EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6358 ISSN 1307-5543 – ejpam.com Published by New York Business Global Theoretical Advances and Improvements of “Generalized Approximation Spaces Generation from Ij-Neighborhoods and Ideals with Application to Chikungunya Disease” Rodyna A. Hosny1, Naglaa M. Madbouly2, Mostafa K. El-Bably3,4,∗ 1 Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt 2 Mathematics Department, Faculty of Science, Helwan University, Helwan 11795, Egypt 3 Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt 4 Jadara University Research Center, Jadara University, Irbid 21110, Jordan Abstract. In their article, “Generalized approximation spaces generation from Ij-neighborhoods and ideals with application to Chikungunya disease,” published in AIMS Mathematics, 9(4) (2024), 10050–10077, T. M. Al-Shami and M. Hosny introduced a novel framework for constructing gener- alized approximation spaces utilizing Ij-neighborhoods and ideals. This current paper undertakes a critical validation of the published work. Our detailed investigation indicates that several core propositions and methodological instances require careful re-examination and precision, including the formulation of key results, underlying derivations, and illustrative examples. To address these areas, we provide rigorous counterexamples that highlight where further theoretical refinement is necessary. We subsequently correct and formally refine the established findings, offer enhanced mathematical formulations for critical concepts, and present new properties and clarifications. The goal of this paper is to strengthen the theoretical foundation of these generalized approximation spaces, ensuring maximal rigor and practical applicability for future research in the field. 2020 Mathematics Subject Classifications: 03E72, 54A05, 68T30, 91B06 Key Words and Phrases: Ij-neighborhoods, IKj -neighborhoods, approximation operators, rough sets, topology ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6358 Email addresses: hrodyna@yahoo.com (R. A. Hosny), naglaa.madbouly@science.helwan.edu.eg (N. M. Madbouly), mkamel_bably@yahoo.com (M. K. El-Bably) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 2 of 24 1. Introduction • Literature Review: The theory of rough sets is a significant and effective methodology for handling uncer- tainty and ambiguity in data, as well as for analyzing and extracting knowledge from it. Introduced by Pawlak in 1982 [1, 2], this theory has served as a foundational framework for effective solutions to decision-making problems. Despite its strengths, the constraints of equivalence relation conditions central to this approach have led many researchers to propose generalizations and extensions to broaden its applications. Several avenues have been explored in this regard. For instance, Yao [3] extended the concept of an equivalence relation to a more general binary relation, introducing the notions of right and left neighborhoods derived from the concepts of after- and before- sets, which were defined in [4]. Similarly, Allam et al. [5, 6] proposed minimal right and minimal left neighborhoods to generate new approximations that generalize Pawlak’s rough set theory. In 2007, Abo Khadra et al. [7] introduced a technique to generalize Pawlak’s approxi- mation space using binary relations. This method constructs dual topologies directly from right and left neighborhoods, without modifying the data within the information system. It asserts that the class T = {A ⊆ U : ℧(s) ⊆ A, ∀s ∈ A} forms a topology on U , where U represents the universe and ℧(s) is the neighborhood of s. By defining these neighborhoods, various topologies on U can be generated. This technique was expanded in El-Bably’s 2008 Master’s thesis [8], which exam- ined additional topological structures and rough set theory generalizations. Later, Abd El-Monsef et al. [9] extended the work of [7, 8] by introducing a new space called j- neighborhood space (j-NS), constructed by defining union and intersection neighborhoods from right and minimal right neighborhoods. This development led to eight types of neigh- borhoods derived from binary relations, which have been widely used in recent studies to define neighborhoods based on these concepts (e.g., [10, 11]), as well as Ej-neighborhoods [12, 13] and initial neighborhoods [14, 15]. Yao [3] presented a special class of neighborhood systems, termed 1-neighborhood sys- tems, which combine right and left neighborhoods without studying topological structures. Kozae et al. [16] further proposed R < x > R as the intersection of minimal right and left neighborhoods to define rough sets, building on the works in [5, 6]. In a subsequent study, Atef et al. [17] introduced the j-adhesion neighborhood space, highlighting the potential of j-NS to extend concepts related to adhesion sets. Later, El-Bably et al. [18] addressed inconsistencies in the findings of Atef et al., providing cor- rected results and additional insights into j-adhesion neighborhoods. Their approach is based on the concept of ”adhesion sets” proposed by Ma in 2012 [19], which is relevant to the theory of covering rough sets. Additionally, Nawar et al. [20] introduced j-adhesion R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 3 of 24 neighborhoods in covering-based rough sets, based on the generalized covering approxi- mation space of Abd El-Monsef et al. [21]. Core neighborhood ideas from [22, 23] were also extended in [24] using techniques from [7, 9]. Furthermore, El-Bably and Al-shami [25] introduced core minimal neighborhoods through binary relations, extending Pawlak rough sets to generalized types for applications, such as in lung cancer research. On the other hand, the application of neighborhood concepts has expanded in new directions, such as advanced neighborhood-based methods used for feature selection—for example, an enhanced evolutionary feature selection method based on extended knowledge from rough approximations [26]. Studies such as An Efficient Approach to Neighborhood Learning in Big Data and Its Application to Medical Diagnosis [27, 28] and Neighborhood Combination Entropy and Multi-Scale Information Fusion-Based Multiple Correlations for Unsupervised Attribute Selection [29, 30] have also employed these methods for feature selection. In addition, NMF-based deep representation algorithms and deep learning al- gorithms based on NMF for multi-view clustering [31, 32] have further demonstrated their usefulness in this context. Furthermore, these approaches have been applied to decision- making problems [33, 34], θβ-ideal approximation spaces [35, 36], primal approximation spaces [37, 38], and medical applications [14, 39]. • Motivations and Objectives: Recently, T. M. Al-Shami and M. Hosny [40] introduced a new type of neighborhood based on the Ij-neighborhood concept [12]. By utilizing the definitions of intersection and union neighborhoods proposed by Abd El-Monsef et al. [9], they constructed eight novel neighborhoods, examined their properties and interrelationships, and presented the- oretical results supported by illustrative examples. They further developed various rough set approximations based on IKj -neighborhoods, demonstrating their potential through a medical case study. However, the objective of their study was not achieved due to several fundamental errors and a lack of precision. This paper aims to address the errors, inconsistencies, and inaccuracies identified in their work. The primary objectives of our study are as follows: (i) Identify and highlight the errors in results such as Theorem 3.4, Proposition 4.3, and Proposition 5.4. (ii) Provide counterexamples to demonstrate inaccuracies in their findings, such as Ex- amples 1, 2, 3, 4, 5, and 6. (iii) Correct the erroneous results, definitions, and properties introduced in their research. (iv) Rectify the incorrect examples and flawed proofs presented in their paper. (v) Enhance the understanding of the research topic by presenting corrected results and expanding the scope of potential applications. R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 4 of 24 (vi) Contribute to the advancement of rough set theory and methodologies through re- fined neighborhood concepts. • Organization of the Manuscript: The remainder of this paper is organized as follows: Section 2 presents essential con- cepts and foundational results necessary for understanding the manuscript. This includes corrections to inaccurate definitions, results, and citations from [40] to ensure coherence. Section 3 systematically identifies and analyzes the errors in [40], addressing inaccuracies in results, definitions, and examples, along with the corresponding corrections. Finally, Section 4 provides a summary of our findings and a concluding discussion. 2. Preliminaries This section establishes fundamental concepts and key results necessary for under- standing both the study and the manuscript’s content. A primary objective is to rectify inaccuracies in definitions and results, as well as to correct erroneous citations of key con- cepts from [40], ensuring clarity and coherence throughout the discussion. A well-established fact is that a binary relation R on a nonempty set (universe) U is a subset of U ×U . Throughout this work, we assume U to be a nonempty finite set, with R representing an arbitrary relation unless specified otherwise. For s, t ∈ U , we write sRt if (s, t) ∈ R. Several key concepts in [40] were inaccurately presented. Below, we provide the corrected definitions to ensure precision and correctness. Definition 1. [3, 4] A relation R on U is classified as follows: (i) Serial: For each s ∈ U , there exists t ∈ U such that sRt. (ii) Reflexive: For all s ∈ U , sRs. (iii) Symmetric: For all s, t ∈ U , sRt if and only if tRs. (iv) Transitive: For all s, t, p ∈ U , if sRp and pRt, then sRt. (v) Preorder (or quasi-order): R is reflexive and transitive. (vi) Equivalence relation: R is reflexive, symmetric, and transitive. In [40], the authors presented Definition 2.2; however, they omitted the essential con- cepts of intersection (and minimal intersection) neighborhoods as well as union (and min- imal union) neighborhoods, which are fundamental to their definitions and results. To rectify this, we provide the necessary definitions below. Definition 2. Let R be a binary relation on a set U , and let s ∈ U . The following types of neighborhoods, based on R, are defined: R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 5 of 24 (i) After (right) neighborhood [3, 4]: ωa(s) = {t ∈ U : sRt}. (ii) Before (left) neighborhood [3, 4]: ωb(s) = {t ∈ U : tRs}. (iii) Minimal-after neighborhood [5, 6]: ω(s) = ∩ {ωa(p) : s ∈ ωa(p)}. (iv) Minimal-before neighborhood [6]: ω(s) = ∩ {ωb(p) : s ∈ ωb(p)}. (v) Intersection neighborhood [3, 9]: ωi(s) = ωa(s) ∩ ωb(s). (vi) Union neighborhood [3, 9]: ωu(s) = ωa(s) ∪ ωb(s). (vii) Minimal-intersection neighborhood [9, 16]: ω(s) = ω(s) ∩ ω(s). (viii) Minimal-union neighborhood [9]: ω(s) = ω(s) ∪ ω(s). For simplicity, the set {a, b,< a >,< b >, i, u,< i >,< u >} will be collectively denoted by Ω. Definition 3. [3, 5, 16] Let R be a binary relation on a finite universe U , and let F ⊆ U be a nonempty subset. For each j ∈ Ω, the approximation operators based on the ωj- neighborhoods are defined as: � Lower approximation: R ωj ⋆ (F ) = {s ∈ U : ωj(s) ⊆ F}. � Upper approximation: R⋆ωj (F ) = {s ∈ U : ωj(s) ∩ F ̸= ∅}. � Boundary region: BND ⋆ωj R (F ) = R⋆ωj (F ) \Rωj ⋆ (F ). � Accuracy measure: ACC ⋆ωj R (F ) = |R ωj ⋆ (F )∩F | |R⋆ωj (F )∪F | , provided that | R⋆ωj (F ) ∪ F |̸= 0. The following definition presents an interesting application of j-NS, as defined by [17, 18]. In this definition, the ωj-neighborhoods are extended to generalized neighbor- hoods called ”adhesion-neighborhoods” (denoted as ρj-neighborhoods). These generalized neighborhoods are utilized in identifying generalized rough sets, which serve as general- izations of Pawlak’s models and their extensions. Definition 4. [17, 18, 20] Let R be a binary relation on a set U , and let s ∈ U . The ρj-neighborhood of s collects all elements that share the same ωj-neighborhood. For each j ∈ Ω, the neighborhoods are defined as follows: R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 6 of 24 (i) After ρ-neighborhood: ρa(s) = {t ∈ U : ωa(t) = ωa(s)}. (ii) Before ρ-neighborhood: ρb(s) = {t ∈ U : ωb(t) = ωb(s)}. (iii) Intersection ρ-neighborhood: ρi(s) = ρa(s) ∩ ρb(s). (iv) Union ρ-neighborhood: ρu(s) = ρa(s) ∪ ρb(s). (v) Minimal-after ρ-neighborhood: ρ(s) = {t ∈ U : ω(t) = ω(s)}. (vi) Minimal-before ρ-neighborhood: ρ(s) = {t ∈ U : ω(t) = ω(s)}. (vii) Minimal-intersection ρ-neighborhood: ρ(s) = ρ(s) ∩ ρ(s). (viii) Minimal-union ρ-neighborhood: ρ(s) = ρ(s) ∪ ρ(s). Definition 5. [17, 18, 20] Let F ⊆ U be a nonempty subset, and let j ∈ Ω. The approximation operators based on ρj-neighborhoods are defined as follows: � Lower approximation: R ρj ⋆ (F ) = {s ∈ U : ρj(s) ⊆ F}. � Upper approximation: R⋆ρj (F ) = {s ∈ U : ρj(s) ∩ F ̸= ∅}. � Boundary region: BND ⋆ρj R (F ) = R⋆ρj (F ) \Rρj ⋆ (F ). � Accuracy measure: ACC ⋆ρj R (F ) = |R ρj ⋆ (F )| |R⋆ρj (F )| , where | R⋆ρj (F ) |̸= 0. The following definition introduces a significant extension of the j-NS concept, as proposed in [12]. It generalizes the notion of ωj-neighborhoods into a broader framework known as Ij-neighborhoods, for each j ∈ Ω. These generalized neighborhoods are employed to define more flexible rough set models that extend beyond Pawlak’s classical framework and support diverse applications, such as those discussed in [40]. Definition 6. [12] The subsequent Ij-neighborhoods of an element s ∈ U , derived from a relation R, are specified as follows: (i) After I-neighborhood: Ia(s) = {t ∈ U : ωa(t) ∩ ωa(s) ̸= ∅}. (ii) Before I-neighborhood: Ib(s) = {t ∈ U : ωb(t) ∩ ωb(s) ̸= ∅}. R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 7 of 24 (iii) Intersection I-neighborhood: Ii(s) = Ia(s) ∩ Ib(s). (iv) Union I-neighborhood: Iu(s) = Ia(s) ∪ Ib(s). (v) Minimal-after I-neighborhood: I(s) = {t ∈ U : ω(t) ∩ ω(s) ̸= ∅}. (vi) Minimal-before I-neighborhood: I(s) = {t ∈ U : ω(t) ∩ ω(s) ̸= ∅}. (vii) Minimal-intersection I-neighborhood: I(s) = I(s) ∩ I(s). (viii) Minimal-union I-neighborhood: I(s) = I(s) ∪ I(s). In [12], Ij-neighborhoods were examined under the designation “Ej-neighborhoods”, j ∈ Ω. Definition 7. [12] For Ij-neighborhoods and for each j ∈ Ω, the approximation operators (both lower and upper), boundary region, and accuracy measure for a nonempty subset F of U are defined as follows: � Lower approximation: R Ij ⋆ (F ) = {s ∈ U : Ij(s) ⊆ F}. � Upper approximation: R⋆Ij (F ) = {s ∈ U : Ij(s) ∩ F ̸= ∅}. � Boundary region: BND ⋆Ij R (F ) = R⋆Ij (F ) \RIj ⋆ (F ). � Accuracy measure: ACC ⋆Ij R (F ) = |R Ij ⋆ (F )∩F | |R⋆Ij (F )∪F | , where | R⋆Ij (F ) ∪ F |̸= 0. Definition 8. [41] An ideal K on a nonempty set U is a nonempty collection of subsets of U that is closed under finite unions and contains all subsets of its elements. By using the concept of ideals, R. A. Hosny et al. [13] have succeeded in proposing a notable application of j-NS. In this context, the concept of ”Ij-neighborhoods” was used alongside the topological structure of ”ideals” to play a crucial role in the identification of generalized rough sets, which encompass and extend Pawlak’s models and their variations. Furthermore, they find applications in diverse settings. Definition 9. [13] Let R be a binary relation and K an ideal on a nonempty set U . The approximation operators (lower and upper), boundary region, and accuracy measure of a nonempty subset F of U , derived from R and K using Ij-neighborhoods, are defined as follows: R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 8 of 24 � Lower approximation: L Ij ⋆ (F ) = {s ∈ U : Ij(s) \ F ∈ K}. � Upper approximation: U⋆Ij (F ) = {s ∈ U : Ij(s) ∩ F ̸∈ K}. � Boundary region: △⋆Ij R (F ) = U⋆Ij (F ) \ LIj ⋆ (F ). � Accuracy measure: M ⋆IKj R (F ) = |L Ij ⋆ (F )∩F | |U⋆Ij (F )∪F | , where | U⋆Ij (F ) ∪ F |̸= 0. Definition 10. [40] Let R be a relation on U , and let K be an ideal on U . The IKj - neighborhoods of an element s ∈ U are specified as follows: (i) IKa (s) = {t ∈ U : ωa(t) ∩ ωa(s) ̸∈ K}. (ii) IKb (s) = {t ∈ U : ωb(t) ∩ ωb(s) ̸∈ K}. (iii) IKi (s) = IKa (s) ∩ IKb (s). (iv) IKu (s) = IKa (s) ∪ IKb (s). (v) IK(s) = {t ∈ U : ω(t) ∩ ω(s) ̸∈ K}. (vi) IK(s) = {t ∈ U : ω(t) ∩ ω(s) ̸∈ K}. (vii) IK(s) = IK(s) ∩ IK(s). (viii) IK(s) = IK(s) ∪ IK(s). Note that if the ideal is given by K = {∅}, then Definition 10 reduces to Definition 6. As a result, the work presented in [40] can be regarded as a genuine extension of the results established in [12]. Furthermore, the notion of the ”I-Generalized approximation space” (corrected and abbreviated as Ij-G approximation space) was not explicitly defined in [40], despite being referenced throughout various definitions and results. To address this gap, we formally introduce it as follows: Definition 11. Let R be a relation on U , and K be an ideal on U . The triplet (U,R,K) is named an Ij-G approximation space. R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 9 of 24 Theorem 1. [40] Let (U,R,K) be an Ij-G approximation space, and let s ∈ U . If R is reflexive, then for each j ∈ {a, b, i, u}, the following inclusion holds: IK(s) ⊆ IKj (s). Definition 12. [40] Let R be a binary relation and K an ideal on a nonempty set U . The improved operators (lower and upper), boundary region, and accuracy measure of a nonempty subset F of U , derived from R and K, are defined as follows: � Lower approximation: R IKj ⋆ (F ) = {s ∈ U : IKj (s) \ F ∈ K}. � Upper approximation: R⋆IKj (F ) = {s ∈ U : IKj (s) ∩ F ̸∈ K}. � Boundary region: BND ⋆IKj R (F ) = R⋆IKj (F ) \R IKj ⋆ (F ). � Accuracy measure: ACC ⋆IKj R (F ) = |R IKj ⋆ (F )∩F | |R⋆IK j (F )∪F | , where | R⋆IKj (F ) ∪ F |̸= 0. Remark 1. (i) According to the principles of Pawlak’s rough set theory, as established in [1], a set is considered definable (or exact) if its lower and upper approximations coincide. In particular, if the lower approximation of the empty set ∅ is equal to its upper approximation, which is also ∅, then ∅ is a definable set, and its accuracy measure is 1. Conversely, if this condition is not satisfied, ∅ is classified as an undefinable (or rough) set, meaning its accuracy measure differs from 1. Addition- ally, the definition of the accuracy measure requires that the upper approximation be nonempty to avoid division by zero. Therefore, if the upper approximation of ∅ is itself ∅, the accuracy measure in this case is considered as an indefinite quantity. (ii) Accordingly, in Definition 12, the empty set ∅ is definable if R IKj ⋆ (∅) = R⋆IKj (∅) = ∅, which implies that ACC ⋆IKj R (∅) = 1. R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 10 of 24 Otherwise, ∅ is an undefinable (rough) set, and the accuracy measure ACC ⋆IKj R (∅) is considered an indefinite quantity. Theorem 2 presents a significant result for deriving a general topology from an arbitrary neighborhood system. This method was initially proposed by Abo Khadra et al. (2007) [7] and later extended by Abd El-Monsef et al. (2014) [9]. Since then, this approach has been widely adopted by researchers to construct various topological frameworks, as demonstrated in Theorem 2. Theorem 2. Let U be a universal set and let j ∈ Ω. Then, (i) Fundamental method [7, 9]: The collection ⊤ωj = {F ⊆ U : ωj(s) ⊆ F, ∀s ∈ F} forms a topology on U . (ii) Adhesion topologies [17, 18, 20]: The collection ⊤ρj = {F ⊆ U : ρj(s) ⊆ F, ∀s ∈ F} defines a topology on U . (iii) Ij-topologies [12]: The collection ⊤Ij = {F ⊆ U : Ij(s) ⊆ F, ∀s ∈ F} constitutes a topology on U . Definition 13. [12] Let ⊤Ij be a topology on U as defined in the theorem above, for all j ∈ Ω, and let F ⊆ U . The interior and closure operators of F in the topological space (U,⊤Ij ), denoted by ⊤Ij (F ) and ⊤Ij (F ), are referred to as the ⊤Ij -lower approximation and ⊤Ij -upper approximation, respectively. Definition 14. [12] The ⊤Ij -boundary and ⊤Ij -accuracy of a subset F in a topological space (U,⊤Ij ) are given as follows: � The ⊤Ij -boundary of F is given by: BND⊤Ij (F ) = ⊤Ij (F ) \ ⊤Ij (F ) � The ⊤Ij -accuracy of F is given by: ACC⊤Ij (F ) = |⊤Ij (F )| |⊤Ij (F )| , where |⊤Ij (F )| ̸= 0. R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 11 of 24 Theorem 3. [40] Let (U,R,K) be an Ij-G approximation space, and let s ∈ U . For all j ∈ Ω, the collection τ I K j = {F ⊆ U : IKj (s) \ F ∈ K, ∀s ∈ F} forms a topology on U . If K = {∅} in Theorem 5.2 [40], then the resulting topologies coincide with those estab- lished in Theorem 2.11 of [12]. Consequently, the findings in [40] serves as a meaningful extension of the work in [12]. Theorem 4. [42] Let (U,R,K) be an ρj-G approximation space and s ∈ U . Then, for all j ∈ Ω, the collection τρ K j = {F ⊆ U : ρj(s) \ F ∈ K, ∀s ∈ F} constitutes a topology on U . 3. Main results This section constitutes the core of our research, where we critically examine the errors identified in the study by T. Al-Shami and M. Hosny [40]. We provide multiple counterex- amples to substantiate these inaccuracies and propose necessary corrections. Additionally, we refine and amend errors in the proofs of certain results while addressing inconsistencies in the examples and comparison tables presented in their study. The following theorem, originally stated as Theorem 3.4 in [40], contains errors requir- ing correction. Theorem 5. [40] Let (U,R,K) be an Ij-G approximation space. If s ∈ U , then the following properties hold: (i) IKi (s) ⊆ IKa (s) ∩ IKb (s) ⊆ IKa (s) ∪ IKb (s) ⊆ IKu (s). (ii) IK(s) ⊆ IK(s) ∩ IK(s) ⊆ IK(s) ∪ IK(s) ⊆ IK(s). (iii) If R is reflexive, then ρj(s) ∪ ωj(s) ⊆ IKj (s) for all j ∈ Ω. (iv) If R is serial, then ρj(s) ⊆ IKj (s) for all j ∈ Ω. (v) If R is transitive, then IKj (s) ⊆ IK(s) for each j ∈ {a, b, i, u}. Remark 2. Firstly, statements (1) and (2) are incorrect, as the equalities cannot be replaced by subset symbols according to Definition 2. The following examples demonstrate errors in the remaining statements of the preceding theorem. Example 1. Consider U = {p, q, s, t} and the reflexive relation R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 12 of 24 R = {(p, s), (p, t), (q, t), (t, q), (p, p), (q, q), (s, s), (t, t)} on U . The associated neighborhoods for the element t are given as follows: � ωa(t) = {q, t}, ωb(t) = {p, q, t}, ωi(t) = {q, t}, ωu(t) = {p, q, t}, � ω(t) = {t}, ω(t) = {q, t}, ω(t) = {t}, ω(t) = {q, t}, � ρa(t) = {q, t}, ρb(t) = {t}, ρi(t) = {t}, ρu(t) = {q, t}, � ρ(t) = {t}, ρ(t) = {q, t}, ρ(t) = {t}, ρ(t) = {q, t}. Now, let K = {∅, {t}}. The corresponding IKj -neighborhoods are: � IKa (t) = {q, t}, IKb (t) = U , IKi (t) = {q, t}, IKu (t) = U . � IK(t) = ∅, IK(t) = {q, t}. IK(t) = ∅, IK(t) = {q, t}. It is evident that for j ∈ {< a >,< i >}, neither ρj(t) ⊆ IKj (t) nor ωj(t) ⊆ IKj (t), thereby demonstrating a counterexample to the corresponding claims. Example 2. Consider the set U = {p, q, s, t} with the serial relation R = {(p, p), (s, p), (t, p), (q, t), (t, q), (t, t)}. The corresponding neighborhoods under ωa, ρa, Ia are given as follows: � ωa(p) = {p}, ωa(q) = {t}, ωa(s) = {p}, ωa(t) = {p, q, t}. � ρa(p) = {p, s}, ρa(q) = {q}, ρa(s) = {p, s}, ρa(t) = {t}. � Ia(p) = {p, s, t}, Ia(q) = {q, t}, Ia(s) = {p, s, t}, Ia(t) = U . Now, let K = {∅, {t}}, then the IKa -neighborhoods are: � IKa (p) = {p, s, t}, IKa (q) = ∅, IKa (s) = {p, s, t}, IKa (t) = {p, s, t}. It is evident that ρa(q) ⊈ IKa (q), thereby demonstrating a counterexample to the claim. Example 3. Consider U = {p, q, s, t} with the transitive relation R = {(p, s), (p, t), (p, q), (t, q), (t, t)}. The corresponding neighborhoods under ωb, ω, ρb, ρ are as follows: � ωb(p) = ∅, ωb(q) = {p, t}, ωb(s) = {p}, ωb(t) = {p, t}. � ω(p) = {p}, ω(q) = ∅, ω(s) = ∅, ω(t) = {p, t}. � ρb(p) = {p}, ρb(q) = {q, t}, ρb(s) = {s}, ρb(t) = {q, t}. � ρ(p) = {p}, ρ(q) = {q, s}, ρ(s) = {q, s}, ρ(t) = {t}. R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 13 of 24 Now, let K = {∅, {t}}. The corresponding IKb - and IK-neighborhoods are: � IKb (p) = ∅, IKb (q) = {q, s, t}, IKb (s) = {q, s, t}, IKb (t) = {q, s, t}. � IK(p) = {p, t}, IK(q) = ∅, IK(s) = ∅, IK(t) = {p, t}. Consequently, IKj (x) ̸= IK(x), ∀x ∈ U . This demonstrates a discrepancy between IKj and IK contradicting any assumption of their equivalence. Example 4. Consider U = {p, q, s} equipped with the preorder relation R = {(p, p), (q, q), (s, s), (p, q), (p, s), (q, s)}. The corresponding neighborhoods under ωa, ρa are given by: � ωa(p) = U , ωa(q) = {q, s}, ωa(s) = {s}. � ρa(p) = {p}, ρa(q) = {q}, and ρa(s) = {s}. Now, let K = {∅, {s}}. The corresponding neighborhoods under IKa are given by: � IKa (p) = {p, q}, IKa (q) = {p, q}, IKa (s) = ∅. It follows that ωa(x) ⊈ IKa (x), for all x ∈ U . Additionally, ρa(s) ⊈ IKa (s). Remark 3. (i) For all j ∈ Ω, and x ∈ U , the inclusion ρj(x) ⊆ IKj (x) does not necessarily hold when R is serial, reflexive, symmetric, transitive, a preorder, or a similarity relation, as demonstrated in Examples 1, 2, 3, and 4. (ii) For all j ∈ Ω and x, y ∈ U , the condition ωj(x) ∩ ωj(y) ̸= ∅ does not imply that ωj(x) ∩ ωj(y) ̸∈ K, as shown in Examples 1, 2, 3, and 4. (iii) For all j ∈ Ω and x, y ∈ U , the intersection IKj (x) ∩ IKj(y) ̸= ∅ does not necessarily imply that IKj (x) ∩ IKj (y) ̸∈ K, as evidenced in Examples 1, 2, 3, and 4. Accordingly, the revised formulation of Theorem 3.4 from [40] is stated as follows: Theorem 6. Let (U,R,K) be an Ij-G approximation space, and let s, t ∈ U . The following statements are valid: (i) For all j ∈ Ω, t ∈ IKj (s) if and only if s ∈ IKj (t). (ii) If R is reflexive, then IK(s) ⊆ IKj (s), for each j ∈ {a, b, i, u}. (iii) If R is preorder, then IK(s) = IKj (s), for each j ∈ {a, b, i, u}. (iv) If R is reflexive, then for all j ∈ Ω, the following equalities hold: R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 14 of 24 ρj(s) ∩ ωj(s) = ρj(s) and ρj(s) ∪ ωj(s) = ωj(s). (v) If R is symmetric, then IKi (s) = IKa (s) = IKb (s) = IKu (s) and IK(s) = IK(s) = IK(s) = IK(s). Proof. The proofs of statements 1, 2, and 5 can be found in [40]. We now proceed to establish the remaining assertions that are not explicitly stated in [40], as follows: 3. According to [43], since R is a preorder, it follows that ω(s) = ωj(s) for all j ∈ Ω. Therefore, by applying statements 1 and 2, we conclude that IK(s) = IKj (s). 4. As noted in [18], the reflexivity of R ensures that ρj(s) ⊆ ωj(s) for all j ∈ Ω. Thus, it follows that ρj(s) ∩ ωj(s) = ρj(s) and ρj(s) ∪ ωj(s) = ωj(s), for all j ∈ Ω. As noted in Remark 3, the proof of item (8) in Theorem 3.4 from [40] contains in- accuracies. To address this, we present the following lemma, accompanied by a detailed proof—absent in [40]—which serves both as a correction to item (8) of Theorem 3.4 in [40] and as a supporting result for the proof of Theorem 7: Lemma 1. Let R be a symmetric and transitive relation on U . Then, for all s, t ∈ U and each j ∈ Ω, ωj(s) ∩ ωj(t) ̸= ∅ if and only if ωj(s) = ωj(t). Proof. We prove the case j = a; the argument for other values of j is identical. Assume ωa(s) ∩ ωa(t) ̸= ∅, and let q ∈ ωa(s) ∩ ωa(t). It suffices to show that ωa(s) = ωa(q) and ωa(t) = ωa(q). (i) ωa(s) = ωa(q). Since q ∈ ωa(s), we have sRq, (1) qRs (by the symmetry of R). (2) If x ∈ ωa(q), then qRx, and by (1) and the transitivity of R, it follows that sRx, so x ∈ ωa(s). Hence, ωa(q) ⊆ ωa(s). R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 15 of 24 Conversely, if y ∈ ωa(s), then sRy, and by (2) and transitivity, qRy, so y ∈ ωa(q). Thus, ωa(s) ⊆ ωa(q), and therefore ωa(s) = ωa(q). (ii) ωa(t) = ωa(q). The same reasoning applies, since q ∈ ωa(t) implies tRq and qRt, and the transitivity argument shows that ωa(t) = ωa(q). (iii) Converse Direction. On the other hand, it is easy to see, using the same reasoning, that if ωj(s) = ωj(t), then ωj(s) ∩ ωj(t) ̸= ∅. Now, the proof of item (8) in Theorem 3.4 from [40] contains errors. Therefore, we provide a revised and corrected version as follows: Theorem 7. [40] Let R be a symmetric and transitive relation on U , and let K be an ideal on U . For any s ∈ U and j ∈ Ω, the following hold: (i) IKj (s) ⊆ ωj(s). (ii) If s ∈ IKj (t) then IKj (s) ⊆ IKj (t). Proof. We again treat the case j = a. 1. Let p ∈ IKa (s). By definition, ωa(p) ∩ ωa(s) ̸∈ K, so in particular ωa(p) ∩ ωa(s) ̸= ∅. Hence there exists q ∈ ωa(p) ∩ ωa(s) with pRq and sRq. Symmetry and transitivity of R imply sRp, i.e. p ∈ ωa(s). Thus IKa (s) ⊆ ωa(s). 2. Suppose s ∈ IKa (t), so ωa(s) ∩ ωa(t) ̸∈ K =⇒ ωa(s) ∩ ωa(t) ̸= ∅. By Lemma 1, ωa(s) = ωa(t) and hence ωa(s) ̸∈ K, ωa(t) ̸∈ K. Now let p ∈ IKa (s), so ωa(p) ∩ ωa(s) ̸∈ K =⇒ ωa(p) ∩ ωa(s) ̸= ∅. Again by Lemma 1, ωa(p) = ωa(s). Therefore ωa(p) ∩ ωa(t) = ωa(s) ̸∈ K, and hence p ∈ IKa (t), completing the proof. R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 16 of 24 The converse of Theorem 7 (1) does not hold in general, as demonstrated by the following example. Example 5. Consider the set U = {p, q, s, t} with the relation R = {(t, t)}, which is both symmetric and transitive. Let K = {∅, {t}} be an ideal on U . Then, we obtain ωa(t) = {t} and IKa (t) = ∅. Consequently, it follows that ωa(t) ⊈ IKa (t). Table 1 in [40] introduces various kinds of neighborhoods, including ωj-neighborhoods, ρj-neighborhoods, Ij-neighborhoods, and IKj -neighborhoods, for each j ∈ Ω. In this work, we address and correct several errors in Table 1, that create ambiguity in the intended meaning. Specifically, while a neighborhood of a point in the universe can be equal to the empty set, it should not be represented as {∅}. Furthermore, Table 2 in [40] provides a comparative analysis of the lower and up- per operators, as well as accuracy measure results for any subset of U , with respect to {a, b, i, u}. According to Remark 1, we identify and clarify several observations and nec- essary corrections related to this table. (i) Since R IKa ⋆ (∅) = {q, s, t}, it follows that R⋆IKa (∅) = ∅. Consequently, ACC ⋆IKa R (∅) is not zero but rather an indefinite quantity. By the same manner, ACC ⋆IKb R (∅), ACC ⋆IKi R (∅), and ACC ⋆IKu R ({p}) should also be considered indefinite quantities. (ii) Since R IKu ⋆ (∅) = R⋆IKu (∅) = ∅, it follows that ACC ⋆IKu R (∅) should be 1, not zero. (iii) Given that R IKb ⋆ (∅) = {q}, which is not equal to ∅, it follows that ACC ⋆IKb R (∅) is also an indefinite quantity. (iv) ACC ⋆IKi R (∅) = 1. (v) Since R IKu ⋆ ({p}) = {q}, this set is nonempty. Item (3) of Proposition 4.3 in [40], which asserts that R⋆IKa (U) ⊇ U , is incorrect. As illustrated in Table 2 of [40], it is observed that R⋆IKa (U) = {p}, which does not satisfy R⋆IKa (U) ⊇ U . Consequently, several observations and necessary corrections have been recorded based on this table. Remark 4.4 in [40] further asserts that the inclusion relations in Proposition 4.3 (3) are generally proper. However, this claim is also inaccurate, as the inclusion R⋆IKa (U) ⊆ U does not hold in this context. According to Remark 1, Table 3 in [40] provides a comparative analysis between the results derived from the proposed Definition 4.1 in [40] and those obtained from Definition 2.7 in [12] for the case j = a. This comparison reveals several observations and necessary corrections, as outlined below: R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 17 of 24 (i) Given that RIKa ⋆ (∅) = {q, s, t}, R⋆IKa (∅) = ∅, it follows that ACC ⋆IKa R (∅) is not zero but an indefinite quantity. (ii) ACC⋆Ia R (∅) should be 1, not zero. According to Remark 1, Table 4 in [40] presents a comparative analysis of results derived from the proposed Definition 4.1 from [40], alongside those of Definition 2 from [3], Definition 3 from [5] and Definitions 4.3, 4.5 in [17] for the case j = b. Several omissions and errors were identified in this comparison, as outlined below. (i) Since R IKb ⋆ (∅) = {q}, which is not equal to ∅, it follows that ∅ is rough, implying that ACC ⋆IKb R (∅) is undefined. (ii) Rwb ⋆ (∅) = ∅, rather than {q}. Thus, Rwb ⋆ (∅) = R⋆ωb(∅) = ∅, indicating that ∅ is exact and hence ACC⋆wb R (∅) = 1. (iii) Rwb ⋆ ({p}) = {p, s}, which differs from {p, q, s}. (iv) Rwb ⋆ ({q}) = ∅, rather than {q}, and consequently, ACC⋆wb R ({q}) = 0. (v) Rwb ⋆ ({s}) = ∅, differing from {q}. (vi) Rwb ⋆ ({p, q}) = {p, s}, which is not equal to U . Also, ACC⋆ωb R ({p, q}) = 1 4 . (vii) Rwb ⋆ ({p, s}) = {p, s}, differing from {p, q, s}. (viii) Rwb ⋆ ({q, s}) = ∅, which does not equal to {q}. Furthermore, ACC⋆wb R ({q, s}) = 0. (ix) Rwb ⋆ ({p, q, s}) = {p, s}, differing from U . Moreover, ACC⋆wb R ({p, q, s}) = 1 2 . (x) Rρb ⋆ (∅) = ∅, differing from {t}. Therefore, Rρb ⋆ (∅) = R⋆ρb(∅) = ∅, implying that ∅ is an exact set, and hence ACC⋆ρb R (∅) = 1. (xi) R⋆ρb(U) = U , rather than {p, q, s}. (xii) Rρb ⋆ ({p}) = ∅, instead of {t}, while R⋆ρb({p}) = {p, s}, differing from {p, q}. (xiii) Rρb ⋆ ({q}) = {q}, not {q, t}. (xiv) Rρb ⋆ ({s}) = ∅, rather than {t}. (xv) R⋆ρb({t}) = {t}, not ∅. (xvi) Rρb ⋆ ({p, q}) = {q}, rather than {q, t}. (xvii) Rρb ⋆ ({p, s}) = {p, s}, differing from {p, s, t}. (xviii) Rρb ⋆ ({p, t}) = {t}, rather than {p, s, t}, while R⋆ρb({p, t}) = {p, s, t}, differing from {p, s}. In addition, ACC⋆ρb R ({p, t}) = 1 3 . R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 18 of 24 (xix) Rρb ⋆ ({q, s}) = {q}, which does not equal to {q, t}. (xx) R⋆ρb({q, t}) = {q, t}, rather than {q}. (xxi) R⋆ρb({s, t}) = {p, s, t}, not {p, s}. (xxii) Rρb ⋆ ({p, q, s}) = {p, q, s}, differing from U . (xxiii) R⋆ρb({p, q, t}) = U , instead of {p, q}. Additionally, ACC⋆ρb R ({p, q, t}) = 1 2 . (xxiv) Rρb ⋆ ({p, s, t}) = {p, s, t}, rather than {p, s}, and R⋆ρb({p, s, t}) = {p, s, t}, differing from {p, s}. Moreover, ACC⋆ρb R ({p, s, t}) = 1. (xxv) R⋆ρb({q, s, t}) = U , differing from {p, q}. Remark 4. In the proof of Theorem 5.1 in [40], the symbol I should be modified into the symbol K. Items (5) and (8) of Proposition 5.4 in [40] are not generally valid, as evidenced by the following example. Example 6. Consider the reflexive relation R = {(p, p), (p, s), (p, t), (q, t), (q, q), (s, s), (t, q), (t, t)} on U = {p, q, s, t}. If the ideal is given by K = {∅, {s}, {t}, {s, t}}, then the associated topologies are as follows: τ I K = 2U (the power set of U), τ I K a = {{p}, {q}, {s}, {p, q}, {p, s}, {q, s}, {q, t}, {p, q, s}, {p, q, t}, {q, s, t}, ∅, U}. Thus, we observe that τ IK ⊈ τ I K a . The next proposition presents a revised formulation to rectify item (5) of Proposition 5.4 in [40]: Proposition 1. Let (U,R,K) be an Ij-G approximation space, and let s ∈ U . If R is reflexive, then for each j ∈ {a, b, i, u}, the inclusion τ I K j ⊆ τ I K holds. Proof. Let F ∈ τ I K j and suppose s ∈ F . By definition, this implies that IKj (s) \F ∈ K. Since R is reflexive, it follows from Theorem 1 that IK(s) ⊆ IKj (s). Therefore, we obtain IK(s) \ F ∈ K, which implies that F ∈ τ I K . Thus, the desired inclusion holds. Remark 5. If R is reflexive, then for every x ∈ U and for all j ∈ Ω, the inclusion ρj(x) ⊆ IKj (x) does not hold in general. As a result, it follows that τρ K j ⊈ τ I K j ∀j ∈ Ω. R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 19 of 24 According to Remark 1, Table 5 in [40] presents a comparative analysis of the boundary region and accuracy measure results for a set F based on Definition 5.7 in [40], where j ∈ {a, b, i, u}. The values of ACCτ I K a R (∅) (and similarly ACCτ I K b R (∅), ACCτ I K i R (∅), andACCτ I K u R (∅)) should be 1, rather than 0. Besides, the indices j = r, j = l have been revised to j = a, j = b respectively. Table 6 in [40] presents a comparison of the boundary region and accuracy measure results for a set F based on the proposed Definition 5.7 and Definition 2.13 from [12] for j = a. The notation τ Ib(L) (and similarly τ Ib(L), BNDτ Ib (L), ACCτ Ib (L)) should be cor- rected to τ Ia(L) (and similarly τ Ia(L), BNDτ Ia (L), ACCτ Ia (L)). Additionally, the values of ACCτ I K a R (∅), and ACCτ Ia R (∅) should be considered indefinite. 4. Conclusion and Discussion This paper presents a systematic critique of the work by T. M. Al-Shami and M. Hosny [40], identifying and resolving fundamental flaws in their formalization of IKj - neighborhoods within rough set theory. Through rigorous analysis, counterexamples, and reconstructed proofs, we have thoroughly examined and rectified critical inaccuracies in their definitions, theoretical results, and illustrative examples. The proposed corrections provide dual benefits: they strengthen the theoretical foundations of neighborhood sys- tems while enhancing their practical applicability within rough set–based frameworks. Our primary contribution lies in dismantling the problematic claims of the original work through comprehensive error identification and replacement with validated and consis- tent alternatives. The revised framework substantially improves the internal coherence and reliability of neighborhood-based approximation methods, particularly through cor- rected examples that now accurately support the theoretical propositions. Moreover, the refined methodology paves the way for new research directions in uncertainty quantifica- tion, offering researchers more robust analytical tools for complex data interpretation, as demonstrated through improved medical application scenarios. In this study, several major errors have been identified in Theorem 3.4, Proposition 4.3, and Proposition 5.4 of [40]. These inaccuracies were demonstrated through various examples, including Examples 1, 2, 3, 4, 5, and 6. Corrections and clarifications were subsequently provided, such as the revised proof of Theorem 3.4, which is now accurately reformulated and presented in Theorems 6 and 7 of this paper. Additional corrected versions of erroneous results, including Theorem 6 and Proposition 1, have also been proposed. Errors within several concepts, definitions, examples, and tables from [40] have been thoroughly identified and corrected. These include revisions to Definitions 1, 2, 11, and 12, as well as major modifications to five tables and the resolution of twenty-five distinct inconsistencies across multiple examples. Furthermore, new theoretical results and properties, including Remark 3 and Lemma 1, have been introduced and justified to reinforce the consistency of the proposed framework. R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 20 of 24 Overall, the findings of this study highlight the necessity of critically examining re- cently proposed methodologies and results in evolving fields such as rough set theory. The corrections and refinements provided herein ensure that the theoretical basis of neigh- borhood systems remains sound, rigorous, and dependable for subsequent research and applications. Moreover, the enhanced understanding of neighborhood-based rough ap- proximations broadens their applicability in decision-making, data analysis, and knowl- edge discovery across diverse scientific and engineering domains. In our future work, we will apply the techniques introduced in the present paper to various fields, such as applications of fuzzy topological spaces [44], rough lattices [45], the study of SC-compactness of rough sets using ideal neighborhoods [46], and the improve- ment of rough approximation and ideal methods introduced in [47]. Finally, to facilitate easy reference, we provide a nomenclature section listing the sym- bols along with their abbreviations. List of symbols and abbreviations U A nonempty universe set. ∅ The empty set. K The ideal. j-NS j-Neighborhood Space, for each j ∈ Ω, where Ω = {a, b,< a >,< b >, i, u,< i >,< u >}. ωj(s) The j-neighborhood of a point s, for each j ∈ Ω. ρj(s) The ρj-neighborhood of a point s, for each j ∈ Ω. Ij(s) The Ij-neighborhood of a point s, for each j ∈ Ω. IKj (s) The IKj -neighborhood of a point s, for each j ∈ Ω. Ij-G approximation space Ij-Generalized approximation space. ⊤hj hj-topology generated by hj-neighborhood, where h ∈ {ω, ρ, I, IK}. R hj ⋆ (F ) The lower approximation of a subset F ⊆ U , where h ∈ {ω, ρ, I, IK}. R⋆hj (F ) The upper approximation of a subset F ⊆ U , where h ∈ {ω, ρ, I, IK}. BND ⋆hj R (F ) The boundary region of a subset F ⊆ U , where h ∈ {ω, ρ, I, IK}. ACC ⋆hj R (F ) The accuracy measure of a subset F ⊆ U , where h ∈ {ω, ρ, I, IK}. Acknowledgements The authors would like to extend their deepest gratitude to the Editor-in-Chief, the Section Editor, and the reviewers for their insightful comments and suggestions, which were instrumental in refining and improving the quality of this work. Special thanks are also due to our esteemed colleagues at the Egyptian School of Topology (in Tanta, Egypt), established by Prof. A. S. Mashhour and Prof. M. E. Abd El-Monsef, and led by Prof. A. M. Kozae. This institution has fostered numerous distinguished researchers both within Egypt and internationally, including scholars from Yemen, Palestine, Jordan, and Saudi Arabia. The authors are particularly thankful for the school’s unwavering support, R. A. Hosny et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6358 21 of 24 encouragement, and the enriching discussions that contributed to this research. Finally, this work is dedicated to the memory of Prof. A. S. Mashhour, and Prof. M. E. Abd El-Monsef — may God’s mercy be upon them. We also dedicate this work to the mem- ory of our dear friend and long-time collaborator in fruitful scientific research, Prof. R. Abu-Gdairi, who passed away in June 2025. Use of AI Tools Declaration: The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of the content of this article. However, AI tools have been utilized solely for grammar and spelling correction support. All the information and data in this paper have been ethically reviewed, edited, and studied. No AI-generated data or results are included. Availability of data and material Not applicable. Competing interests The authors declare that they have no competing interests. Funding There is no funding available. References [1] Z. Pawlak. Rough sets. International Journal of Information and Computer Sciences, 11:341–356, 1982. [2] Z. Pawlak. Rough sets: Theoretical aspects of reasoning about data. Springer, Dor- drecht, 1991. [3] Y. Y. Yao. 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