr }}^{{\pounds }^{**}}$-lower and ${{\mathcal{M}}_{\wr }}^{{\pounds }^{**}}$-upper approximations of a set V are

$${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{**}}\left(V\right))=\cup \{{\mathcal{M}}_{\wr }\left({♭}_5\right):{\mathcal{M}}_{\wr }\left({♭}_5\right)\cap V^{\prime }\in \pounds \},$$

$${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{**}}\left(V\right)={\left[{\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{**}}\left(V^{\prime }\right)\right]}^{\prime }.$$

Definition 6 

([18,32,33,34]). Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ The ${{\mathcal{M}}_{\wr }}^{{\pounds }^{***}}$-lower and ${{\mathcal{M}}_{\wr }}^{{\pounds }^{***}}$-upper approximations of a set V are

$${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{***}}\left(V\right)=\cup \{{\mathcal{M}}_{\wr }\left({♭}_5\right):{\mathcal{M}}_{\wr }\left({♭}_5\right)\cap V\notin \pounds \},$$

$${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{***}}\left(V\right))={\left[{\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{***}}\left(V^{\prime }\right)\right]}^{\prime }.$$

Theorem 3. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS. Then, ${\mathfrak{J}}_{{\mathcal{M}}_r}=\{V\subseteq \mathcal{Q}:{\mathcal{M}}_r\left({♭}_5\right)\subseteq V,\forall {♭}_5\in V\}$ [35], and ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}=\{V\subseteq \mathcal{Q}:{\mathcal{M}}_{\wr }\left({♭}_5\right)\subseteq V,\forall {♭}_5\in V\},\forall \wr \in \{\mathfrak{l},i,u,\langle r\rangle ,\langle \mathfrak{l}\rangle ,\langle i\rangle ,\langle u\rangle \}$ [36] represent ${\mathcal{M}}_{\wr }$-topology on $\mathcal{Q}.$

Definition 7 

([36]). Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS. The ${\mathcal{M}}_{\wr }$-lower approximation, ${\mathcal{M}}_{\wr }$-upper approximation, ${\mathcal{M}}_{\wr }$-accuracy, and ${\mathcal{M}}_{\wr }$-boundary of V are as follows:

$${\underline{⨿}}_{{\mathcal{M}}_{\wr }}\left(V\right)=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}:S\subseteq V\}=In{t}_{{\mathcal{M}}_{\wr }}\left(V\right),$$

$${\overline{⨿}}_{{\mathcal{M}}_{\wr }}\left(V\right)=\cap \{X:X^{\prime }\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}:V\subseteq X\}=C{l}_{{\mathcal{M}}_{\wr }}\left(V\right),$$

$${\mathcal{A}}_{{\mathcal{M}}_{\wr }}\left(V\right)={\displaystyle \frac{\mid {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\mid }{\mid {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\mid }}, \mathit{where} V\ne \varnothing ,$$

$${\mathcal{B}}_{{\mathcal{M}}_{\wr }}\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\setminus {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right).$$

3. Idealization of Topologies Derived from Various Kinds of Maximal Neighborhoods

Topologies constructed based on the right maximal neighborhoods were introduced in [35]. Additionally, Hosny [36] proposed the remaining seven topologies derived from different maximal neighborhoods as an extension of [35]. This section generalizes these topologies through ideals and examines their relationships.

In Theorem 4, the topologies are formed by merging the maximal neighborhoods and ideals, serving as a conceptual expansion of the topologies in [35,36].

Theorem 4. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ The collection

$${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }=\{V\subseteq \mathcal{Q}:{\mathcal{M}}_{\wr }\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\mathit{for}\mathit{all}{♭}_5\in V\}$$

(1)

represents an ${\mathcal{M}}_{\wr }^{\pounds }$-topology on $\mathcal{Q}$ regarding the ideal $\pounds .$

Proof. 

(i)

Clearly, $\mathcal{Q},\varnothing \in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }.$

(ii)

Let $V_i\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }(\mathrm{for}\mathrm{all}i\in I)$ and ${♭}_5\in {\bigcup }_{i\in I}{V}_i$. Then, $\exists i_0\in I\mathrm{such}\mathrm{that}{♭}_5\in V_{i_0}.$

$\Rightarrow {\mathcal{M}}_{\wr }\left({♭}_5\right)\cap {\left(V_{i_0}\right)}^{\prime }\in \pounds .$

$\Rightarrow {\mathcal{M}}_{\wr }\left({♭}_5\right)\cap \left({\bigcup }_{i\in I}{V}_i\right),$ and so ${\bigcup }_{i\in I}{V}_i\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }.$

(iii)

Let $V_1,V_2\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds },$ and ${♭}_5\in V_1\cap V_2.$

$\Rightarrow {\mathcal{M}}_{\wr }\left({♭}_5\right)\cap {V_1}^{\prime }\in \pounds$ and ${\mathcal{M}}_{\wr }\left({♭}_5\right)\cap {V_2}^{\prime }\in \pounds .$

$\Rightarrow ({\mathcal{M}}_{\wr }\left({♭}_5\right)\cap {V_1}^{\prime })\cup ({\mathcal{M}}_{\wr }\left({♭}_5\right)\cap {V_2}^{\prime })\in \pounds .$

$\Rightarrow ({\mathcal{M}}_{\wr }\left({♭}_5\right)\cap {(V_1\cap V_2)}^{\prime })\in \pounds .$

$\Rightarrow V_1\cap V_2\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }.$

Therefore, ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$ is an ${\mathcal{M}}_{\wr }^{\pounds }$-topology on $\mathcal{Q}$ regarding the ideal $\pounds .$ □

Definition 8. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q}$, and $V\subseteq \mathcal{Q}.$ If $V\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$, then it is known as ${\pounds }_{{\mathcal{M}}_{\wr }}$-open, and if $V^{\prime }\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$, then it is called ${\pounds }_{{\mathcal{M}}_{\wr }}$-closed. All ${\pounds }_{{\mathcal{M}}_{\wr }}$-closed subsets of $\mathcal{Q}$ are represented by ${\mathfrak{I}}_{{\mathcal{M}}_{\wr }}^{\pounds }$.

Definition 9. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ The $\begin{array}{c}\hfill \pounds \hfill \\ \hfill {\mathcal{M}}_{\wr }\hfill \end{array}$-interior and $\begin{array}{c}\hfill \pounds \hfill \\ \hfill {\mathcal{M}}_{\wr }\hfill \end{array}$-closure of $V\subseteq \mathcal{Q}$ are

$$In{t}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }:S\subseteq V\} \mathit{and}$$

$$C{l}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)=\cap \{X\in {\mathfrak{I}}_{{\mathcal{M}}_{\wr }}^{\pounds }:V\subseteq X\}.$$

The previous topologies in [35,36] are coarser than the current ones as evidenced by Theorem 5.

Theorem 5. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ Then, ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }.$

Proof. 

Let $V\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}$. Then, ${\mathcal{M}}_{\wr }\left({♭}_5\right)\subseteq V\mathrm{for}\mathrm{all}{♭}_5\in V$. Thus, ${\mathcal{M}}_{\wr }\left({♭}_5\right)\cap V^{\prime }\in \pounds \mathrm{for}\mathrm{all}{♭}_5\in V$. Therefore, $V\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$. Hence, ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }.$ □

Remark 1. 

One should be aware of the following:

1. 

When $\pounds =\{\varnothing \}$ in an ${\mathcal{M}}_{\wr }^{\pounds }$-topology (1), ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}={\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }.$ Thus, the proposed manner embodies a more inclusive representation of [35,36].

2. 

${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}⊊{\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$, as manifested in Example 1.

Example 1. 

Let $⨿=\{({♭}_1,{♭}_1),({♭}_2,{♭}_3),({♭}_2,{♭}_4),({♭}_3,{♭}_4),({♭}_3,{♭}_5),({♭}_4,{♭}_1),({♭}_4,{♭}_5),({♭}_5,{♭}_2),({♭}_5,{♭}_5)\}$ be a relation on $\mathcal{Q}=\{{♭}_1,{♭}_2,{♭}_3,{♭}_4,{♭}_5\}$. Then, all ${\mathcal{M}}_{\wr }$-neighborhoods are in

Table 1. ${\mathcal{M}}_{\wr }$-neighborhoods.

	${♭}_1$	${♭}_2$	${♭}_3$	${♭}_4$	${♭}_5$
${\mathcal{N}}_r$	$\left\{{♭}_1\right\}$	$\{{♭}_3,{♭}_4\}$	$\{{♭}_4,{♭}_5\}$	$\{{♭}_1,{♭}_5\}$	$\{{♭}_2,{♭}_5\}$
${\mathcal{N}}_{\mathfrak{l}}$	$\{{♭}_1,{♭}_4\}$	$\left\{{♭}_5\right\}\}$	$\left\{{♭}_2\right\}$	$\{{♭}_2,{♭}_3\}$	$\{{♭}_3,{♭}_4,{♭}_5\}$
${\mathcal{M}}_r$	$\{{♭}_1,{♭}_5\}$	$\{{♭}_2,{♭}_5\}$	$\{{♭}_3,{♭}_4\}$	$\{{♭}_3,{♭}_4,{♭}_5\}$	$\{{♭}_1,{♭}_2,{♭}_4,{♭}_5\}$
${\mathcal{M}}_{\mathfrak{l}}$	$\{{♭}_1,{♭}_4\}$	$\{{♭}_2,{♭}_3\}$	$\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}$	$\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\}$	$\{{♭}_3,{♭}_4,{♭}_5\}$
${\mathcal{M}}_i$	$\left\{{♭}_1\right\}$	$\left\{{♭}_2\right\}$	$\{{♭}_3,{♭}_4\}$	$\{{♭}_3,{♭}_4,{♭}_5\}$	$\{{♭}_4,{♭}_5\}$
${\mathcal{M}}_u$	$\{{♭}_1,{♭}_4,{♭}_5\}$	$\{{♭}_2,{♭}_3,{♭}_5\}$	$\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}$	$\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\}$	$\mathcal{Q}$
${\mathcal{M}}_{\langle r\rangle }$	$\{{♭}_1,{♭}_5\}$	$\{{♭}_2,{♭}_5\}$	$\{{♭}_3,{♭}_4\}$	$\left\{{♭}_4\right\}$	$\left\{{♭}_5\right\}$
${\mathcal{M}}_{\langle \mathfrak{l}\rangle }$	$\{{♭}_1,{♭}_4\}$	$\{{♭}_2,{♭}_3\}$	$\left\{{♭}_3\right\}$	$\left\{{♭}_4\right\}$	$\{{♭}_3,{♭}_4,{♭}_5\}$
${\mathcal{M}}_{\langle i\rangle }$	$\left\{{♭}_1\right\}$	$\left\{{♭}_2\right\}$	$\left\{{♭}_3\right\}$	$\left\{{♭}_4\right\}$	$\left\{{♭}_5\right\}$
${\mathcal{M}}_{\langle u\rangle }$	$\{{♭}_1,{♭}_4,{♭}_5\}$	$\{{♭}_2,{♭}_3,{♭}_5\}$	$\{{♭}_3,{♭}_4\}$	$\left\{{♭}_4\right\}$	$\{{♭}_3,{♭}_4,{♭}_5\}$

.

Let $\pounds =\{\varnothing ,\left\{{♭}_4\right\},\left\{{♭}_5\right\},\{{♭}_4,{♭}_5\}\}.$ Consequently, the following statements are true:

1. 

${\mathfrak{J}}_{{\mathcal{M}}_r}=\{\mathcal{Q},\varnothing \}$ and ${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }=\{\mathcal{Q},\varnothing ,\left\{{♭}_1\right\},\left\{{♭}_2\right\},\left\{{♭}_3\right\},\{{♭}_1,{♭}_2\},\{{♭}_1,{♭}_3\},\{{♭}_2,{♭}_3\},\{{♭}_3,{♭}_4\},$ $\{{♭}_1,{♭}_2,{♭}_3\},\{{♭}_1,{♭}_2,{♭}_5\},\{{♭}_1,{♭}_3,{♭}_4\},\{{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_5\}\};$

2. 

${\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}=\{\mathcal{Q},\varnothing \}$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }=\{\mathcal{Q},\varnothing ,\left\{{♭}_1\right\},\left\{{♭}_3\right\},\{{♭}_2,{♭}_3\},\{{♭}_1,{♭}_2,{♭}_3\},\{,{♭}_3,{♭}_5\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_5\}\};$

3. 

${\mathfrak{J}}_{{\mathcal{M}}_i}=\{\mathcal{Q},\varnothing ,\left\{{♭}_1\right\},\left\{{♭}_2\right\},\{{♭}_1,{♭}_2\},\{{♭}_3,{♭}_4,{♭}_5\},\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\},\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}\}$ and ${\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }=\{\mathcal{Q},\varnothing ,\left\{{♭}_1\right\},\left\{{♭}_2\right\},\left\{{♭}_3\right\},\left\{{♭}_5\right\},\{{♭}_1,{♭}_2\},\{{♭}_1,{♭}_3\},\{{♭}_1,{♭}_5\},\{{♭}_2,{♭}_3\},\{{♭}_2,{♭}_5\},\{{♭}_3,{♭}_4\},\{{♭}_3,{♭}_5\},$ $\{{♭}_1,{♭}_2,{♭}_3\},\{{♭}_1,{♭}_2,{♭}_5\},\{{♭}_1,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_3,{♭}_5\},\{{♭}_2,{♭}_3,{♭}_4\},\{{♭}_2,{♭}_3,{♭}_5\},\{{♭}_3,{♭}_4,{♭}_5\},$$\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_5\},\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\},\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}\};$

4. 

${\mathfrak{J}}_{{\mathcal{M}}_u}=\{\mathcal{Q},\varnothing \}$ and ${\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }=\{\mathcal{Q},\varnothing ,\left\{{♭}_1\right\},\{{♭}_2,{♭}_3\},\{{♭}_1,{♭}_2,{♭}_3\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_5\}\};$

5. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}=\{\mathcal{Q},\varnothing ,\left\{{♭}_4\right\},\left\{{♭}_5\right\},\{{♭}_1,{♭}_5\},\{{♭}_2,{♭}_5\},\{{♭}_3,{♭}_4\},\{{♭}_4,{♭}_5\},\{{♭}_1,{♭}_2,{♭}_5\},\{{♭}_1,{♭}_5,{♭}_4\},$$\{{♭}_2,{♭}_4,{♭}_5\},\{{♭}_3,{♭}_4,{♭}_5\},\{{♭}_1,{♭}_2,{♭}_4,{♭}_5\},\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\},\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}\}$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }=P\left(\mathcal{Q}\right)$;

6. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}=\{\mathcal{Q},\varnothing ,\left\{{♭}_3\right\},\left\{{♭}_4\right\},\{{♭}_1,{♭}_4\},\{{♭}_2,{♭}_3\},\{{♭}_3,{♭}_4\},\{{♭}_1,{♭}_3,{♭}_4\},\{{♭}_2,{♭}_3,{♭}_4\},\{{♭}_3,{♭}_4,{♭}_5\},$$\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\},\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}\}$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }=\{\mathcal{Q},\varnothing ,\left\{{♭}_1\right\},\left\{{♭}_3\right\},\left\{{♭}_4\right\},\{{♭}_1,{♭}_3\},$$\{{♭}_1,{♭}_4\},\{{♭}_2,{♭}_3\},\{{♭}_3,{♭}_4\},\{{♭}_3,{♭}_5\},\{{♭}_1,{♭}_2,{♭}_3\},\{{♭}_1,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_3,{♭}_5\},\{{♭}_2,{♭}_3,{♭}_4\},\{{♭}_2,{♭}_3,{♭}_5\},\{{♭}_3,{♭}_4,{♭}_5\},$$\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_5\},\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\},\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}\};$

7. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}={\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }=P\left(\mathcal{Q}\right);$

8. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}=\{\mathcal{Q},\varnothing ,\left\{{♭}_4\right\},\{{♭}_3,{♭}_4\},\{{♭}_3,{♭}_4,{♭}_5\},\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\},\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}\}$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }=P\left(\mathcal{Q}\right).$

Remark 2. 

Example 1 indicates that the method described in this section is distinct from those in [17,26,27,28]. For instance, ${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }=\{\mathcal{Q},\varnothing ,\left\{{♭}_1\right\},\left\{{♭}_2\right\},\left\{{♭}_3\right\},\{{♭}_1,{♭}_2\},\{{♭}_1,{♭}_3\},\{{♭}_2,{♭}_3\},$$\{{♭}_3,{♭}_4\},\{{♭}_1,{♭}_2,{♭}_3\},\{{♭}_1,{♭}_2,{♭}_5\},\{{♭}_1,{♭}_3,{♭}_4\},\{{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_5\}\}\ne {\mathfrak{J}}_{{\mathcal{N}}_r}^{\pounds }=$ $\{\mathcal{Q},\varnothing ,\left\{{♭}_1\right\},\left\{{♭}_2\right\},\left\{{♭}_3\right\},\{{♭}_1,{♭}_3\},\{{♭}_1,{♭}_4\},\{{♭}_2,{♭}_3\},\{{♭}_1,{♭}_2,{♭}_3\},\{{♭}_1,{♭}_3,{♭}_4\},\{{♭}_2,{♭}_3,{♭}_5\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_5\}\}.$

In the ensuing conclusions, the correlations among the different sorts of ${\mathcal{M}}_{\wr }^{\pounds }$-topologies are scrutinized.

Proposition 2. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ Then, the following statements are true:

1. 

${\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }$ and ${\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds };$

2. 

${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds };$

3. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds };$

4. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }.$

Proof. 

(1)

Let $V\in {\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }$. Then, ${\mathcal{M}}_u\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V.$ Thus, $({\mathcal{M}}_r\left({♭}_5\right)\cup {\mathcal{M}}_l\left({♭}_5\right))\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V.$ Hence, ${\mathcal{M}}_r\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V$ and ${\mathcal{M}}_l\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V$. Therefore, $V\in {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }$ and $V\in {\mathfrak{J}}_l^{\mathcal{M}}.$ Hence, ${\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }$ and ${\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }\subseteq {\mathfrak{J}}_l^{\mathcal{M}}.$ Statement 3 can be evidenced in a similar manner.

(2)

Let $V\in {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }$. Then, ${\mathcal{M}}_r\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V.$ Thus, $({\mathcal{M}}_r\left({♭}_5\right)\cap {\mathcal{M}}_l\left({♭}_5\right))\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V.$ Consequently, ${\mathcal{M}}_i\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V.$ Hence, $V\in {\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }$. Therefore, ${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }.$ Statement 4 can be evidenced in a similar manner.

□

Corollary 1. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ Then, the following statements are true.

1. 

${\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds };$

2. 

${\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_l}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds };$

3. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\subseteq {\mathfrak{J}}_{\langle r\rangle }^{\mathcal{M}}\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds };$

4. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }.$

Theorem 6 presents a unique characterization of the proposed topologies by comparing ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\langle \wr \rangle }}^{\pounds }.$ Consequently, Corollary 2 identifies the smallest ${\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }$ and largest ${\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }$ among all types. The prior approach [17,26,27,28] did not offer this characterization and was limited.

Theorem 6. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ Then, ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle \wr \rangle }}^{\pounds },\wr \in \{r,\mathfrak{l},i,u\}.$

Proof. 

Let $V\in {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }.$ Then ${\mathcal{M}}_r\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V$, and consequently, ${\mathcal{M}}_{\langle r\rangle }\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V$ by Theorem 2. Therefore, $V\in {\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }.$ Hence, ${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }.$ The other statements can be evidenced in a similar manner. □

Corollary 2. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ Then, ${\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds },$ $\wr \in \{r,\mathfrak{l},i,\langle r\rangle ,\langle \mathfrak{l}\rangle ,\langle u\rangle \}.$

Remark 3. 

Example 1 confirms that the significant distinctions between the current approach and those in [17,26,27,28] is that ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle \wr \rangle }}^{\pounds },\wr \in \{r,\mathfrak{l},i,u\},$ even though, for instance, ${\mathfrak{J}}_{{\mathcal{N}}_{\wr }}^{\pounds }$ and ${\mathfrak{J}}_{{\mathcal{N}}_{\langle \wr \rangle }}^{\pounds }$ are incomparable as in [17]. Furthermore, it verifies that the converses of Proposition 2, of Theorem 2, and of Corollaries 1 and 6 are not always true.

1. 

${\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds },{\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds },{\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds };$

2. 

${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds };$

3. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds },{\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds };{\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds },$

4. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds };$

5. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds };$

6. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds };$

7. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds };$

8. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds };$

9. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds };$

10. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds };$

11. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds };$

12. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds };$

13. 

${\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }⊈{\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }$ (it is obvious if we set $\pounds =\{\varnothing ,\left\{{♭}_4\right\}\}$);

14. 

${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }$ is not a dual of ${\mathfrak{J}}_{{\mathcal{M}}_l}^{\pounds }$ (it is obvious if we set $\pounds =\{\varnothing ,\left\{{♭}_4\right\}\}$).

Theorem 7 outlines the conditions necessary for establishing equivalences among the suggested topologies.

Theorem 7. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ Then,

(i) 

${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }$ when ⨿ is symmetric;

(ii) 

${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle \wr \rangle }}^{\pounds },\forall \wr \in \{r,\mathfrak{l},i,u\}$ when ⨿ is symmetric and transitive;

(iii) 

${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }$ when ⨿ is equivalence.

Proof. 

(i)

Let $V\in {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }.$ Then, ${\mathcal{M}}_r\left({♭}_5\right)\subseteq V\mathrm{for}\mathrm{all}{♭}_5\in V.$

$\iff {\mathcal{M}}_l\left({♭}_5\right)\subseteq V,{\mathcal{M}}_i\left({♭}_5\right)\subseteq V{\mathcal{M}}_u\left({♭}_5\right)\subseteq V,\mathrm{for}\mathrm{all}{♭}_5\in V$ (by Theorem 2).

It follows that ${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }$, and consequently, ${\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle l\rangle }}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }.$

(ii)

Let $V\in {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }.$ Then, ${\mathcal{M}}_r\left({♭}_5\right)\subseteq V,\forall {♭}_5\in V.$

$\iff {\mathcal{M}}_{\langle r\rangle }\left({♭}_5\right),\forall {♭}_5\in V$ (by Theorem 2).

It follows that ${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }={\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds };$ the remaining cases are comparable.

(iii)

The proof is straightforward.

□

Remark 4. 

Example 1 is scrutinized, showing that, in Theorem 7, the following statements are true:

1. 

Similarity in (i) is crucial because ${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }\ne {\mathfrak{J}}_{{\mathcal{M}}_l}^{\pounds }\ne {\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }\ne {\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds },$ and ${\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\ne {\mathfrak{J}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\ne {\mathfrak{J}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds };$

2. 

Similarity and transitivity in (ii) are crucial because ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }\ne {\mathfrak{J}}_{{\mathcal{M}}_{\langle \wr \rangle }}^{\pounds },\wr \in \{r,\mathfrak{l},i,\langle r\rangle ,\langle \mathfrak{l}\rangle ,\langle u\rangle \};$

3. 

Equivalence in (iii) is crucial because not all topologies are always equal in the absence of this restriction.

To explore the property of monotonicity in the fourth section, which is crucial, we must first examine the relationship between the two topologies generated from the two subset relations. This is addressed in the subsequent key proposition.

Proposition 3. 

Let $(\mathcal{Q},{⨿}_1,{\varsigma }_{1\wr })$ and $(\mathcal{Q},{⨿}_2,{\varsigma }_{2\wr })$ be two ≀-NS, £ be an ideal on $\mathcal{Q}$, and ${⨿}_1\subseteq {⨿}_2.$ Then, ${\mathfrak{J}}_{2\wr }^{\mathcal{M}}\subseteq {\mathfrak{J}}_{1\wr }^{\mathcal{M}},\wr \in \{r,\mathfrak{l},i,u\}.$

Proof. 

Let $V\in {\mathfrak{J}}_{2r}^{\mathcal{M}}.$ Then, ${\mathcal{M}}_{2r}\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V.$ Thus, ${\mathcal{M}}_{1r}\left({♭}_5\right)\cap V^{\prime }\in \pounds ,\forall {♭}_5\in V$ (by Proposition 1). Consequently, $V\in {\mathfrak{J}}_{1r}^{\mathcal{M}}$, and hence, ${\mathfrak{J}}_{2r}^{\mathcal{M}}\left({♭}_5\right)\subseteq {\mathfrak{J}}_{1r}^{\mathcal{M}}\left({♭}_5\right);$ the others can be proven in a similar manner. □

4. Idealization of Approximate Models

This section proposes rough models utilizing ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$-topologies and details their key properties. These models maintain the property of monotonicity without constraints. In contrast, this property may be either lost or preserved under strict conditions in some previous methods [26,27,28].

Definition 10 uses the topologies generated in Section 3 to provide approximations.

Definition 10. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ The £-${\mathcal{M}}_{\wr }$-lower approximation ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }$ and £-${\mathcal{M}}_{\wr }$-upper approximation ${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }$ of $V\subseteq \mathcal{Q}$ are

$${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }:S\subseteq V\}=In{t}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right),$$

$${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)=\cap \{X\in {\mathfrak{I}}_{{\mathcal{M}}_{\wr }}^{\pounds }:V\subseteq X\}=C{l}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right).$$

The most important features of the suggested approximations are considered in the description of the subsequent results.

Proposition 4. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS; £ be an ideal on $\mathcal{Q}$; and $V,S\subseteq \mathcal{Q}$. Then, the following statements are true:

(i) 

${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq V$;

(ii) 

${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }(\varnothing )=\varnothing$;

(iii) 

${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(\mathcal{Q}\right)=\mathcal{Q}$;

(iv) 

If $V\subseteq S$, then ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)$;

(v) 

${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }(V\cap S)={\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\cap {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right);$

(vi) 

${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V^{\prime }\right)={\left({\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\right)}^{\prime };$

(vii) 

${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left({\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\right)={\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$.

Proof. 

(i), (ii), and (iii) are straightforward to prove.

(iv)

Let $V\subseteq S$. Then, $\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }:S\subseteq V\}\subseteq \cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }:S\subseteq S\}$, and so ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)$.

(v)

${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }(V\cap S)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\cap {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)$ by (iv). Since ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq V$ and ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)\subseteq S,$ it follows that ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\cap {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)\subseteq V\cap S.$ Consequently, ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }({\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\cap {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right))\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }(V\cap S).$ Then, ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\cap {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right))\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }(V\cap S).$ Thus, ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }(V\cap S)={\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\cap {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right).$

(vi)

Let ${♭}_1\in {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V^{\prime }\right)$. Then, $\exists S\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$ such that ${♭}_1\in S\subseteq V^{\prime },$ and so $S\cap V=\varnothing$. Therefore, ${♭}_1\notin {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$. Thus, ${♭}_1\in {\left({\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\right)}^{\prime }$. Let ${♭}_1\in {\left({\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\right)}^{\prime }$. Then, ${♭}_1\notin {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$, and so $\exists U\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$ such that ${♭}_1\in S$ and $S\cap V=\varnothing$. Therefore, ${♭}_1\in S\subseteq V^{\prime }$. Hence, ${♭}_1\in {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V^{\prime }\right)$.

(vii)

From (i) we obtain ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left({\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$. Conversely, let ${♭}_1\in {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$. Then, $\exists S\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$ such that ${♭}_1\in S\subseteq V$. ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$ (by (iv)). By Definition 10, we observe that $S={\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)$, so $\left\{{♭}_1\right\}\in {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left({\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\right)$. Thus, ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left({\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\right)$.

□

Corollary 3. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS; £ be an ideal on $\mathcal{Q}$; and $V,S\subseteq \mathcal{Q}$. Then, ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\cup {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }(V\cup S)$.

Proof. 

It can be directly inferred from (iv) of Proposition 4. □

Remark 5. 

In Example 1, the following statements are true:

(i) 

${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(\left\{{♭}_4\right\}\right)=\varnothing \subset \left\{{♭}_4\right\}$;

(ii) 

${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(\left\{{♭}_4\right\}\right)=\varnothing \subset \left\{{♭}_2\right\}={\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(\left\{\mathcal{M}\right\}\right)$ but $\left\{{♭}_4\right\}⊈\left\{{♭}_2\right\}$;

(iii) 

${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(\left\{{♭}_5\right\}\right)\cup {\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\}\right)=\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\}\subset \mathcal{Q}={\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }(\left\{{♭}_5\right\}\cup \{{♭}_1,{♭}_2,{♭}_3,{♭}_4\})$.

Proposition 5. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS; £ be an ideal on $\mathcal{Q}$; and $V,S\subseteq \mathcal{Q}$. Then, the following statements are true:

(i) 

$V\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$;

(ii) 

${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }(\varnothing )=\varnothing$;

(iii) 

${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)=V$;

(iv) 

If $V\subseteq S$, then ${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)$;

(v) 

${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\cup {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right)={\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }(V\cup S)$;

(vi) 

${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V^{\prime }\right)={\left({\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\right)}^{\prime }$;

(vii) 

${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left({\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\right)={\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$.

Proof. 

This proof is similar to the proof of Proposition 4. □

Corollary 4. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS; £ be an ideal on $\mathcal{Q}$; and $V,S\subseteq \mathcal{Q}$. Then, ${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }(V\cap S)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\cap {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(S\right).$

Proof. 

It can be directly inferred from (iv) of Proposition 4. □

Remark 6. 

In Example 1, the following statements are true:

(i) 

$\{{♭}_1,{♭}_4\}\subset {\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(\{{♭}_1,{♭}_4\}\right)=\{{♭}_1,{♭}_4,{♭}_5\}$;

(ii) 

${\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(\left\{{♭}_5\right\}\right)=\left\{{♭}_5\right\}\subset \{{♭}_2,{♭}_5\}={\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(\left\{{♭}_2\right\}\right),$ but $\left\{{♭}_5\right\}⊈\left\{{♭}_5\right\}$;

(iii) 

${\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }(\left\{{♭}_3\right\}\cap \left\{{♭}_4\right\})=\varnothing \subset \left\{{♭}_4\right\}={\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(\left\{{♭}_3\right\}\right)\cap {\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(\left\{{♭}_4\right\}\right)$.

Definition 11. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ The £-${\mathcal{M}}_{\wr }$-accuracy and £-${\mathcal{M}}_{\wr }$-roughness of $V\subseteq \mathcal{Q}$ are

$${\mathcal{A}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)={\displaystyle \frac{\mid {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\mid }{\mid {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\mid }}, \mathit{where} V\ne \varnothing ,$$

$${\mathfrak{L}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)=1-{\mathcal{A}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right).$$

Proposition 6 guarantees that the lower approximation does not diminish as additional data are introduced. Likewise, the upper approximation does not decrease. Therefore, the proposed model has the property of monotonicity. This property ensures that the approximations either gain precision or stay consistent, never decreasing, with new information.

Proposition 6. 

Let $(\mathcal{Q},{⨿}_1,{\varsigma }_{1\wr })$ and $(\mathcal{Q},{⨿}_2,{\varsigma }_{2\wr })$ be two ≀-NSs, £ be an ideal on $\mathcal{Q}$, and ${⨿}_1\subseteq {⨿}_2.$ Then $,\forall \wr \in \{r,l,i,u\},V\subseteq \mathcal{Q}$, and the following statements are true:

1. 

${\overline{⨿}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right);$

2. 

${\underline{⨿}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right);$

3. 

${\mathcal{A}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right);$

4. 

${\mathfrak{L}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)\le {\mathfrak{L}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right).$

Proof. 

• ${\overline{⨿}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right))=\cap \{V\in {\mathfrak{I}}_{1\wr }^{\mathcal{M}}:V\subseteq V\}\subseteq \cap \{V\in {\mathfrak{I}}_{2\wr }^{\mathcal{M}}:V\subseteq V\}={\overline{⨿}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right)$ (by Proposition 3). Hence, ${\overline{⨿}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right).$
• Let ${\underline{⨿}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right)=\cup \{S\in {\mathfrak{J}}_{2\wr }^{\mathcal{M}}:S\subseteq V\}\subseteq =\cup \{S\in {\mathfrak{J}}_{1\wr }^{\mathcal{M}}:S\subseteq V\}={\underline{⨿}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)$ (by Proposition 3). Hence, ${\underline{⨿}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right).$
• $$\begin{array}{cc}\hfill {\mathcal{A}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)& \hfill =|{\displaystyle \frac{{\underline{⨿}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)|}{{\overline{⨿}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)|}}|.\hfill \\ \hfill & \hfill \le {\displaystyle \frac{|{\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)|}{|{\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)|}}.\hfill \\ \hfill & \hfill ={\mathcal{A}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right).\hfill \end{array}$$
• The proof is straightforward using statement 3.

□

Definition 12. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ The £-${\mathcal{M}}_{\wr }$-positive, £-${\mathcal{M}}_{\wr }$-boundary, and £-${\mathcal{M}}_{\wr }$-negative regions of $V\subseteq \mathcal{Q}$ are

$${{⨿}_{{\mathcal{M}}_{\wr }}^{\pounds }}^{+}\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right),$$

$${\mathcal{B}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\setminus {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right), \mathit{and}$$

$${{⨿}_{{\mathcal{M}}_{\wr }}^{\pounds }}^{-}\left(V\right)=\mathcal{Q}\setminus {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right).$$

Proposition 7. 

Let $(\mathcal{Q},{⨿}_1,{\varsigma }_{1\wr })$ and $(\mathcal{Q},{⨿}_2,{\varsigma }_{2\wr })$ be two ≀-NSs, £ be an ideal on $\mathcal{Q}$, and ${⨿}_1\subseteq {⨿}_2.$ Then $,\forall \wr \in \{r,l,i,u\},V\subseteq \mathcal{Q}$, and the following statements are true:

(i) 

${\mathcal{B}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right),$

(ii) 

${⨿}_{{\mathcal{M}}_{2\wr }}^{{\pounds }^{-}}\left(V\right)\subseteq {⨿}_{{\mathcal{M}}_{1\wr }}^{{\pounds }^{-}}\left(V\right)$.

Proof. 

(i)

Let ${♭}_1\in {\mathcal{B}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right).$ Then, ${♭}_1\in {\overline{⨿}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)\setminus \underline{⨿}{\pounds }_{{\mathcal{M}}_{1\wr }}\left(V\right).$ Therefore, ${♭}_1\in {\overline{⨿}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)$ and ${♭}_1\in {\left({\underline{⨿}}^{{\mathcal{M}}_{1\wr }}\left(V\right)\right)}^{\prime }.$ Thus, ${♭}_1\in {\overline{⨿}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right)$ and ${♭}_1\in {\left({\underline{⨿}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right)\right)}^{\prime }$. Hence, ${♭}_1\in {\mathcal{B}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right).$ Therefore, ${\mathcal{B}}_{{\mathcal{M}}_{1\wr }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_{2\wr }}^{\pounds }\left(V\right).$

(ii)

This is obtained from Proposition 6.

□

Definition 13. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ $V\subseteq \mathcal{Q}$ is £-${\mathcal{M}}_{\wr }$-exact if ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)=V;$ otherwise, it is £-${\mathcal{M}}_{\wr }$-rough.

Proposition 8. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS and £ be an ideal on $\mathcal{Q}.$ $V\subseteq \mathcal{Q}$ is £-${\mathcal{M}}_{\wr }$-exact iff ${\mathcal{B}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)=\varnothing$.

Proof. 

Let V be any £-${\mathcal{M}}_{\wr }$-exact set. Then, ${\mathcal{B}}_{\wr }^{\mathcal{M}}\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\setminus {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)=\varnothing$. Conversely, ${\mathcal{B}}_{\wr }^{\mathcal{M}}\left(V\right)=\varnothing ;$ therefore, ${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\setminus {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)=\varnothing$, and so ${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$. However, ${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$. Thus, ${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$. Hence, V is £-${\mathcal{M}}_{\wr }$-exact. □

5. Comparisons of the Proposed Methods and Their Advantages over Earlier Ones

This section clarifies the possibility of comparing all the approximations proposed in Section 4. It is noteworthy that in earlier studies, comparisons were limited to specific types of approximations only [17,26,27,28]. It also indicates that these approximations offer greater benefits than those in [36].

5.1. Relationships Among Different Sorts of the Current Approximations

The relationships between the newly proposed approximations are detailed in the next results.

Proposition 9. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q}$, and $V\subseteq \mathcal{Q}.$ Then, the following statements are true:

(i) 

${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$;

(ii) 

${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$;

(iii) 

${\underline{⨿}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)$;

(iv) 

${\underline{⨿}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\langle l\rangle }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)$;

(v) 

${\overline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)$;

(vi) 

${\overline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_l}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)$;

(vii) 

${\overline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)$;

(viii) 

${\overline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\langle l\rangle }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)$.

Proof. 

To scrutinize (i) and (ii), let ${♭}_1\in {\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)$. Then, $\exists S\in {\mathfrak{J}}_u^{\mathcal{N}}$ and ${♭}_1\in S\subseteq V$. We find $S\in {\mathfrak{J}}_r^{\mathcal{N}}$ and $S\in {\mathfrak{J}}_{\mathfrak{l}}^{\mathcal{N}}$ (by Proposition 2). Thus, ${♭}_1\in In{t}_r^{\mathcal{M}}\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)$ and ${♭}_1\in In{t}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)$. Hence, ${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)$ and ${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)$. Similarly, ${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$ and ${\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$ are proved. The other proofs can be expounded similarly. □

Corollary 5. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q}$, and $V\subseteq \mathcal{Q}.$ Then, the following statements are true:

(i) 

${\mathcal{B}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)$;

(ii) 

${\mathcal{B}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)$;

(iii) 

${\mathcal{B}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)$;

(iv) 

${\mathcal{B}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)$;

(v) 

${\mathcal{A}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$;

(vi) 

${\mathcal{A}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_l}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$;

(vii) 

${\mathcal{A}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)$;

(viii) 

${\mathcal{A}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{\langle l\rangle }}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)$.

Proof. 

(v): ${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$ and ${\overline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)$ (by Proposition 9), and so we obtain

$$\mid {\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\mid \le \mid {\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\mid \le \mid {\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\mid$$

(2)

and

$${\displaystyle \frac{1}{\mid {\overline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\mid }}\le {\displaystyle \frac{1}{\mid {\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\mid }}\le {\displaystyle \frac{1}{\mid {\overline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\mid }}.$$

(3)

By (2) and (3), we obtain

${\displaystyle \frac{\mid {\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\mid }{\mid {\overline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\mid }}\le {\displaystyle \frac{\mid {\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\mid }{\mid {\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\mid }}\le {\displaystyle \frac{\mid {\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\mid }{\mid {\overline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\mid }}$, which is equivalent to ${\mathcal{A}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$.

In a comparable way, we can expound the other cases. □

The outcomes presented herein elucidate the feasibility of comparing all the proposed approximations outlined in Section 4. Notably, earlier studies were restricted to comparing only certain types of approximations [17,26,27,28].

Theorem 8. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q}$, and $V\subseteq \mathcal{Q}.$ Then, the following statements are true:

(i) 

${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right);$

(ii) 

${\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right);$

(iii) 

${\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right);$

(iv) 

${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right);$

(v) 

${\overline{⨿}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right);$

(vi) 

${\overline{⨿}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right);$

(vii) 

${\overline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right);$

(viii) 

${\overline{⨿}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right).$

Proof. 

${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }:S\subseteq V\}\subseteq \cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }:S\subseteq V\}={\underline{⨿}}_{\langle r\rangle }^{\mathcal{M}}\left(V\right)$ by Theorem 6. Hence, ${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right).$ In a comparable way, we can expound the other cases. □

Corollary 6. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q}$, and $V\subseteq \mathcal{Q}.$ Then, the following statements are true:

(i) 

${\mathcal{B}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right);$

(ii) 

${\mathcal{B}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_l}^{\pounds }\left(V\right);$

(iii) 

${\mathcal{B}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right);$

(iv) 

${\mathcal{B}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)$;

(v) 

${\mathcal{A}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right);$

(vi) 

${\mathcal{A}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)$;

(vii) 

${\mathcal{A}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right);$

(viii) 

${\mathcal{A}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right).$

Corollary 7. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q}$, and $V\subseteq \mathcal{Q}.$ Then, $\wr \in \{r,\mathfrak{l},i,\langle r\rangle ,\langle \mathfrak{l}\rangle ,\langle u\rangle \},$ and the following statements are true:

(i) 

${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right);$

(ii) 

${\overline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right);$

(iii) 

${\mathcal{B}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right);$

(iv) 

${\mathcal{A}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\le {\mathcal{A}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right).$

Remark 7. 

Example 1 validates the results obtained in Proposition 9, Corollaries 5–7, and Theorem 8. At the same time, it highlights that the converse statements are not valid in general. Some examples are as follows:

(i) 

If $V=\{{♭}_3,{♭}_5\},$ then ${\overline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)=\{{♭}_3,{♭}_4,{♭}_5\}⊈\{{♭}_3,{♭}_5\}={\overline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right);$

(ii) 

If $V=\{{♭}_3,{♭}_4\},$ then ${\underline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)=\{{♭}_4,{♭}_5\}⊈\left\{{♭}_5\right\}={\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right);$

(iii) 

If $V=\{{♭}_1,{♭}_3\},$ then ${\mathcal{A}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)=1\nleqq {\displaystyle \frac{2}{3}}={\mathcal{A}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)\nleqq ={\displaystyle \frac{1}{5}}={\mathcal{A}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right).$

The results detailed below provide equivalences between the proposed approximations under certain conditions.

Proposition 10. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q},V\subseteq \mathcal{Q}$, and ⨿ be symmetric. Then, the following statements are true:

(i) 

${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$ and ${\overline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$;

(ii) 

${\underline{⨿}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)$ and ${\overline{⨿}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)$.

Proof. 

${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_u}^{\pounds }:S\subseteq V\}=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }:S\subseteq V\}={\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right),$ ${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }:S\subseteq V\}=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }:S\subseteq V\}={\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right),$ and ${\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }:S\subseteq V\}=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_i}^{\pounds }:S\subseteq V\}={\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)$ (by Theorem 7). Hence, ${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right).$ The other proofs can be derived similarly. □

Corollary 8. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q},V\subseteq \mathcal{Q}$, and ⨿ be symmetric. Then, the following statements are true:

(i) 

${\mathcal{B}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_l}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)$;

(ii) 

${\mathcal{B}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle l\rangle }}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right);$

(iii) 

${\mathcal{A}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_l}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)$;

(iv) 

${\mathcal{A}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right).$

Proposition 11. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q},V\subseteq \mathcal{Q}$, and ⨿ be symmetric and transitive. Then, the following statements are true:

(i) 

${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right));$

(ii) 

${\overline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)$.

Proof. 

The proof is exactly analogous to the proof of Proposition 10. □

Corollary 9. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q},V\subseteq \mathcal{Q}$, and ⨿ be symmetric and transitive. Then, the following statements are true:

(i) 

${\mathcal{B}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_l}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle l\rangle }}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right);$

(ii) 

${\mathcal{A}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_l}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right).$

Proposition 12. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q},V\subseteq \mathcal{Q}$, and ⨿ be equivalence. Then, the following statements are true:

(i) 

${\underline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)=$ ${\overline{⨿}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\langle l\rangle }}^{\pounds }\left(V\right)={\underline{⨿}}^{{\mathcal{M}}_{\langle r\rangle }}\left(V\right)={\underline{⨿}}^{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}\left(V\right)={\underline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right)=$ ${\overline{⨿}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)={\overline{⨿}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right);$

(ii) 

${\mathcal{B}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)={\mathcal{B}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right);$

(iii) 

${\mathcal{A}}_{{\mathcal{M}}_i}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\mathfrak{l}}}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_u}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle i\rangle }}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle r\rangle }}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle \mathfrak{l}\rangle }}^{\pounds }\left(V\right)={\mathcal{A}}_{{\mathcal{M}}_{\langle u\rangle }}^{\pounds }\left(V\right).$

Proof. 

The proof is exactly analogous to the proof of Proposition 10. □

5.2. Evaluating Our Approach Against Previous Methods

This section confirms that the suggested approximations provide greater benefits than those in [36]. Additionally, it indicates that the current approximations, based on maximal neighborhoods and ideals, are not comparable to the earlier ones that depended on neighborhoods and ideals [17].

The correlations between the recommended approximations in Definition 10 and that in Definition 7 [36] are proposed in the next findings and in

Table 2. Definitions 7 [36], 10, and 11 for $\wr =r$.

V	Definitions 10 and 11 for $\wr =\mathit{r}$			Definitions 7 [36] for $\wr =\mathit{r}$
	${\underline{⨿}}_{{\mathcal{M}}_{\mathbf{r}}}^{\pounds }\left(\mathbf{V}\right)$	${\overline{⨿}}_{{\mathcal{M}}_{\mathbf{r}}}^{\pounds }\left(\mathbf{V}\right)$	$_{{\mathcal{M}}_{\mathbf{r}}}^{\pounds }\left(\mathbf{V}\right)$	${\underline{⨿}}_{{\mathcal{M}}_{\mathbf{r}}}\left(\mathbf{V}\right)$	${\overline{⨿}}_{{\mathcal{M}}_{\mathbf{r}}}\left(\mathbf{V}\right)$	${\mathcal{M}}_{\mathbf{r}}\left(\mathbf{V}\right)$
$\left\{{♭}_1\right\}$	$\left\{{♭}_1\right\}$	$\{{♭}_1,♭\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\left\{{♭}_1\right\}$	$\left\{{♭}_1\right\}$	$\{{♭}_1,{♭}_5\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\left\{{♭}_2\right\}$	$\left\{{♭}_2\right\}$	$\{{♭}_2,{♭}_5\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\left\{{♭}_3\right\}$	$\left\{{♭}_3\right\}$	$\{{♭}_3,{♭}_4\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\left\{{♭}_4\right\}$	∅	$\left\{{♭}_4\right\}$	0	∅	$\mathcal{Q}$	0
$\left\{{♭}_5\right\}$	∅	$\left\{{♭}_5\right\}$	0	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_2\}$	$\{{♭}_1,{♭}_2\}$	$\{{♭}_1,{♭}_2,{♭}_5\}$	${\displaystyle \frac{2}{3}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_3\}$	$\{{♭}_1,{♭}_3\}$	$\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_4\}$	$\left\{{♭}_1\right\}$	$\{{♭}_1,{♭}_4,{♭}_5\}$	${\displaystyle \frac{1}{3}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_5\}$	$\left\{{♭}_1\right\}$	$\{{♭}_1,{♭}_5\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\{{♭}_2,{♭}_3\}$	$\{{♭}_2,{♭}_3\}$	$\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\{{♭}_2,{♭}_4\}$	$\left\{{♭}_2\right\}$	$\{{♭}_2,{♭}_4,{♭}_5\}$	${\displaystyle \frac{1}{3}}$	∅	$\mathcal{Q}$	0
$\{{♭}_2,{♭}_5\}$	$\left\{{♭}_2\right\}$	$\{{♭}_2,{♭}_5\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\{{♭}_3,{♭}_4\}$	$\{{♭}_3,{♭}_4\}$	$\{{♭}_3,{♭}_4\}$	1	∅	$\mathcal{Q}$	0
$\{{♭}_3,{♭}_5\}$	$\left\{{♭}_3\right\}$	$\{{♭}_3,{♭}_4,{♭}_5\}$	${\displaystyle \frac{1}{3}}$	∅	$\mathcal{Q}$	0
$\{{♭}_4,{♭}_5\}$	∅	$\{{♭}_4,{♭}_5\}$	0	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_2,{♭}_3\}$	$\{{♭}_1,{♭}_2,{♭}_3\}$	$\mathcal{Q}$	${\displaystyle \frac{3}{5}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_2,{♭}_4\}$	$\{{♭}_1,{♭}_2\}$	$\{{♭}_1,{♭}_2,{♭}_4,{♭}_5\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_2,{♭}_5\}$	$\{{♭}_1,{♭}_2,{♭}_5\}$	$\{{♭}_1,{♭}_2,{♭}_5\}$	1	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_3,{♭}_4\}$	$\{{♭}_1,{♭}_3,{♭}_4\}$	$\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\}$	${\displaystyle \frac{3}{4}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_3,{♭}_5\}$	$\{{♭}_1,{♭}_3\}$	$\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_4,{♭}_5\}$	$\left\{{♭}_1\right\}$	$\{{♭}_1,{♭}_4,{♭}_5\}$	${\displaystyle \frac{1}{3}}$	∅	$\mathcal{Q}$	0
$\{{♭}_2,{♭}_3,{♭}_4\}$	$\{{♭}_2,{♭}_3,{♭}_4\}$	$\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}$	${\displaystyle \frac{3}{4}}$	∅	$\mathcal{Q}$	0
$\{{♭}_2,{♭}_3,{♭}_5\}$	$\{{♭}_2,{♭}_3\}$	$\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}$	${\displaystyle \frac{1}{2}}$	∅	$\mathcal{Q}$	0
$\{{♭}_2,{♭}_4,{♭}_5\}$	$\left\{{♭}_2\right\}$	$\{{♭}_2,{♭}_4,{♭}_5\}$	${\displaystyle \frac{1}{3}}$	∅	$\mathcal{Q}$	0
$\{{♭}_3,{♭}_4,{♭}_5\}$	$\{{♭}_3,{♭}_4\}$	$\{{♭}_3,{♭}_4,{♭}_5\}$	${\displaystyle \frac{2}{3}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\}$	$\{{♭}_1,{♭}_2,{♭}_3,{♭}_4\}$	$\mathcal{Q}$	${\displaystyle \frac{4}{5}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_2,{♭}_3,{♭}_5\}$	$\{{♭}_1,{♭}_2,{♭}_3,{♭}_5\}$	$\mathcal{Q}$	${\displaystyle \frac{4}{5}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_2,{♭}_4,{♭}_5\}$	$\{{♭}_1,{♭}_2,{♭}_5\}$	$\{{♭}_1,{♭}_2,{♭}_4,{♭}_5\}$	${\displaystyle \frac{3}{4}}$	∅	$\mathcal{Q}$	0
$\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\}$	$\{{♭}_1,{♭}_3,{♭}_4\}$	$\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\}$	${\displaystyle \frac{3}{4}}$	∅	$\mathcal{Q}$	0
$\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}$	$\{{♭}_2,{♭}_3,{♭}_4\}$	$\{{♭}_2,{♭}_3,{♭}_4,{♭}_5\}$	${\displaystyle \frac{3}{4}}$	∅	$\mathcal{Q}$	0

, which relies on Example 1.

Theorem 9. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q},$ and $V\subseteq \mathcal{Q}$. Then, the following statements are true:

1. 

${\underline{⨿}}_{{\mathcal{M}}_{\wr }}\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right);$

2. 

${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\wr }}\left(V\right).$

Proof. 

(1)

${\underline{⨿}}_{{\mathcal{M}}_{\wr }}\left(V\right)=\cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}:S\subseteq V\}\subseteq \cup \{S\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }:S\subseteq V\}={\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)$ (by Theorem 5). The proof of (2) can be expounded similarly.

□

Corollary 10. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q},$ and $V\subseteq \mathcal{Q}$. Then, the following statements are true:

1. 

${\mathcal{B}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\mathcal{B}}_{{\mathcal{M}}_{\wr }}\left(V\right);$

2. 

${\mathcal{A}}_{{\mathcal{M}}_{\wr }}\left(V\right)⩽{\mathcal{A}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right);$

3. 

Every ${\mathcal{M}}_{\wr }$-exact subset in $\mathcal{Q}$ is £-${\mathcal{M}}_{\wr }$-exact;

4. 

Every £-${\mathcal{M}}_{\wr }$-rough subset in $\mathcal{Q}$ is ${\mathcal{M}}_{\wr }$-rough.

Theorem 9 and Corollary 10 validate that the proposed approximations are superior to their predecessors. Moreover, Table 2 shows that using the previous approximations resulted in a consistent accuracy of zero, while the accuracy with the current approximations ranged from zero to one. This represents a noticeable and tangible improvement.

Since the definition of maximal neighborhoods (3) [29,30] depends on the definition of neighborhoods (1) [24,25,38,39], this motivated us to conduct an assessment of the suggested approximations based on maximal neighborhoods and ideals, as well as the other approximations that rely solely on neighborhoods and ideals introduced by Hosny [17].

Remark 8. 

The approximations rely on topologies created from various kinds of maximal neighborhoods and ideals, differing significantly from those in [17], which utilized a topology formed by neighborhoods and ideals. In Example 1:

1. 

If $V=\{{♭}_1,{♭}_4\},$ then

• ${\underline{⨿}}_{{\mathcal{M}}_r}\left(V\right)=\{{♭}_1,{♭}_4\}⊈\left\{{♭}_1\right\}={\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right);$
• ${\overline{⨿}}_{{\mathcal{M}}_r}\left(V\right)=\{{♭}_1,{♭}_4\}⊉\{{♭}_1,{♭}_4,{♭}_5\}={\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right).$

2. 

If $V=\{{♭}_1,{♭}_2\},$ then

• ${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)=\{{♭}_1,{♭}_2\}⊈\left\{{♭}_1\right\}={\underline{⨿}}_{{\mathcal{M}}_r}\left(V\right);$
• ${\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)=\{{♭}_1,{♭}_2,{♭}_5\}⊉\{{♭}_1,{♭}_2,{♭}_4,{♭}_5\}={\overline{⨿}}_{{\mathcal{M}}_r}\left(V\right).$

There are many studies that have used the idea of ideals and several types of maximal neighborhoods directly to define the approximations in Definitions 4–6 [18,32,33,34]. We compare these studies to the present one in the results below.

Proposition 13. 

Let $(\mathcal{Q},⨿,{\varsigma }_{\wr })$ be a ≀-NS, £ be an ideal on $\mathcal{Q},$ and $V\subseteq \mathcal{Q}$. Then, the following statements are true:

1. 

${\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right)\subseteq {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{*}}\left(V\right);$

2. 

${\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{*}}\left(V\right)\subseteq {\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right).$

Proof. 

• Let ${♭}_5\in {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right).$ Then, $\exists S\in {\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }$ such that ${♭}_5\in S\subseteq V.$ Thus, ${\mathcal{M}}_{\wr }\left({♭}_5\right)\cap S^{\prime }\in \pounds .$ Consequently, ${\mathcal{M}}_{\wr }\left({♭}_5\right)\cap V^{\prime }\in \pounds .$ Hence, ${♭}_5\in {\underline{⨿}}_{{\mathcal{M}}_{\wr }}^{{\pounds }^{*}}\left(V\right).$
• This proof can be elaborated in the same manner as the proof of (1).

□

Remark 9. 

Example 1 clarifies the following:

1. 

Proposition 13’s converse is not always true.

• If we let $V=\left\{{♭}_3\right\}$, then ${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)=\left\{{♭}_3\right\}⊉\{{♭}_3,{♭}_4\}={\underline{⨿}}_{{\mathcal{M}}_r}^{{\pounds }^{*}}\left(V\right);$
• If we let $V=\{{♭}_1,{♭}_2,{♭}_4,{♭}_5\}$, then ${\overline{⨿}}_{{\mathcal{M}}_r}^{{\pounds }^{*}}\left(V\right)=\{{♭}_1,{♭}_2,{♭}_5\}⊉\{{♭}_1,{♭}_2,{♭}_4,{♭}_5\}={\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right).$

2. 

The prior approximations in Definitions 5 and 6 [18,32,33,34] and the present ones are incomparable. For instance, in the case of Definition 5 [18,32,33,34]:

• If we let $V=\left\{{♭}_2\right\}$, then ${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(V\right)=\left\{{♭}_2\right\}\ne \left\{{♭}_4\right\}={\underline{⨿}}_{{\mathcal{M}}_r}^{{\pounds }^{**}}\left(V\right);$
• If we let $V=\{{♭}_1,{♭}_3,{♭}_4,{♭}_5\}$, then ${\overline{⨿}}_{{\mathcal{M}}_r}^{{\pounds }^{**}}\left(V\right)=\{{♭}_1,{♭}_2,{♭}_3,{♭}_5\}\ne \{{♭}_1,{♭}_3,{♭}_4,{♭}_5\}={\overline{⨿}}_{{\mathcal{M}}_{\wr }}^{\pounds }\left(V\right).$

Remark 10. 

It is important to highlight that while the previous approximations in Definition 4 [18,32,33,34] are superior to the current ones (refer to Proposition 13), the proposed methods meet all of Pawlak’s criteria. In contrast, the earlier Definition 4 [18,32,33,34] do not fulfill these criteria and necessitate constraints on the relations to do so, which limits their applicability.

6. An Application of the Suggested Method to Diagnose Dengue Sickness

Dengue fever, a mosquito-borne viral illness, is triggered by the dengue virus, a member of the flavivirus family, and is widespread in tropical and subtropical areas globally.

It often starts with a sudden high fever, and the rest of the symptoms may be quite debilitating. The condition is sometimes referred to as “breakbone fever” due to the intense pain it can cause.

Efforts to combat dengue include ongoing research into vaccines and improved mosquito control measures to reduce the incidence of this potentially serious disease.

This section assesses the performance of our models in dengue disease management, demonstrating how our approach improves decision-making and showcasing the effectiveness of our topological method in identifying key symptoms.

We analyze the data from eight patients with this sickness, $\mathcal{Q}=\{{♭}_1,{♭}_2,{♭}_3,{♭}_4,{♭}_5,{♭}_6,{♭}_7,{♭}_8\}$, as represented in

Table 3. Information system of dengue fever.

V	H	P	J	B	Dengue Fever
${♭}_1$	■	□	■	■	■
${♭}_2$	□	□	■	□	□
${♭}_3$	□	□	■	□	■
${♭}_4$	□	■	□	□	□
${♭}_5$	■	□	□	■	□
${♭}_6$	■	■	■	□	■
${♭}_7$	■	■	□	□	□
${♭}_8$	■	■	■	□	■

. The symptoms are $\{H,P,J,B\}$, where H, P, J, and B, respectively, denote severe headache, pain behind the eyes, joint and muscle pain, and mild bleeding. The values of these variables are represented by two symbols: ■ indicates that the patient exhibits symptoms, while □ signifies the absence of symptoms. The decision is also represented using the same two symbols, indicating whether the patient has dengue fever or not.

$${♭}_i⨿{♭}_j⟺{♭}_{i}and{♭}_{j}exhibitatleasttwosimilarsymptoms.$$

(4)

Therefore, $⨿=\{({♭}_1,{♭}_1),({♭}_1,{♭}_5),({♭}_1,{♭}_6),({♭}_1,{♭}_8),({♭}_5,{♭}_1),({♭}_5,{♭}_5),({♭}_6,{♭}_1),({♭}_6,{♭}_6),({♭}_6,{♭}_7),({♭}_6,{♭}_8),$ $({♭}_7,{♭}_6),({♭}_7,{♭}_7),({♭}_7,{♭}_8),({♭}_8,{♭}_1),({♭}_8,{♭}_6),({♭}_8,{♭}_7),({♭}_8,{♭}_8)\}$. Then, ${\mathfrak{J}}_{{\mathcal{M}}_r}=\{\mathcal{Q},\varnothing ,\left\{{♭}_2\right\},\left\{{♭}_3\right\},\left\{{♭}_4\right\},$ $\{{♭}_2,{♭}_3\},\{{♭}_2,{♭}_4\},\{{♭}_3,{♭}_4\},\{{♭}_2,{♭}_3,{♭}_4\},\{{♭}_1,{♭}_5,{♭}_6,{♭}_7,{♭}_8\},\{{♭}_1,{♭}_2,{♭}_5,{♭}_6,{♭}_7,{♭}_8\},\{{♭}_1,{♭}_3,{♭}_5,{♭}_6,{♭}_7,{♭}_8\},$$\{{♭}_1,{♭}_4,{♭}_5,{♭}_6,{♭}_7,{♭}_8\},\{{♭}_1,{♭}_2,{♭}_3,{♭}_5,{♭}_6,{♭}_7,{♭}_8\},\{{♭}_1,{♭}_2,{♭}_4,{♭}_5,{♭}_6,{♭}_7,{♭}_8\},\{{♭}_1,{♭}_3,{♭}_4,{♭}_5,{♭}_6,{♭}_7,{♭}_8\}\}.$

Finally, let $\pounds =\{\varnothing ,\left\{{♭}_1\right\},\left\{{♭}_5\right\},\left\{{♭}_6\right\},\left\{{♭}_7\right\},\left\{{♭}_8\right\},\{{♭}_1,{♭}_5\},\{{♭}_1,{♭}_6\},\{{♭}_1,{♭}_7\},\{{♭}_1,{♭}_8\},\{{♭}_5,{♭}_6\},$ $\{{♭}_5,{♭}_7\},\{{♭}_5,{♭}_8\},\{{♭}_6,{♭}_7\},\{{♭}_6,{♭}_8\},\{{♭}_7,{♭}_8\},\{{♭}_1,{♭}_5,{♭}_6\},\{{♭}_1,{♭}_5,{♭}_7\},\{{♭}_1,{♭}_5,{♭}_8\},\{{♭}_1,{♭}_6,{♭}_7\},$ $\{{♭}_1,{♭}_6,{♭}_8\},\{{♭}_1,{♭}_7,{♭}_8\},\{{♭}_5,{♭}_6,{♭}_7\},\{{♭}_5,{♭}_6,{♭}_8\},\{{♭}_5,{♭}_7,{♭}_8\},\{{♭}_6,{♭}_7,{♭}_8\},\{{♭}_1,{♭}_5,{♭}_6,{♭}_7\},$ $\{{♭}_1,{♭}_5,{♭}_6,{♭}_8\},\{{♭}_1,{♭}_5,{♭}_7,{♭}_8\},\{{♭}_1,{♭}_6,{♭}_7,{♭}_8\},\{{♭}_5,{♭}_6,{♭}_7,{♭}_8\},\{{♭}_1,{♭}_5,{♭}_6,{♭}_7,{♭}_8\}\}$ be an ideal on $\mathcal{Q}.$ Then, ${\mathfrak{J}}_{{\mathcal{M}}_r}^{\pounds }=P\left(\mathcal{Q}\right).$

The infected patients are $S=\{{♭}_1,{♭}_3,{♭}_6,{♭}_8\}$, whereas the uninfected patients are $X=\{{♭}_2,{♭}_4,{♭}_5,{♭}_7\}$. Their approximation, accuracy, and boundary with the prior approach in [36] and the current methodology are computed as follows.

Case 1: The infected patients $S=\{{♭}_1,{♭}_3,{♭}_6,{♭}_8\}$.

1.

Hosny’s method [36] in Definition 7:

• ${\underline{⨿}}_{{\mathcal{M}}_r}\left(S\right)=\left\{{♭}_3\right\};$
• ${\overline{⨿}}_{{\mathcal{M}}_r}\left(S\right)=\mathcal{Q};$
• ${\mathcal{A}}_{{\mathcal{M}}_r}\left(S\right)={\displaystyle \frac{1}{8}};$
• ${\mathcal{B}}_{{\mathcal{M}}_r}\left(S\right)=\mathcal{Q}\setminus \left\{{♭}_3\right\}.$

2.

The proposed Definitions 10–12:

• ${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(S\right)=S;$
• ${\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(S\right)=S;$
• ${\mathcal{A}}_{{\mathcal{M}}_r}^{\pounds }\left(S\right)=1;$
• ${\mathcal{B}}_{{\mathcal{M}}_r}^{\pounds }\left(S\right)=\varnothing .$

Case 2: The uninfected patients $X=\{{♭}_2,{♭}_4,{♭}_5,{♭}_7\}$.

1.

Hosny’s method [36] in Definition 7:

• ${\underline{⨿}}_{{\mathcal{M}}_r}\left(X\right)=\{{♭}_2,{♭}_4\};$
• ${\overline{⨿}}_{{\mathcal{M}}_r}\left(X\right)=\mathcal{Q};$
• ${\mathcal{A}}_{{\mathcal{M}}_r}\left(X\right)={\displaystyle \frac{1}{4}};$
• ${\mathcal{B}}_{{\mathcal{M}}_r}\left(X\right)=\mathcal{Q}\setminus \{{♭}_2,{♭}_4\}.$

2.

The proposed Definitions 10–12:

• ${\underline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(X\right)=X;$
• ${\overline{⨿}}_{{\mathcal{M}}_r}^{\pounds }\left(X\right)=X;$
• ${\mathcal{A}}_{{\mathcal{M}}_r}^{\pounds }\left(X\right)=1;$
• ${\mathcal{B}}_{{\mathcal{M}}_r}^{\pounds }\left(X\right)=\varnothing .$

Thus, Hosny’s boundaries [36] for infected and uninfected individuals are $\mathcal{Q}\setminus \left\{{♭}_3\right\}$ and $\mathcal{Q}\setminus \{{♭}_2,{♭}_4\}$, respectively, leading to uncertainty and reduced decision accuracy. In contrast, the present approach yields an empty boundary, effectively reducing vagueness and enhancing accuracy.

7. Discussions: Merits and Demerits

This section details the benefits and drawbacks of the present models relative to the prior ones.

• The merits of the new models are as follows:

  (i)

  The prior topologies [35,36] are weaker than the present one. That is, the current topologies are stronger and include a lot of information that we need when studying rough set. These topologies are more effective for handling large samples and are crucial for making more accurate decisions in cases where the framework is appropriate, such as in infectious diseases, where the spread of infection depends on the sample size.

  (ii)

  It is possible to know the smallest and largest of all topologies. In contrast, previous topologies [17,26,27,28] were comparable only under restrictions.

  (iii)

  ${\mathfrak{J}}_{{\mathcal{M}}_{\wr }}^{\pounds }\subseteq {\mathfrak{J}}_{{\mathcal{M}}_{\langle \wr \rangle }}^{\pounds },\wr \in \{r,\mathfrak{l},i,u\},$ even though ${\mathfrak{J}}_{{\mathcal{N}}_{\wr }}^{\pounds }⊈,⊉{\mathfrak{J}}_{{\mathcal{N}}_{\langle \wr \rangle }}^{\pounds },\wr \in \{r,\mathfrak{l},i,u\}$ as in [17].

  (iv)

  The current method opens the door to many important real-life applications because it does not impose constraints on the relations, unlike its predecessors, which were limited in their applications because they used restricted relations [29,30].

  (v)

  The present methods can compare all types of approximations, while this ability is lacking in [17,26,27,28].

  (vi)

  The proposed technique upholds the property of monotonicity, which might be either retained or lost under stringent conditions in some prior methods [26,27,28]

  (vii)

  The current technique satisfies all of Pawlak’s criteria without conditions, whereas the methods in some earlier studies either failed to meet all of Pawlak’s characteristics or achieved them only under certain constraints [18].

• The demerits of the present models are as follows:

  (i)

  The prior approximations [17,27] are superior to the present one with a reflexive relation.

  (ii)

  The present methods require many calculations.

8. Conclusions

Rough set theory is a mathematical approach aimed at tackling uncertainty. Ideals can extend this theory, acting as a powerful tool for reducing vagueness by facilitating a broader approximation. A central focus within the examination of rough sets is the reduction of boundary to enhance accuracy. Ideals are among the most effective techniques for achieving this. Therefore, various methods utilizing ideals for constructing different topologies have been proposed.The prior topologies employing ideals were coarser than the present ones. In other words, the current topologies are more comprehensive and encompass a wealth of information that can be valuable for studying rough sets. This enhances its applicability in fields that require large samples, such as global epidemics. The properties of these topologies were scrutinized, including comparisons among them. The smallest and largest of all topologies were identified; this was not performed in previous studies. Additionally, new approximations were suggested utilizing these proposed topologies as an extension of the prior one. Moreover, the proposed models distinguish themselves from earlier ones by their ability to compare all sorts of approximations, fulfill all of Pawlak’s criteria, and exhibit monotonicity across any relations. Lastly, a real-world example was proposed to demonstrate how the present models outperform the previous ones and to illustrate their relevance in tackling real-world issues.

An exciting path for future research will encompass the following:

• Introducing two ideals in lieu of one to generalize the present method;
• Generating supra-topology to extend the current work;
• Generalizing the current paper to soft and fuzzy sets.