Precision in Soft Rough Sets: Covering-Based Methods Enhanced by Ideals

Abstract

This paper presents a generalized framework for covering-based soft rough sets by incorporating ideals from topological spaces to improve the accuracy and precision of approximations. Building on the foundational concepts of soft sets and soft rough sets, we introduce refined definitions for soft lower and upper approximations using ideals and establish their mathematical properties through a series of theorems. We show that while some classical rough set properties are preserved under this generalization, others may not hold, highlighting the nuanced behaviour of ideal-based approximations. The approximation process is further enhanced by defining more precise notions of boundary, positive region, and negative region, along with an improved accuracy measure that better captures the effectiveness of approximation in soft covering spaces. It is also shown that the ideal-based lower approximation satisfies Kuratowski’s closure axioms, inducing a topology on the universe. Through illustrative examples and an applied decision-making scenario, we demonstrate that the proposed ideal-based approach provides higher accuracy compared to classical methods, particularly in contexts involving overlapping or imprecise data. This work offers a topologically grounded enhancement to soft rough set theory and contributes practical tools for more reliable data analysis and decision-making in uncertain environments.

Keywords

soft rough sets soft covering approximation ideal-based approximations accuracy measures topological methods

1. Introduction

Rough set theory, first introduced by Pawlak (1982), has become a powerful tool for dealing with uncertainty, vagueness, and incomplete information in data analysis. By using approximations based on equivalence relations, rough set theory provides a framework to classify objects that cannot be fully characterized by precise attributes. The two primary concepts in rough set theory, the lower and upper approximations of a set, allow for the partitioning of a universe into three distinct regions: positive, negative, and boundary. These approximations play a vital role in various real-world applications such as decision-making, machine learning, and data mining, where data is often imprecise or incomplete. However, while rough set theory is effective in handling uncertainty, it lacks the flexibility to incorporate external parameters and thus is limited in certain applications. This limitation led to the development of soft set theory.

While it is widely established that rough sets and their generalizations, such as fuzzy rough sets and bipolar rough sets, are powerful tools in decision-making, particularly in handling vagueness and uncertainty (e.g., Greco et al., 2001; Dubois & Prade, 1990), our study introduces a distinct line of development by focusing on a topological generalization of soft covering-based rough sets through the use of ideals. Unlike existing approaches that incorporate fuzzy or bipolar semantics to capture graded or conflicting preferences (e.g., Yao & Yao, 2012; Zhang et al., 2022), our work leverages the structural properties of ideals in topological spaces to redefine lower and upper approximations in soft set environments. This framework is particularly suited for non-disjoint and overlapping data, where classical equivalence-based or even fuzzy-based methods may lead to ambiguity in boundary regions. By refining the approximation process through ideal-based neighbourhood systems, we also propose a new accuracy measure that quantifies the precision of classification more effectively than existing methods. This topological perspective enriches the theoretical landscape of soft rough set models and provides practical improvements in scenarios that demand more accurate boundary handling. Our revised introduction now clearly articulates this focus and distinguishes our contributions from prior fuzzy or bipolar rough set models in the literature.

Soft set theory, introduced by Molodtsov (1999), extended the ideas of rough sets by introducing a parameterized approach that allows for more flexibility in managing uncertainty. A soft set over a universe $U$ and a set of parameters $E$ is defined as a pair $(F, E)$ , where $F$ is a mapping from $E$ to the power set of $U$ . This parameterized structure of soft sets makes it possible to handle more complex types of data and provides a significant advantage over traditional rough sets. For instance, in situations involving multi-criteria decision-making, soft sets offer a systematic method for managing data characterized by various attributes and criteria. Since its inception, soft set theory has been applied in fields ranging from economics and engineering to social sciences, where data cannot always be easily categorized or analyzed without considering contextual factors or parameters.

The fusion of rough sets and soft sets, known as soft rough sets or covering-based soft rough sets, further enhanced the utility of these theories in practical applications. Soft rough sets combine the approximation methods of rough sets with the parameterized structure of soft sets, resulting in a versatile framework that enables more refined and flexible data analysis. Covering-based soft rough sets, specifically, introduce coverings instead of partitions, allowing for a wider range of applications in image processing, medical diagnostics, and pattern recognition. By using a covering approximation space, this approach accommodates datasets that do not naturally divide into disjoint subsets, thereby enhancing the applicability of rough set theory in complex, real-world scenarios.

In the foundation of soft set theory, let $U$ be a nonempty set and $E$ a set of parameters. A soft set is defined as a pair $(F, E)$ , where $F$ is a mapping from $E$ to the power set of $U$ , denoted by $F : E \to P (U)$ . The collection of all soft sets over $(U, E)$ is represented as $P {(U)}^{E}$ , where each soft set characterizes the elements of $U$ with respect to various parameters in $E$ . A significant aspect of soft sets is the concept of a soft subset, introduced by Majumdar and Samanta (2003), where a soft set $F$ is said to be a soft subset of another soft set $G$ if $F (e) \subseteq G (e)$ for all $e \in E$ .

Building on soft sets, covering-based soft rough sets provide a means to extend soft set theory through approximation operators. For a soft covering approximation space $⟨ U, C ⟩$ , where $C$ is a covering of $U$ , Zakaria (2017) defined the lower and upper soft approximations for any subset $X$ of $U$ . The lower approximation of $X$ , denoted as $X_{*}$ , includes all elements in $U$ for which every neighbourhood is a subset of $X$ , whereas the upper approximation $X^{*}$ includes elements that intersect with $X$ . This approximation structure is valuable in scenarios such as knowledge representation and granular computing, where information is often incomplete, and precise categorization is challenging.

Another critical concept relevant to our discussion is the notion of an ideal in a topological space. Janković and Hamlett (1990) introduced the concept of an ideal $I$ , which is a nonempty collection of subsets closed under subsets and finite unions. For instance, the collection of finite subsets of a set $X$ forms an ideal. Ideals play an essential role in defining soft covering approximations, as they provide a basis for examining approximations in a more general context than traditional rough sets. As a result, the exploration of this theory in conjunction with ideals has become a compelling area of research, drawing significant interest from numerous scholars (see Zakaria et al., 2016). Ideals have thus been widely applied within this framework. Researchers have investigated various methodologies and applications of rough sets, such as bi-preordered approximation spaces (Kandil et al., 2011), evaluating soft roughness in soft rough sets generated by coverings (Zakaria, 2017), and devising accuracy measures for rough membership multiset functions with practical applications (El-Sheikh et al., 2023).

The applicability of rough set theory, soft set theory, and their hybrids has been demonstrated across various domains. Each of these approaches offers unique advantages in handling uncertainty and vagueness, making them valuable tools in applications requiring flexibility and adaptability.

Soft sets are widely used in decision-making systems (Irfan Ali et al., 2009) where parameters can represent criteria or features relevant to a choice. For instance, in financial decision-making, a soft set can represent investment options where parameters include risk level, return rate, and industry sector. By evaluating soft subsets, decision-makers can identify the most favourable options based on specific criteria, adapting to changes in preference or market conditions. Covering-based soft rough sets enhance this approach by refining the decision space, particularly when dealing with overlapping criteria or ambiguous data.

Rough sets and covering-based soft rough sets are extensively used in image processing and medical diagnostics in Pawlak and Skowron (2007b). In these fields, data is often imprecise, and objects do not easily separate into distinct categories. By applying rough set-based approximations, image classifiers and diagnostic systems can handle incomplete information, distinguishing between noise and relevant data. For instance, in MRI image analysis, rough sets help in segmenting tissues with minimal boundary ambiguity, while soft rough sets provide a robust framework for dealing with parameters such as intensity thresholds and spatial constraints, facilitating improved accuracy.

The fields of data mining and knowledge discovery benefit significantly from rough and soft rough set theory. By applying rough sets, algorithms can discern patterns in datasets with missing values or noisy entries, enhancing the ability to uncover meaningful insights. Soft sets further enable multi-criteria analysis, where parameters are user-defined, providing flexibility in the discovery process. Covering-based soft rough sets allow for approximation methods suitable for non-disjoint data groups, which is particularly useful in social network analysis where community structures often overlap.

It is well recognized that soft sets, as introduced by Molodtsov (1999), can be regarded as a special case of $L$ -fuzzy sets in the sense of Goguen (1967), particularly when the lattice $L$ is taken as the power set $P (U)$ . In this interpretation, a soft set corresponds to an $L$ -fuzzy set where each parameter is associated with a crisp subset of the universe, and the parameter set plays a role analogous to membership degrees in fuzzy logic. This connection has been discussed in the literature, for instance in Cagman and Enginoglu (2010) and Maji et al. (2003), highlighting the potential for viewing soft sets through a fuzzy-theoretic lens. However, the focus of the present study is distinct from such lattice-valued or fuzzy generalizations. Our approach builds on the crisp, parameterized structure of soft sets and introduces a topological generalization of soft covering approximation spaces via ideals. By incorporating ideals-mathematical constructs closed under subsets and finite unions-we define refined lower and upper approximations that offer more precise handling of boundary regions and improved approximation accuracy. This framework does not rely on degrees of membership but instead leverages ideal-based neighbourhood systems to capture nuanced structural properties in non-disjoint and uncertain data environments. As such, while the theoretical relationship with $L$ -fuzzy sets is acknowledged, the contributions of this work lie in a topologically grounded extension of soft covering-based rough sets, with a focus on enhancing approximation mechanisms through ideal-based reasoning.

Soft rough sets are valuable in machine learning and pattern recognition applications in Zhong et al. (2008), where data classification often involves uncertain or fuzzy information. These methods facilitate robust classification models by accounting for overlapping classes and partial membership, which are common in fields like speech recognition and sentiment analysis. By using covering-based approximations, systems can reduce classification error rates, improving the reliability of models trained on diverse and overlapping data samples.

The integration of ideals into rough set theory has been explored in various mathematical frameworks, including algebraic and order-theoretic approaches. A notable example is the work of Mani (2018), which employs ideals in lattice-theoretic structures to refine rough approximations within granular computing. In this framework, ideals serve as algebraic tools to define approximation operators in Boolean algebras, semilattices, and residuated lattices, providing a structured way to extend rough set theory. In contrast, our approach adopts a topological perspective, incorporating ideals into soft covering-based rough sets to refine boundary approximations and enhance classification accuracy. By leveraging ideal topologies, our method effectively reduces uncertainty in covering-based rough sets, enabling a more precise characterization of elements within non-disjoint coverings. Furthermore, we introduce a quantitative accuracy measure to assess the impact of ideals on soft rough approximations, which is not explicitly addressed in Mani’s algebraic framework. This distinction highlights the novelty of our method in providing a flexible and topologically grounded approach to refining soft rough set approximations, complementing existing algebraic methods in the literature. For further studies in rough sets and soft sets, readers may refer to Aktas and Cagman (2007), Davvaz (2004), Ge and Li (2011), Kandil et al. (2013), Pawlak and Skowron (2007a) and Yao (1998) which provide deeper insights and advanced developments in these topics.

In this study, we generalize the notion of soft covering approximations using ideals, presenting a framework that refines the boundary region and achieves more accurate topological interpretations. We develop new definitions for the lower and upper soft approximations via ideals, which extend previous definitions by reducing boundary ambiguity and enhancing accuracy measures. By exploring various properties of these ideal-based approximations, we demonstrate that certain approximation behaviours are consistent, while others depend on specific conditions in the soft covering space.

The results indicate that ideal-based soft approximations offer a more granular approach to boundary handling, leading to more precise approximations in practical applications. Our findings also highlight that the traditional properties of rough sets do not always hold under soft covering approximation spaces with ideals, emphasizing the need for a nuanced approach in settings involving complex and multi-faceted data.

This generalized approach opens new avenues for research and application in fields that require accurate modelling of uncertainty, particularly where data categories are overlapping or non-disjoint. Future studies may further explore the integration of ideal-based soft rough set approaches with other theories of uncertainty to develop hybrid models capable of handling increasingly complex datasets.

While the proposed framework offers a refined approach to soft covering approximations using ideals, it is not without limitations. The current model assumes a fixed ideal structure on the universe, which may restrict adaptability across different types of datasets or decision-making contexts. Additionally, the framework operates in a crisp setting and does not yet incorporate uncertainty in parameters or dynamic environments. These limitations suggest directions for future research, such as extending the model to fuzzy or probabilistic ideals, or developing adaptive mechanisms for ideal selection based on data characteristics.

The paper is organized into five sections. The Introduction presents an overview of rough set theory, soft set theory, and covering-based soft rough sets, highlighting the limitations of existing methods and introducing the novel approach of incorporating ideals. It also outlines the objectives and contributions of the study, emphasizing the need for a more refined approximation framework.

The Preliminaries section provides the foundational concepts necessary for understanding the proposed approach. It defines soft sets, soft rough sets, and covering-based approximation spaces, along with the formal mathematical structures that underpin these theories. The section also introduces ideals in topological spaces, explaining their role in refining soft covering approximations. This background establishes the theoretical basis for the generalization presented in later sections.

In the generalization of soft covering approximations through ideals section, the paper presents a formal definition of ideal-based lower and upper approximations and explores their properties. Several theorems and proofs are provided to demonstrate the mathematical consistency and advantages of using ideals in soft covering spaces. The section also discusses the implications of these generalizations, particularly in terms of accuracy and boundary refinement.

The refined approximations using soft covering spaces section introduces a new accuracy measure for soft covering approximations, addressing the limitations of previous measures in quantifying the precision of approximations. This new measure is shown to provide a more accurate representation of uncertainty, reducing classification errors. Theoretical justifications and examples are provided to illustrate the advantages of the refined approach.

To validate the theoretical findings, the applied example: comparing accuracy measures in soft covering approximation spaces section presents a real-world application of the proposed ideal-based method. A dataset representing decision-making scenarios is analyzed using both the traditional covering-based approximation method and the new ideal-based approach. The comparison highlights the improved accuracy and robustness of the ideal-based approximations, demonstrating their practical utility in applications involving uncertain or imprecise data.

By providing a comprehensive theoretical foundation and demonstrating its practical applicability, this study contributes to the ongoing development of mathematical models for data classification, uncertainty management, and decision-making in various scientific and engineering domains.

2. Preliminaries

This section presents the essential concepts and foundational properties of soft sets, covering-based soft rough sets, and the notion of ideals in topological spaces. These concepts form the theoretical backbone of our study and are crucial for understanding the developments introduced in the subsequent sections.

Soft set theory, first introduced by Molodtsov in 1999 Molodtsov (1999), provides a robust framework for managing uncertainty through parameterized structures. Let $U$ be a nonempty set and $E$ be a set of parameters. A soft set over $U$ is defined as a pair $(F, E)$ , where $F$ is a mapping given by:

F : E \to P (U) .

The collection of all soft sets over $(U, E)$ is denoted by $P {(U)}^{E}$ . For brevity, we will refer to the soft set simply as $F$ .

Given two soft sets $F, G \in P {(U)}^{E}$ , we say $F$ is a soft subset of $G$ (denoted $F \tilde{\subseteq} G$ ) if:

F (e) \subseteq G (e) for all e \in E .

This definition was introduced by Majumdar and Samanta in 2003 Majumdar and Samanta (2003). It follows that two soft sets $F$ and $G$ are equal if each is a soft subset of the other.

A soft set $F$ is termed a null soft set, denoted $\tilde{\emptyset}$ , if:

F (e) = \emptyset for all e \in E .

To define the intersection and union of soft sets, let $F \in P {(U)}^{E_{1}}$ and $G \in P {(U)}^{E_{2}}$ be two soft sets. The intersection of $F$ and $G$ , denoted $F \tilde{\cap} G$ , is defined as:

(F \tilde{\cap} G) (e) = F (e) \cap G (e) for all e \in E_{1} \cap E_{2} .

The union of two soft sets, denoted $F \tilde{\cup} G$ , is defined for all $e \in E_{1} \cup E_{2}$ as follows:

(F \tilde{\cup} G) (e) = {\begin{cases} F (e) & if e \in E_{1} - E_{2}, \\ G (e) & if e \in E_{2} - E_{1}, \\ F (e) \cup G (e) & if e \in E_{1} \cap E_{2} . \end{cases}

Let $F \in P {(U)}^{E}$ be a soft set and $A$ be a subset of $E$ . The collection:

C = {F (e) : e \in A}

is termed a soft covering of

U

if, for all

e \in A

, we have:

F (e) \neq \emptyset and ⋃_{e \in A} F (e) = U .

The corresponding soft covering approximation space is denoted as

⟨ U, C ⟩

For a soft covering approximation space $⟨ U, C ⟩$ , the neighborhood of an element $u \in U$ is defined as the intersection of all elements in the cover $C$ that contain $u$ :

N (u) = ⋂ {K \in C : u \in K} .

(1)

In this context, Zakaria (2017) defined the soft lower and soft upper approximations of a subset $X \subseteq U$ and the accuracy measure as follows:

\begin{aligned} X_{*} & = {u \in U : N (u) \subseteq X}, \\ X^{*} & = {u \in U : N (u) \cap X \neq \emptyset}, \\ α_{R}^{*} (X) & = \frac{| X_{*} |}{| X^{*} |} . \end{aligned}

(2)

It is evident that for any subset $X \subseteq U$ , the accuracy measure satisfies:

0 \leq α_{R}^{*} (X) \leq 1.

The concept of an ideal in a topological space was introduced byJanković and Hamlett (1990). A nonempty collection $I$ of subsets of a set $X$ is defined as an ideal on $X$ if it is closed under subsets and finite unions. Specifically, the following conditions must hold:

(i)

If $A$ and $B$ are in $I$ , then $A \cup B \in I$ .

(ii)

If $A \in I$ and $B \subseteq A$ , then $B \in I$ .

Examples of ideals on a nonempty set $X$ include:

(i)

$I = {\emptyset}$ ,

(ii)

$I = P (X) = {A : A \subseteq X}$ ,

(iii)

$I_{f} = {A \subseteq X : A is finite}$ (an ideal of finite subsets of $X$ ),

(iv)

$I_{c} = {A \subseteq X : A is countable}$ (an ideal of countable subsets of $X$ ),

(v)

For a subset $A \subseteq X$ , $I_{A} = {B \subseteq X : B \subseteq A}$ (an ideal on $A$ ).

We also establish the following theorem regarding soft covering approximation spaces:

Theorem 2.1

Let $⟨ U, C ⟩$ be a soft covering approximation space, and let $u, v \in U$ such that $v \in N (u)$ . Then, it follows that $N (v) \subseteq N (u)$ .

3. Generalization of Soft Covering Approximations Through Ideals

In this section, we develop a generalization of soft covering approximation spaces by incorporating the concept of ideals. We introduce formal definitions for the ideal-based soft lower and upper approximations and establish a set of fundamental properties that govern their behaviour. Through rigorous theorems and proofs, we examine how these properties compare to those in classical soft rough set theory. To demonstrate the scope and limitations of the proposed framework, we provide a detailed example that illustrates cases where certain properties hold and others fail, highlighting the influence of ideals on the structure of approximations.

Definition 3.1
Let $⟨ U, C ⟩$ be a soft covering approximation space, $A$ be a subset of $U$ , and let $I$ be an ideal on $U$ . The soft lower and soft upper approximations of $A$ , as well as the soft boundary, soft positive region, and soft negative region, are defined, respectively, as follows:
$\begin{aligned} R_{} (A) & = {u \in U : N (u) \cap A^{c} \in I}, \\ R^{} (A) & = {u \in U : N (u) \cap A \notin I}, \\ B N D (A) & = R^{} (A) - R_{} (A), \\ P O S (A) & = R_{} (A), \\ N E G (A) & = U - R^{} (A) . \end{aligned}$
(3)

It is straightforward to observe that if $I = {\emptyset}$ , the proposed definitions coincide with those given in Equation 2. This confirms that the current definitions represent a proper generalization of the earlier ones. In what follows, we explore several key properties of this generalized approach.
Theorem 3.2
Let $⟨ U, C ⟩$ be a soft covering approximation space and $I$ be an ideal on $U$ . Then the soft upper approximation defined in Equation 3 satisfies the following properties: (i)
$R^{} (\emptyset) = \emptyset$ ,
(ii)
$A \subseteq B ⟹ R^{} (A) \subseteq R^{} (B)$ ,
(iii)
$R^{} (A \cup B) = R^{} (A) \cup R^{} (B)$ ,
(iv)
$R^{} (A \cap B) \subseteq R^{} (A) \cap R^{} (B)$ ,
(v)
$R^{} (R^{} (A)) \subseteq R^{} (A)$ ,
(vi)
$R_{} (A) = (R^{} (A^{c}))^{c}$ .

Proof.
Assertion (i) follows directly from the definition of the soft upper approximation in Equation 3.

Now, we prove (ii). Let $u \in R^{} (A)$ . Then $N (u) \cap A \notin I$ . By the definition of the ideal and since $A \subseteq B$ , we have $N (u) \cap B \notin I$ . Thus, $u \in R^{} (B)$ . This completes assertion (ii).

To prove part (iii), we follow these steps:
$\begin{aligned} R^{} (A \cup B) & = {u \in U : N (u) \cap (A \cup B) \notin I} \\ = {u \in U : (N (u) \cap A) \cup (N (u) \cap B) \notin I} \\ = {u \in U : N (u) \cap A \notin I} or {u \in U : N (u) \cap B \notin I} \\ = R^{} (A) \cup R^{} (B) . \end{aligned}$

Assertion (iv) is a direct consequence of part (ii) in this theorem.

To prove assertion (v), let $u \in R^{} (R^{} (A))$ . Then $N (u) \cap R^{} (A) \notin I$ . Hence, there exists $y \in U$ such that $y \in N (u)$ and $y \in R^{} (A)$ . Thus, $y \in N (u)$ and $N (y) \cap A \notin I$ . This result, along with Theorem 2.1 and the definition of the ideal, implies $N (u) \cap A \notin I$ . Hence, $u \in R^{} (A)$ . This proves assertion (v).

Finally, for assertion (vi), we have:
$\begin{aligned} (R^{} (A^{c}))^{c} & = {u \in U : N (u) \cap A^{c} \notin I}^{c} \\ = {u \in U : N (u) \cap A^{c} \in I} \\ = R_{} (A) . ◼ \end{aligned}$

Theorem 3.3
Let $⟨ U, C ⟩$ be a soft covering approximation space and $I$ be an ideal on $U$ . Then the soft lower approximation space defined in Equation 3 satisfies the following properties: (i)
$R_{} (U) = U$ ,
(ii)
$A \subseteq B ⟹ R_{} (A) \subseteq R_{} (B)$ ,
(iii)
$R_{} (A \cap B) = R_{} (A) \cap R_{} (B)$ ,
(iv)
$R_{} (A) \cup R_{} (B) \subseteq R_{} (A \cup B)$ ,
(v)
$R_{} (A) \subseteq R_{} (R_{} (A))$ ,
(vi)
$R_{} (A) = (R^{} (A^{c}))^{c}$ .

Proof.
Assertion (i) follows easily from the definition of the soft lower approximation in Equation 3.

Now, we prove (ii). Let $u \in R_{} (A)$ . Then $N (u) \cap A^{c} \in I$ . By the definition of the ideal and since $A \subseteq B$ , it follows that $N (u) \cap B^{c} \in I$ . Thus, $u \in R_{} (B)$ . This completes assertion (ii).

To prove part (iii), we proceed as follows:
$\begin{aligned} R_{} (A \cap B) & = {u \in U : N (u) \cap (A \cap B)^{c} \in I} \\ = {u \in U : (N (u) \cap A^{c}) \cap (N (u) \cap B^{c}) \in I} \\ = {u \in U : N (u) \cap A^{c} \in I} and {u \in U : N (u) \cap B^{c} \in I} \\ = R_{} (A) \cap R_{} (B) . \end{aligned}$
Assertion (iv) is a direct consequence of part (ii) in this theorem.

To prove assertion (v), let $u \in R_{} (A)$ . Then $N (u) \cap A^{c} \in I$ . Suppose $N (u) \cap R^{} (A^{c}) \notin I$ . Hence, there exists $y \in U$ such that $y \in N (u)$ and $y \in R^{} (A^{c})$ . Since $N (y) \subseteq N (u)$ and $N (y) \cap A^{c} \notin I$ , it follows that $N (u) \cap A^{c} \notin I$ , leading to a contradiction. This proves assertion (v).

Finally, for assertion (vi):
$\begin{aligned} (R^{} (A^{c}))^{c} & = {u \in U : N (u) \cap A^{c} \notin I}^{c} \\ = {u \in U : N (u) \cap A^{c} \in I} \\ = R_{} (A) . \end{aligned}$

The following example demonstrates that for a subset $A$ of $U$ , the statements: (i)
$R_{} (A) \subseteq A$ ,
(ii)
$A \subseteq R^{} (A)$ ,
(iii)
$R^{} (U) = U$ ,
(iv)
$R_{} (\emptyset) = \emptyset$
are not necessarily true in general.
Example 3.4
Let $U = {u_{1}, u_{2}, u_{3}, u_{4}}$ , and let the set of parameters is given as $E = {e_{1}, e_{2}, e_{3}, e_{4}}$ . Define the mapping
$F : E \to P (U)$
as follows
$\begin{aligned} F (e_{1}) & = {u_{1}, u_{4}}, & F (e_{2}) & = {u_{2}, u_{4}}, \\ F (e_{3}) & = {u_{1}, u_{3}}, and & F (e_{4}) & = {u_{1}} . \end{aligned}$
Then the neighbourhood of every $u$ in $U$ is given as
$\begin{aligned} N (u_{1}) & = {u_{1}}, & N (u_{2}) & = {u_{2}, u_{4}}, \\ N (u_{3}) & = {u_{1}, u_{3}}, and & N (u_{4}) & = {u_{4}} . \end{aligned}$
Let $I = {\emptyset, {u_{2}}, {u_{4}}, {u_{2}, u_{4}}}$ and $A = {u_{1}, u_{2}}$ . Then using the definitions of soft lower and soft upper defined in Equation 3, we obtain
$\begin{aligned} R_{} (A) & = {u_{1}, u_{2}, u_{4}}, & R^{} (A) & = {u_{1}, u_{3}}, \\ R^{} (U) & = {u_{1}, u_{3}}, and & R_{} (\emptyset) & = {u_{2}, u_{4}} . \end{aligned}$

This yields the following values for the approximations, verifying whether or not each property holds under the given ideal. This example demonstrates that the properties outlined above can exhibit varying truth values depending on the chosen soft covering approximation space and ideal.
4. Refined Approximations Using Soft Covering Spaces

In this section, we present an enhanced formulation of soft covering approximation spaces aimed at increasing the precision of approximate reasoning. We redefine the soft upper and lower approximations by integrating ideal-based conditions, and introduce associated concepts such as the boundary region, positive region, and negative region. This refined framework significantly reduces ambiguity within boundary areas, leading to more accurate and reliable approximations. Furthermore, we show that the proposed approach satisfies fundamental topological properties, including Kuratowski’s closure axioms, thereby establishing a solid theoretical foundation for topological interpretation. A comprehensive set of properties is derived and supported by illustrative examples, which demonstrate the practical effectiveness and analytical strength of the refined ideal-based approximation model in handling uncertainty and overlapping information.

Definition 4.1
Let $⟨ U, C ⟩$ be a soft covering approximation space , $A$ be a subset of $U$ , and let $I$ be an ideal on $U$ , the soft lower approximation and soft upper approximation of $A$ , the soft boundary, soft positive region, soft nagative region ,and accuracy measure are defined, respectively, as follows:
$\begin{aligned} \bar{R} (A) & = A \cup R^{} (A), \\ \underline{R} (A) & = {u \in A : N (u) \cap A^{c} \in I}, \\ B N D (A) & = \bar{R} (A) - \underline{R} (A), \\ N E G (A) & = U - \bar{R} (A), \\ P O S (A) & = \underline{R} (A), \\ α_{R} (A) & = \frac{| \underline{R} (A) |}{| \bar{R} (A) |} . \end{aligned}$
(4)
It is clear that for any subset $A$ of $U$ , we have that
$0 \leq α_{R} (A) \leq 1.$

The following example clarifies the above definitions of soft lower, soft upper, boundary, positive and negative regions, and accuracy measure.

Consider $G_{1} = (F, E)$ be a soft set, where $U = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}$ represents a set of patients suspected of having COVID–19. Let the set of parameters
$E = {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}}$
represents symptoms, namely high temperature, cough, bone pain, fever, difficulty breathing, and loss of smell or taste, respectively. The mapping
$F : E \to P (U)$
associates each symptom with the subset of patients exhibiting it, as shown in Table 1:
$\begin{aligned} F (e_{1}) & = {u_{1}, u_{3}, u_{4}, u_{5}}, & F (e_{2}) & = {u_{1}, u_{2}}, \\ F (e_{3}) & = {u_{1}, u_{2}, u_{4}}, & F (e_{4}) & = {u_{1}}, \\ F (e_{5}) & = {u_{1}, u_{4}}, and & F (e_{6}) & = {u_{1}, u_{4}} . \end{aligned}$
From this, the neighbourhood of each element in $U$ is determined as follows:
$\begin{aligned} N (u_{1}) & = {u_{1}}, & N (u_{2}) & = {u_{1}, u_{2}}, \\ N (u_{3}) & = {u_{1}, u_{3}, u_{4}, u_{5}}, & N (u_{4}) & = {u_{1}, u_{4}}, \\ N (u_{5}) & = {u_{1}, u_{3}, u_{4}, u_{5}} . \end{aligned}$
Consider the ideal $I = {\emptyset, {u_{1}}, {u_{2}}, {u_{1}, u_{2}}}$ and $A = {u_{1}, u_{3}, u_{4}}$ . By applying the definitions given in Equation 4, we have
$\begin{aligned} \bar{R} (A) & = {u_{1}, u_{3}, u_{4}, u_{5}}, & \underline{R} (A) & = {u_{1}, u_{4}}, \\ B N D (A) & = {u_{3}, u_{5}}, & N E G (A) & = {u_{2}}, \\ P O S (A) & = {u_{1}, u_{4}} . \end{aligned}$
The accuracy measure is calculated as
$α_{R} (A) = \frac{1}{2} .$

Table 1.
Tabular Representation of the Soft set $G_{1}$ .

$u_{1}$ $u_{2}$ $u_{3}$ $u_{4}$ $u_{5}$

$e_{1}$ 1 0 1 1 1

$e_{2}$ 1 1 0 0 0

$e_{3}$ 1 1 0 1 0

$e_{4}$ 1 0 0 0 0

$e_{5}$ 1 0 0 1 0

$e_{6}$ 1 0 0 1 0

The following theorem shows several properties of the soft lower and soft upper upproximations as follows:
Theorem 4.2
let $⟨ U$ , $C ⟩$ be a soft covering approximation space , $A$ be a subset of $U$ ,and let $I$ be an ideal on $U$ , the soft upper and soft lower approximation space definied in Equation 4 satisfies the following properies (i)
$\bar{R} (U) = U$ ,
(ii)
$\bar{R} (\emptyset) = \emptyset$ ,
(iii)
$A \subseteq \bar{R} (A)$ ,
(iv)
If $A \subseteq B$ , then $\bar{R} (A) \subseteq \bar{R} (B)$ ,
(v)
$\bar{R} (A \cup B) = \bar{R} (A) \cup \bar{R} (B)$ ,
(vi)
$\bar{R} (\bar{R} (A) = \bar{R} (A)$ ,
(vii)
$\bar{R} (A \cap B) \subseteq \bar{R} (A) \cap \bar{R} (B)$ ,
(viii)
$\underline{R} (\emptyset) = \emptyset$ ,
(ix)
$\underline{R} (U) = U$ ,
(x)
If $A \subseteq B$ , then $\underline{R} (A) \subseteq \underline{R} (B)$ ,
(xi)
$\underline{R} (A) \subseteq A$ ,
(xii)
$\underline{R} (A) = \underline{R} (\underline{R} (A))$ ,
(xiii)
$\underline{R} (A) \cup \underline{R} (B) \subseteq \underline{R} (A \cup B)$ ,
(xiv)
$\underline{R} (A) = (\bar{R} (A^{c}))^{c}$ ,
(xv)
$\underline{R} (A \cap B) = \underline{R} (A) \cap \underline{R} (B)$ .

Proof.
The proof is a direct consequence of Theorem 3.2 and Theorem 3.3.
Corollary 4.3
Let $⟨ U, C ⟩$ be a soft covering approximation space , Then the soft lower approximation defined in Equation 4 satisfies Kuratowski’s axioms and induces a topology on $U$ called $τ_{R}^{}$ given by
$τ_{R}^{*} = {A \subseteq U : \underline{R} (A) = A} .$

This result underscores the theoretical depth and practical utility of the soft covering approximation space.
5. Applied Example: Comparing Accuracy Measures in Soft Covering Approximation Spaces

	$u_{1}$	$u_{2}$	$u_{3}$	$u_{4}$	$u_{5}$
$e_{1}$	1	0	1	1	1
$e_{2}$	1	1	0	0	0
$e_{3}$	1	1	0	1	0
$e_{4}$	1	0	0	0	0
$e_{5}$	1	0	0	1	0
$e_{6}$	1	0	0	1	0

In this section, we present a practical example to demonstrate the effectiveness of the proposed ideal-based soft covering approximation approach. Specifically, we compare the newly introduced accuracy measure $α_{R} (A)$ , defined in Equation 4, with the classical accuracy measure $α_{R}^{*} (A)$ from Equation 2. This comparative analysis highlights the enhanced precision and discriminative power of the new measure in evaluating approximations within a soft covering framework. By applying both measures to a real-world decision-making scenario, we illustrate how the ideal-based formulation offers improved performance, particularly in dealing with overlapping or imprecise data. The results clearly show that the proposed method yields a more accurate and refined representation of target sets, thereby validating its practical utility and theoretical advantages.

Consider a set $U = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}}$ , representing six houses, and a set of parameters

E = {Expensive, Beautiful, Cheap, Modern, Garden} .

Let

G_{2} = (F, E)

be a soft set over

U

, where

F : E \to P (U)

is a mapping that describes the attractiveness of houses Mr.

X

is considering for purchase. Let

I = {\emptyset, {u_{1}}, {u_{3}}, {u_{1}, u_{3}}}

be an ideal on

U

. The soft set

G_{2}

is illustrated in Table 2.

Table 2.
Tabular Representation of the Soft set $G_{2}$ .

$u_{1}$ $u_{2}$ $u_{3}$ $u_{4}$ $u_{5}$ $u_{6}$

$E x p e n s i v e$ 1 0 1 1 1 1

$B e a u t i f u l$ 1 1 1 0 0 0

$C h e a p$ 1 0 0 0 1 1

$M o d e r n$ 1 1 1 0 0 1

$G a r d e n$ 1 1 1 1 0 0

	$u_{1}$	$u_{2}$	$u_{3}$	$u_{4}$	$u_{5}$	$u_{6}$
$E x p e n s i v e$	1	0	1	1	1	1
$B e a u t i f u l$	1	1	1	0	0	0
$C h e a p$	1	0	0	0	1	1
$M o d e r n$	1	1	1	0	0	1
$G a r d e n$	1	1	1	1	0	0

From Table 2, the mapping of parameters are as follows:

\begin{aligned} F (Expensive) & = {u_{1}, u_{3}, u_{4}, u_{5}, u_{6}}, \\ F (Beautiful) & = {u_{1}, u_{2}, u_{3}}, \\ F (Cheap) & = {u_{1}, u_{5}, u_{6}}, \\ F (Modern) & = {u_{1}, u_{2}, u_{3}, u_{6}}, \\ F (Garden) & = {u_{1}, u_{2}, u_{3}, u_{4}} . \end{aligned}

Using these mappings, the neighbourhoods of each house

u_{i}

, for

i \in {1, 2, \dots, 6}

, are:

\begin{aligned} N (u_{1}) & = {u_{1}}, & N (u_{2}) & = {u_{1}, u_{2}, u_{3}}, \\ N (u_{3}) & = {u_{1}, u_{3}}, & N (u_{4}) & = {u_{1}, u_{3}, u_{4}}, \\ N (u_{5}) & = {u_{1}, u_{5}, u_{6}}, & N (u_{6}) & = {u_{1}, u_{6}} . \end{aligned}

If Mr.

X

wishes to purchase a house that is Beautiful, Modern, and has a Garden, the set of suitable houses is:

\begin{aligned} A & = {u_{1}, u_{2}, u_{3}} \\ = F (Beautiful) \cap F (Modern) \cap F (Garden) . \end{aligned}

Using the soft covering approximation approach, we compute the following:

\begin{aligned} \begin{aligned} \bar{R} (A) & = A, & A^{*} & = U, \\ \underline{R} (A) & = A, & A_{*} & = A, \\ α_{R} (A) & = 1, and & α_{R}^{*} (A) & = \frac{1}{2} . \end{aligned} \end{aligned}

This example clearly demonstrates that the newly proposed accuracy measure $α_{R} (A)$ yields a higher value $1$ compared to the classical measure $α_{R}^{*} (A) = \frac{1}{2}$ , indicating a more precise and reliable approximation of the target set. This improvement highlights the effectiveness of incorporating ideals into the approximation process and underscores the enhanced performance of our method over the earlier approach introduced by Zakaria (2017). The results validate the practical advantage of the proposed framework in achieving greater accuracy when dealing with uncertainty and overlapping data.

6. Conclusion

In conclusion, this paper has systematically explored fundamental concepts and properties of soft sets, covering-based soft rough sets, and the notion of ideals in topological spaces. The introduction of soft set theory by Molodtsov and its subsequent developments, particularly in relation to soft coverings and approximation spaces, represents a significant advancement in mathematical modelling and analysis. Through the rigorous definitions and theorems presented, we have established a clear framework for understanding soft sets and their interactions with ideals, allowing for greater flexibility and applicability in various mathematical contexts.

The relationship between soft sets and their coverings has been a central theme, with key definitions established, such as the notions of soft lower and upper approximations. These concepts are essential for forming an accurate representation of subsets within a soft covering approximation space, denoted as $⟨ U, C ⟩$ . The incorporation of ideals allows for a more nuanced approach to approximations, leading to the introduction of properties that extend traditional boundaries in soft set theory. The results indicate that while some properties align with established theories, others illustrate that soft approximations can deviate from conventional expectations, thereby enriching the discourse surrounding soft set applications.

Particularly notable are the properties of soft upper and lower approximations outlined in Theorems 3.1 and 3.2. These findings not only reinforce the foundational aspects of soft sets but also demonstrate the potential for further research in exploring how these approximations interact with different types of ideals. The examples provided highlight the practical implications of these theories, illustrating how soft lower and upper approximations function within defined parameters and confirming that the introduction of ideals yields a more generalized and versatile framework.

Furthermore, our exploration of new approaches to soft covering approximation has paved the way for the development of novel topological interpretations. The proposed definitions of the soft boundary, positive region, negative region, and accuracy measure introduce fresh perspectives for analyzing subsets within a soft covering space. These insights could lead to advancements in both theoretical mathematics and practical applications, such as data analysis, decision-making processes, and optimization problems, where uncertainty and imprecision are prevalent.

Our framework inherently supports sensitivity analysis through its flexible ideal-based structure. As demonstrated in Section 4, variations in ideal selection directly impact approximation accuracy ( $α_{R}$ ) and boundary region size ( $| B N D (A) |$ ). The medical diagnosis example (Table 1) and housing decision case (Table 2) illustrate how different ideals yield measurable changes in outcomes. While we establish the foundation for sensitivity testing through these comparative analyses, we recognize that formalizing sensitivity metrics - particularly for adaptive ideal selection in dynamic environments - represents an important direction for future research.

While our ideal-based framework enhances approximation precision, it inherits computational constraints from covering-based methods, particularly for large datasets. These limitations suggest several immediate improvements including sampled ideal construction, parallelized approximation operators, and dimensionality reduction techniques. Beyond computational optimization, the framework’s hybrid potential remains largely unexplored - particularly integration with fuzzy rough sets for graded uncertainty and deep learning architectures for adaptive ideal selection. Promising application domains include real-time decision systems with dynamic adjustments, biomedical data fusion requiring multi-covering approximations, and granular computing in hierarchical knowledge representation. The topological foundations further suggest connections with sheaf-theoretic approaches to distributed data analysis, opening new theoretical pathways.

As we look to the future, the integration of soft set theory with other mathematical frameworks presents a rich avenue for exploration. The potential for interdisciplinary applications, particularly in areas like fuzzy logic, artificial intelligence, and decision-making under uncertainty, remains a compelling field of inquiry. Researchers are encouraged to delve deeper into the implications of soft coverings and their properties, potentially leading to the discovery of new mathematical principles and innovative applications that can address complex real-world challenges.

In summary, this paper contributes to the burgeoning field of soft set theory by elucidating its foundational concepts, exploring its intersection with ideals, and proposing new methods for approximation. The findings underscore the versatility and robustness of soft set theory, providing a solid groundwork for future research that seeks to expand its reach and application across diverse mathematical and practical domains.

Footnotes

ORCID iDs

Sobhy El-Sheikh

Amr Zakaria

Eman Said

Alyaa Adel

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Precision in Soft Rough Sets: Covering-Based Methods Enhanced by Ideals

Abstract

Keywords

1. Introduction

2. Preliminaries

Table 2. Tabular Representation of the Soft set G 2 . u 1 u 2 u 3 u 4 u 5 u 6 E x p e n s i v e 1 0 1 1 1 1 B e a u t i f u l 1 1 1 0 0 0 C h e a p 1 0 0 0 1 1 M o d e r n 1 1 1 0 0 1 G a r d e n 1 1 1 1 0 0

Footnotes

ORCID iDs

Funding

Declaration of Conflicting Interests

References

Table 2.
Tabular Representation of the Soft set $G_{2}$ .

$u_{1}$ $u_{2}$ $u_{3}$ $u_{4}$ $u_{5}$ $u_{6}$

$E x p e n s i v e$ 1 0 1 1 1 1

$B e a u t i f u l$ 1 1 1 0 0 0

$C h e a p$ 1 0 0 0 1 1

$M o d e r n$ 1 1 1 0 0 1

$G a r d e n$ 1 1 1 1 0 0