EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7160 ISSN 1307-5543 – ejpam.com Published by New York Business Global Exploring the Applications of Grill Rough Topological Structures Generated by Different Minimal Neighborhood Types A. A. Azzam1,∗, B. Alreshidi1, M. Aldawood1 1 Mathematics Department, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia Abstract. A variety of grill-based topologies are developed and contrasted with earlier topologies. The results demonstrate that the present ones exceed their predecessors. This study distinguishes itself by highlighting the advantages of certain topologies and identifying both the minimum and maximum values. These structural topologies are later utilized to conduct a more thorough inves- tigation of extended rough sets. Compared to earlier models, the suggested approximate models reduce vagueness and uncertainty, which makes them especially important when applied to rough sets (rhss). Furthermore, the suggested models differ from their predecessors in that they exhibit all of Pawlak’s properties, including the capability of contrasting various approximations (Aprs), and have the quality of monotonicity across all relations. In addition, the importance of new discoveries was highlighted by demonstrating their use for human health. Besides examining its limitations, the benefits of the chosen technique were assessed. The paper ends with a summary of the main ideas of the proposed methodology and recommendations for future research paths. 2020 Mathematics Subject Classifications: 54A05, 54C55, 54D80 Key Words and Phrases: Upper-lower Apr, grill, minimal neighborhood, (rhss) 1. Introduction In the 1980s, Pawlak [1, 2] created rough set theory (rhst) to deal with uncertainty in medical and technical data processing, as well as other domains. It is very beneficial for assessing inadequate or confusing information systems, as well as categorizing data. According to the facts at hand, each class comprises a collection of items that are com- parable to one another. To deal with uncertainty and confusion, the theory makes use of two main (Aprs). In situations where exact limits are unclear, this distinction aids in data management and analysis. Through (Aprs) of groups that cannot be properly specified, it also offers a formal method of handling ambiguous or incomplete information. rhst ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7160 Email addresses: azzam0911@yahoo.com (A. A. Azzam), b.alreshidi@psau.edu.sa (B. Alreshidi), m.aldawood@psau.edu.sa (M. Aldawood) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 2 of 22 has been expanded in a number of ways to handle a greater variety of uncertainty and complexity in data processing [3-6]. Although the equivalence relation (eqr) is the main emphasis of classic rhst, its generalizations seek to improve and broaden this methodology to handle more complicated kinds of data and do away with the equivalency requirement. rhst can now handle a wider variety of complicated and varied datasets thanks to these generalizations. This also allows it to be used for a broader range of issues in domains such as data analysis, machine learning, decision support systems, smart cities, and medicine, among others. Practitioners as well as researchers can obtain more precise and significant insights from data with different levels of uncertainty by broadening the use of rhst, which also makes decision-making easier. Topology was used to achieve one of these generaliza- tions. The connection between rough sets (rhss) and topology was initially identified by [7, 8]. Here, the closure is linked to the upper (up) approximation (Apr), whereas the topological notion of the interior is linked to the lower (lw) Apr [9–14]. A grill is a nonempty collection of closed sets with hereditary property and limited additivity [15]. This idea was first put forth by Choquet [15]. The grill and ideas, nets, and filters have a number of similarities. Many theories and facets have been covered by [16, 17] and [18]. It facilitates the enhancement of the topological framework, which substitutes numerical values for evaluating attributes such as affection, intellect, aesthetic appeal, and academic achievement. Furthermore, by employing the notion of grill modifications in the border region (Br), up, and lw-Aprs, it expands the topological structure and creates novel lim- its in nano topological spaces [17]. When it comes to removing ambiguity from raw sets, grills are proven to be helpful [19, 20]. Therefore, the introduction of novel grill-based rhs approaches is one of the main reasons for this effort. In other words, when the grill is the universal set, it generates a unique scenario for its generic rhs model counterparts. As a result, some interesting research looked into the rhst that grills describe [21]. Neighbor- hoods (nbs) are a fundamental topological notion to understand and evaluate set Aprs. nbs and any relation was used to generate Apr spaces in [22, 23]. Neighborhood (nb) systems are used to generalize rhst by describing Aprs through a nb rather than an equivalency class. Some of the nb types used to define the lw and up-Apr were union, intersection [24, 25], equal nbs [26, 27], minimal and maximal nbs [28, 29], cardinality nbs [30, 31], right and left nbs [22, 23], rough nb ideal [32], and nbs with minimal left(MNL) and minimal right(MNR) [33, 34]. In the interim, Abo-Tabl [35] created the apps using MNR nbs, which are created using reflexive relations (rfr) that form the basis of topological space. Three more Apr categories were created more recently by Dai et al. [36] via maximal right nbs determined by similarity relations (sir). Stated differently, they offer a broad framework that is unrestricted in terms of the types of binary relations (br) that can be established. Interestingly, it rhst has proven to be a valuable tool for describing information content in a wide range of frameworks and applications in a wide range of domains [37-41]. The topological properties rhss were studied in [42]. This led to the combination of topological and rhs theories, which became the focus of a number of researchs [43-47]. Furthermore, topological generalizations such as minimal structures [48], supra topology [49], infra topology [50], nano-topology [17], and bitopology [51] were engaged in this relationship, and rh Apr spaces using grills A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 3 of 22 and maximal rh-nbs [18]. This study investigates the purpose behind the specialized expansion utilizing grills by distinct forms of minimal nbs to reduce the boundary regions. This work focuses on building different topologies using grills and highlights the linkages between these topologies and rhss, acknowledging the crucial role grills play in influencing topological rhs difficulties. The six sections that make up this article are presented as follows: A list of important definitions is provided in Section 2. The many topologies that grills can create are examined in Section 3. Unlike previous methods [36, 52], it presents comparisons among diverse topologies and determines the smallest and largest, whereas other methods just compare topologies within distinct sets. Conditions for determining equivalencies between these topologies are established in the section’s conclusion. Section 4 describes the attributes of the new apps, which are examined using the recommended topologies. In contrast to the previous ones, they have the property of monotonicity and satisfy all of Pawlak’s properties without limitations [53, 54]. In Section 5, a medical use is also suggested. Consequently, these techniques make it simple and very accurate for medical professionals to diagnose heart failure. The usefulness and effectiveness of the suggested models are demonstrated, highlighting the crucial part grills play in decision- making. Consequently, these techniques enable physicians to make a quick and accurate diagnosis of heart failure. The discussion is in Section 6, and the conclusion section marks the end of this study. 2. Preliminaries Definition 1. [25, 37, 38, 39] Assume that π1 ∈ Π and let × be any (bir) on a finite set Π ̸= ϕ. The subsequent terms are elucidated: (1) Nr(π1) = {π2 ∈ Π : π1×π2}. (2) Nl(π1) = {π2 ∈ Π : π2×π1}. (3) N⟨r⟩(π1) = { ⊓π1∈Nr(π2)Nr(π2) : ∃ Nr(π2) containing π1, ϕ : otherwise. (4) N⟨l⟩(π1) = { ⊓π1∈Nl(π2)Nl(π2) : ∃ Nl(π2) containing π1, ϕ : otherwise. . (5) Ni(π1) = Nr(π1) ⊓Nl(π1). (6) Nu(π1) = Nr(π1) ⊔Nl(π1). A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 4 of 22 (7) N⟨i⟩(π1) = N⟨r⟩(π1) ⊓N⟨l⟩(π1). (8) N⟨u⟩(π1) = N⟨r⟩(π1) ⊔N⟨l⟩(π1). Definition 2. [25] The triple (Π,×, ζι) is referred to as a ι-nb space (shortened to ι-NS). ι ∈ {r, ⟨r⟩, l, ⟨l⟩, i, ⟨i⟩, u, ⟨u⟩}, and ζι is a map from Π to 2Π which links each member of Π to its ι-NS. Theorem 1. [16] For a ι-NS (Π,×, ζι), and L be a grill. The family τLNι = {C ⊑ Π : ∀π1 ∈ C,Nι(π1) ⊓ Cc /∈ L} a NL ι topology on Π where Cc represents C’s complementary set. Definition 3. [55] Consider (Π,×, ζι) is a ι-NS, and L be a grill on Π. The L-Nι-up app ×L Nι , and L-Nι-lo app ×L Nι of C ⊑ Π are ×L Nι (C) = ⊓{Y : Y c ∈ τLNι : C ⊑ Y } = CLL Nι (C). ×L Nι (C) = ⊔{G ∈ τLNι : G ⊑ C} = IntLNι (C). Definition 4. [56] Let the relation × be arbitrary (arr) on a universe Π. The maximal right nb of an π1 ∈ Π is delineated as such. Mr(π1) = ⊔π1∈Nr(π2)Nr(π2) Definition 5. [55] Let (Π,×, ζι) be a ι-NS and L be a grill on Π. The NL∗ ι lower and ×L∗ ι -up apps of the set C delineated as such.. ×L∗ Mι (C) = {π1 ∈ Π : Mι(π1) ⊓ Cc /∈ L}. ×L∗ Mι (C) = {π1 ∈ Π : Mι(π1) ⊓ C ∈ L}. Definition 6. [55] Let (Π,×, ζι) be a ι-NS and L be a grill on Π. The NL∗∗ ι lo and ×L∗∗ ι -up apps of the set C is delineated as such. ×L∗∗ Mι (C) = {π1 ∈ Π : Mι(π1) ⊓ Cc /∈ L}. ×L∗∗ Mι (C) = C ⊔ ×L∗ Mι (C). Definition 7. [55] Let (Π,×, ζι) be a ι-NS and L be a grill on Π. The NL∗∗∗ ι lo and ×L∗∗∗ ι -up apps of the set C articulated as follows. ×L∗∗∗ Mι (C) = ⊔{Mι(π1) : Mι(π1) ⊓ Cc ∈ L}. ×L∗∗∗ Mι (C) = [×L∗∗∗ Mι (Cc)]c. A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 5 of 22 Definition 8. [52] Assume that π1 ∈ Π and let × be a br on a finite set Π ̸= ϕ. The minimal nb(MN ) of π1 ∈ Π are as follows: (i) MN r(π1) = ⊓π2{Nr(π2) : π2×π1}. (2) MN l(π1) = ⊓π2{Nl(π2) : π1×π2}. (3) MN u(π1) = MN r(π1) ⊔MN l(π1). (4) MN i(π1) = MN r(π1) ⊓MN l(π1). (5) MN ⟨r⟩(π1) = ⊓π1∈MN r(π2)MN r(π2). (6) MN ⟨l⟩(π1) = ⊓π1∈MN l(π2)MN l(π2). (7) MN ⟨i⟩(π1) = MN ⟨r⟩(π1) ⊓MN ⟨l⟩(π1). (8) MN ⟨u⟩(π1) = MN ⟨r⟩(π1) ⊔MN ⟨l⟩(π1). Definition 9. [52] Let MN ι(π1) be a MN system with br ×, where π1 ∈ Π, and ι ∈ {r, l, u, i}. Then, (Π,×,MN ι) is an app space (shortly, MN ι-app space). Lemma 1. [52] If π2 ∈ MN ι(π1), then MN ι(π2) ⊑ MN ι(π1), where π1, π2 ∈ Π, and ι ∈ {r, l, i}. Lemma 2. [52] Let × be a symmetric relation (sr) and π2 ∈ MN u(π1). Then, MN u(π2) ⊑ MN u(π1), for all π1, π2 ∈ Π. Proposition 1. [52] If × is reflexive relation(rr) and π2 ∈ Π, then (1) MN r(π2) ⊑ Nr(π2) (2) MN l(π2) ⊑ Nl(π2) (3) MN i(π2) ⊑ Nr(π2) (4) MN i(π2) ⊑ Nl(π2) (5) MN u(π2) ⊑ Nr(π2) ⊔Nl(π2) Definition 10. [52] In (Π,×,MN ι) with C ⊑ Π, then MN ι-lower and MN ι-up apps of C are defined by A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 6 of 22 ×MN ι (C) = {π1 ∈ Π : MN ι(π1) ⊑ C}. ×MN ι(C) = {π1 ∈ Π : MN ι(π1) ⊓ C ̸= ϕ}. Theorem 2. [52] If (Π,×,MN ι) is a MN ι-app space and × is a br, the families τMN ι = {C ⊑ Π : MN ι(π1) ⊑ C, π1 ∈ C} are topology on Π, ∀ι ∈ {r, l, u, i, ⟨r⟩, ⟨l⟩, ⟨u⟩, ⟨i⟩}. Definition 11. [52] Let (Π,×,MN ι) be a MN ι-app space, the MN ι-lo app, MN ι-up app, MN ι-accuracy, and MN ι-boundary of C are as follows: ×MN ι (C) = ⊔{G ∈ τMN ι : G ⊑ C} = IntMN ι(C), ×MN ι(C) = ⊓{Y : Y c ∈ τMN ι : C ⊑ Y } = ClMN ι(C), AMN ι(C) = |×MN ι (C)| |×MN ι (C)| , where C ̸= ϕ, BMN ι(C) = ×MN ι(C)\×MN ι (C). 3. Grill topology induced by various types of minimum neighborhoods [52] proposed topologies based on right minimal nbds, and they also presented three other topologies based on distinct minimal nbds. This part looks at the links between various topologies and generalizes them using grills. As a conceptual extension of the topologies in [52], Theorem 3.5 creates the topologies by combining the minimal nbds and grills. Also, this section’s objective is to provide three types of rh app extensions. Definition 12. Let (Π,×,MN ι) be a MN ι-app space and L be a grill on Π. The MNL∗ ι - lo and MNL∗ ι -up app of C are defined by ×L∗ MN ι (C) = {π1 ∈ Π : MN ι(π1) ⊓ Cc /∈ L}. ×L∗ MN ι (C) = {π1 ∈ Π : MN ι(π1) ⊓ C ∈ L}. AL∗ MN ι (C) = |×L∗ MN ι (C)| |×L∗ MN ι (C)| , where ×L∗ MN ι (C) ̸= ϕ,. BL∗ MN ι (C) = ×L∗ MN ι (C)\×L∗ MN ι (C). Definition 13. Let (Π,×,MN ι) be a MN ι-app space and L be a grill on Π. The MNL∗∗ ι - lw and MNL∗∗ ι -up-apps of C are defined by ×L∗∗ MN ι (C) = {π1 ∈ C : MN ι(π1) ⊓ Cc /∈ L}. A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 7 of 22 ×L∗∗ MN ι (C) = C ⊔ ×L∗ MN ι (C). AL∗∗ MN ι (C) = |×L∗∗ MN ι (C)| |×L∗∗ MN ι (C)| , where ×L∗∗ MN ι (C) ̸= ϕ,. BL∗∗ MN ι (C) = ×L∗∗ MN ι (C)\×L∗∗ MN ι (C). Definition 14. Let (Π,×,MN ι) be a MN ι-app space and L be a grill on Π. The MNL∗∗∗ ι -up and MNL∗∗∗ ι -lw-apps of C are defined by ×L∗∗∗ MN ι (C) = ⊔{MN ι(π1) : MN ι(π1) ⊓ Cc ∈ L}. ×L∗∗∗ MN ι (C) = [×L∗∗∗ MN ι (Cc)]c. AL∗∗∗ MN ι (C) = |×L∗∗∗ MN ι (C)| |×L∗∗∗ MN ι (C)| , where ×L∗∗∗ MN ι (C) ̸= ϕ,. BL∗∗∗ MN ι (C) = ×L∗∗∗ MN ι (C)\×L∗∗∗ MN ι (C). Theorem 3. [52] Let (Π,×, ζι) be a ι-NS and π1 ∈ Π. Then the following statement are true: (1) MN ⟨ι⟩(π1) ⊑ MN ι(π1), ι ∈ {r, l, i, u}; (2) MN r(π1) = MN l(π1) = MN i(π1) = MN u(π1), and MN ⟨r⟩(π1),MN ⟨l⟩(π1) = MN ⟨i⟩(π1) = MN ⟨u⟩(π1) when × is symmetric; (3) MN ⟨ι⟩(π1) = MN ι(π1) ∀ι ∈ {r, l, i, u} at what time × exhibits symmetry and transi- tivity.; (4) Al types of MN ⟨ι⟩(π1) are equal when × is equivalence. Theorem 4. If (Π,×,MN ι) is a MN ι-NS, L is a grill on Π, and × is a br, the families τLMN ι = {C ⊑ Π : MN ι(π1) ⊓Cc /∈ L,∀π1 ∈ C} represents as MNL ι -topology on Π about the grill L, ∀ι ∈ {r, l, u, i}. Proof. (1) Π, ϕ clearly belongs to τLMN ι . (2) let C1, C2 ∈ τLMN ι , π1 ∈ C1 ⊓ C2. Then MN ι(π1) ⊓ Cc 1 /∈ L, and MN ι(π1) ⊓ Cc 2 /∈ L ⇒ (MN ι(π1) ⊓ Cc 1) ⊔ (MN ι(π1) ⊓ Cc 2) /∈ L A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 8 of 22 ⇒ MN ι(π1) ⊓ (Cc 1 ⊔ Cc 2) /∈ L ⇒ MN ι(π1) ⊓ (C1 ⊓ C2) c /∈ L ⇒ C1 ⊓ C2 ∈ L. (3) Let Ci ∈ τLMN ι ∀i ∈ I, and π1 ∈ ⊔i∈ICi. Then, exist i0 ∈ I such that π1 ∈ Ci0 , i.e., MN ι ⊓ (Ci0) c /∈ L. ⇒ MN ι ⊓ (⊔i∈ICi) /∈ L ⇒ C1 ⊓ C2 ∈ τLMN ι . τLMN ι indicates a MNL ι -topology on Π with respect to the grill L. Definition 15. Let (Π,×,MN ι) be a MN ι-app space, L be a grill on Π, and C ⊑ Π. If C ∈ τLMN ι , then it is known as LMN ι-open, and if Cc ∈ τLMN ι , that is closed LMN ι-closed. All LMN ι-closed subset of Π are revered by τLMN ι . Definition 16. Let (Π,×,MN ι) be a MN ι-app space, and L be a grill on Π. The LMN ι-interior and LMN ι-closure of C ⊑ Π are IntLMN ι (C) = ⊔{G ∈ τLMN ι : G ⊑ C}, and ClLMN ι (C) = ⊓{Y : Y C ∈ τLMN ι : C ⊑ Y }. The prior topologies in [34, 56] are weaker than the present ones, as shown by Theorem 3.4. Theorem 5. If (Π,×,MN ι) is a MN ι-NS, and L is a grill on Π. Then, τMN ι ⊑ τLMN ι . Proof. Consider C ∈ τMN ι . Thus, MN ι(π1) ⊑ C ∀π1 ∈ C. Therefore, MN ι(π1) ⊓ Cc /∈ L. So, C ∈ τLMN ι . Therefore, τMN ι ⊑ τLMN ι . Remark 1. The following should be understood: (1) At L = {Π} in an τMN ι-topology, then τMN ι = τLMN ι . (2) τMN ι ⊑ τLMN ι , as Observed in Example 3.9. Example 1. Let × = {{π1, π1}, {π1, π2}, {π2, π2}, {π3, π3}, {π2, π3}, {π3, π2}} be a br on Π = {π1, π2, π3, π4}. Table 1 contains all MN ι - nbds. A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 9 of 22 Table 1: MN ι - nbds π1 π2 π3 π4. Nr {π1, π2} {π2, π3} {π2, π3} ϕ Nl {π1} {π1, π2, π3} {π2, π3} ϕ Nu {π1, π2} {π1, π2, π3} {π2, π3} ϕ Ni {π1} {π2, π3} {π2, π3} ϕ MN r {π1, π2} {π2} {π2, π3} ϕ MN l {π1} {π2, π3} {π2, π3} ϕ MN u {π1, π2} {π2, π3} {π2, π3} ϕ MN i {π1} {π2, π3} {π2, π3} ϕ MN ⟨r⟩ {π1, π2} {π2} {π2, π3} ϕ MN ⟨l⟩ {π1} {π2, π3} {π2, π3} ϕ MN ⟨u⟩ {π1, π2} {π2, π3} {π2, π3} ϕ MN ⟨i⟩ {π1} {π2} {π2, π3} ϕ Let L = {{π2}, {π1, π2}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4},Π}. As a result, the following claims are accurate: (1) τMN r = {Π, ϕ, {π2}, {π4}, {π1, π2}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4}}, and τLMN r = {{π2}, {π4}, {π1, π2}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4}, Π, ϕ}. (2) τMN l = {Π, ϕ, {π1}, {π4}, {π1, π4}, {π2, π3}, {π1, π2, π3}, {π2, π3, π4}}, and τLMN l = {{π1}, {π2}, {π4}, {π1, π2}, {π1, π4}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4},Π, ϕ}. (3) τMNu = {Π, ϕ, {π4}, {π2, π3}, {π1, π2, π3}, {π2, π3, π4}}, and τLMNu = {{π2}, {π4}, {π1, π2}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4}, Π, ϕ}. (4) τMN i = {Π, ϕ, {π1}, {π2}, {π4}, {π1, π2}, {π1, π4}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4}}, and A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 10 of 22 τLMN i = {{π1}, {π2}, {π4}, {π1, π2}, {π1, π4}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4},Π, ϕ}. (5) τMN ⟨r⟩ = {Π, ϕ, {π2}, {π4}, {π1, π2}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4}}, and τLMN ⟨r⟩ = {{π2}, {π4}, {π1, π2}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4}, Π, ϕ}. (6) τMN ⟨l⟩ = {Π, ϕ, {π1}, {π4}, {π1, π4}, {π2, π3}, {π1, π2, π3}, {π2, π3, π4}}, and τLMN ⟨l⟩ = {{π1}, {π2}, {π4}, {π1, π2}, {π1, π4}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4},Π, ϕ}. (7) τMN ⟨u⟩ = {Π, ϕ, {π4}, {π2, π3}, {π1, π2, π3}, {π2, π3, π4}}, and τLMN ⟨u⟩ = {{π2}, {π4}, {π1, π2}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4}, Π, ϕ}. (8) τMN ⟨i⟩ = {Π, ϕ, {π1}, {π2}, {π4}, {π1, π2}, {π1, π4}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4}}, and τLMN ⟨i⟩ = {{π1}, {π2}, {π4}, {π1, π2}, {π1, π4}, {π2, π3}, {π2, π4}, {π1, π2, π3}, {π1, π2, π4}, {π2, π3, π4},Π, ϕ}. Remark 2. Example 3.9 indicates that the method described in this section in distinct from those in [16, 19, 52]. Proposition 2. While L is a grill on Π, (Π,×,MN ι) is a MN ι-NS. Consequently, the following claims are accurate: (1) τLMNu ⊑ τLMN r ; (2) τLMNu ⊑ τLMN l ; (3) τLMN r ⊑ τLMN i ; A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 11 of 22 (4) τLMN l ⊑ τLMN i ; (5) τLMN ⟨u⟩ ⊑ τLMN ⟨r⟩ ; (6) τLMN ⟨u⟩ ⊑ τLMN ⟨l⟩ ; (7) τLMN ⟨r⟩ ⊑ τLMN ⟨i⟩ ; (8) τLMN ⟨l⟩ ⊑ τLMN ⟨i⟩ . Proof. Let C ∈ τLMNu . Consequently, MN u(π1) ⊓ Cc /∈ L for every π1 ∈ C. Consequently, (MN r(π1)⊔MN l(π1))⊓Cc /∈ L,∀π1 ∈ C. Therefore, For every π1 ∈ C, MN r(π1)⊓Cc /∈ L, and MN l(π1) ⊓ Cc /∈ L. Consequently, C ∈ MN r for all π1 ∈ C, and C ∈ MN l for all π1 ∈ C. Therefore, τLMNu ⊑ τLMN r and this demonstrates (1), (2), (3) and (4). Similar evidence can be used to support statements (5), (6), (7), and (8). Corollary 1. Let (Π,×, ζι) denote a ι-NS, and let L represent a grill on Π. Subsequently, the ensuing statements are accurate: (1) τLMNu ⊑ τLMN r ⊑ τLMN i ; (2) τLMNu ⊑ τLMN l ⊑ τLMN i ; (3) τLMN ⟨u⟩ ⊑ τLMN ⟨r⟩ ⊑ τ ⟨L⟩ MN ⟨i⟩ ; (4) τLMN ⟨u⟩ ⊑ τLMN ⟨l⟩ ⊑ τLMN ⟨i⟩ . Theorem 3.13 offers a distinctive characterisation of the suggested topologies by con- trasting τLMN ι and τLMN ⟨ι⟩ . Thus, Corollary 3.14 delineates both the smallest τLMN r and the greatest τLMN ⟨l⟩ . Theorem 6. If (Π,×, ζι) constitutes a ι-NS, and L represents a grill on Π. Consequently, τLMN ι ⊑ τLMN ⟨ι⟩ , where ι ∈ {r, l, i, u}. Proof. Let C ∈ τLMN r . Then, for every π1 ∈ C, it holds that MN r(π1) ⊓ Cc /∈ L, and consequently, MN ⟨r⟩(π1) ⊓ Cc /∈ L for all π1 ∈ C. Consequently, C ∈ τLMN ⟨r⟩ . Therefore, τLMN r ⊑ τLMN ⟨r⟩ The remaining assertions can be substantiated in a same way. Corollary 2. If (Π,×, ζι) constitutes a ι-NS, and L represents a grill on Π. Consequently, τLMNu ⊑ τLMN ι ⊑ τLMN ⟨i⟩ , where ι ∈ {r, l, i, ⟨r⟩, ⟨l⟩, ⟨i⟩}. A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 12 of 22 Remark 3. Example 3.9 illustrates that the notable differences between the current method- ology and those presented in [16, 18] are that τLMN ι ⊑ τLMN ⟨ι⟩ , where ι ∈ {r, l, i, u}, despite the fact that τLNι and τLN⟨ι⟩ are not comparable. Moreover, it confirms that the converses of Proposition 3.11 and Corollary 3.12 are not universally valid. (1) τLMN i ⊈ τLMN r ; (2) τLMN i ⊈ τLMNu ; (3) τLMN l ⊈ τLMNu ; (4) τLMN l ⊈ τLMN r ; (5) τLMN ⟨i⟩ ⊈ τLMN ⟨r⟩ ; (6) τLMN ⟨i⟩ ⊈ τLMN ⟨u⟩ ; (7) τLMN ⟨l⟩ ⊈ τLMN ⟨u⟩ ; (8) τLMN ⟨l⟩ ⊈ τLMN ⟨r⟩ . Theorem 3.17 specifies the necessary criteria to determine equivalents among the sug- gested topologies. Theorem 7. If (Π,×, ζι) constitutes a ι-NS, and L represents a grill on Π. Then, (1) τLMN r = τLMN i = τLMN l = τLMNu and τLMN ⟨l⟩ = τLMN ⟨r⟩ = τLMN ⟨i⟩ = τLMN ⟨u⟩ at × is symmetric; (2) τLMN r = τLMN i = τLMN l = τLMNu = τLMN ⟨l⟩ = τLMN ⟨r⟩ = τLMN ⟨i⟩ = τLMN ⟨u⟩ at × is equivalence. Proof. (1) Let C ∈ τLMN r . Consequently, MN r(π1) ⊑ C, ∀π1 ∈ C. ⇔MN i(π1) ⊑ C, MN l(π1) ⊑ C, MN u(π1) ⊑ C, ∀π1 ∈ C (From Theorem 3.4). Consequently τLMN r = τLMN l = τLMN i = τLMNu , and τLMN ⟨r⟩ = τLMN ⟨l⟩ = τLMN ⟨i⟩ = τLMN ⟨u⟩ . (2) The evidence is simple. To investigate the characteristic of monotonicity in the fourth section, which is essen- tial, we must initially analyze the relationship between the two topologies produced by the two subset relations. This is discussed in the following major proposition. Proposition 3. Let (Π,×., ζ1ι), and (Π,×.., ζ2ι) be two ι-NS, L be a grill on Π, and ×. ⊑ ×... Then, τMN 2ι ⊑ τMN 1ι , ι ∈ {r, l, i, u}. A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 13 of 22 Proof. Let C ∈ τMN 2r . Then, MN 2r(π1) ⊓ Cc /∈ L, ∀π1 ∈ C. Thus, MN 1r(π1) ⊓ Cc /∈ L, ∀π1 ∈ C. Consequently, C ∈ MN 1r, and hence, τMN 2r (π1) ⊑ τMN 1r (π1); The remaining cases can be demonstrated in a comparable way. 4. Approximate Models Grill This part presents preliminary models employing τLMN ι -topologies and elucidates their principal characteristics. These mathematical representations preserve the characteristic of monotonicity unencumbered by limitations. Conversely, it may be either forfeited or maintained under stringent terms in certain prior methodologies [16, 52]. Definition 4.1 employs the topologies established in Section 3 to furnish apps. Definition 17. If (Π,×, ζι) constitutes a ι-NS, and L represents a grill on Π. The L- MN ι-lw-app ×L MN ι , and L-MN ι-up-app ×L MN ι of C ⊑ Π are ×L MN ι (C) = ⊔{G ∈ τLMN ι : G ⊑ C} = IntLMN ι , ×L MN ι (C) = ⊓{Y ∈ τLMN ι : C ⊑ Y } = ClLMN ι . The principal characteristics of the proposed apps are addressed in the exposition of the ensuing outcomes. Proposition 4. If (Π,×, ζι) forms a ι-NS, L denotes a grill on Π, and C,G ∈ Π. Sub- sequently, the ensuing assertions are accurate: (1) ×L MN ι (C) ⊑ C; (2) ×L MN ι (ϕ) = ϕ; (3) ×L MN ι (Π) = Π; (4) If C ⊑ G, then ×L MN ι (C) ⊑ ×L MN ι (G); (5) ×L MN ι (C ⊓G) = ×L MN ι (C) ⊓ ×L MN ι (G); (6) ×L MN ι (Cc) = (×L MN ι (C))c; (7) ×L MN ι (×L MN ι (C)) = ×L MN ι (C). Proof. Given that 1, 2, and 3 are readily demonstrable, we shall commence with the proof of 4. A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 14 of 22 (4) Let C ⊑ G. Then, ⊔{G ∈ τLMN ι : G ⊑ C} ⊑ ⊔{G ∈ τLMN ι : G ⊑ G}, and so ×L MN ι (C) ⊑ ×L MN ι (G). (5) From (4), ×L MN ι (C ⊓ G) ⊑ ×L MN ι (C) ⊓ ×L MN ι (G) . Since ×L MN ι (C) ⊑ C, and ×L MN ι (G) ⊑ G, it follows that ×L MN ι (C) ⊓ ×L MN ι (G) ⊑ C ⊓G. Consequently, ×L MN ι (×L MN ι (C)⊓×L MN ι (G)) ⊑ ×L MN ι (C⊓G). Then, ×L MN ι (C)⊓×L MN ι (G) ⊑ ×L MN ι (C⊓ G). Thus, ×L MN ι (C ⊓G) = ×L MN ι (C) ⊓ ×L MN ι (G). (6) Let π1 ∈ ×L MN ι (Cc). At hence, ∃ G ∈ τLMN ι such that π1 ∈ G ⊑ Cc, and so G ⊓ C = ϕ. Therefore, π1 /∈ ×L MN ι (C). Thus, π1 ∈ (×L MN ι (C))c. Conversely, let π1 ∈ (×L MN ι (C))c. Then, π1 /∈ ×L MN ι (C), and so ∃O ∈ τLMN ι that π1 ∈ G, and C ⊓G = ϕ. So, π1 ∈ G ⊑ Cc. Hence, π1 ∈ ×L MN ι (Cc). (7) ×L MN ι (×L MN ι (C)) ⊑ ×L MN ι (C) by (1). On the other hand, let π1 ∈ ×L MN ι (C). Then, ∃G ∈ τLMN ι that π1 ∈ G ⊑ C, ×L MN ι (G) ⊑ ×L MN ι (C) (from(4)). It is observed that G = ×L MN ι (G) from Definition 4.1. So {π1} ∈ ×L MN ι (G) ⊑ ×L MN ι (×L MN ι (C)). Consequently, ×L MN ι (×L MN ι (C)) ⊒ ×L MN ι (C). Corollary 3. If (Π,×, ζι) forms a ι-NS, L denotes a grill on Π, and C,G ∈ Π. Then, ×L MN ι (C) ⊔ ×L MN ι (G) ⊑ ×L MN ι (C ⊔G). Proof. This can be directly deduced from (5) of Proposition 4.2. Remark 4. In Example 3.10, the subsequent propositions are accurate: (1) ×L MN r ({π3}) = ϕ ⊑ {π3}; (2) ×L MN r ({π3}) = ϕ ⊑ {π4} = ×L MN r ({π4}) but {π3} ⊈ {π4}; (3) ×L MN r ({π2})⊔×L MN r ({π1, π2, π3}) = {π1, π2, π3}) = ×L MN r ({π2}⊔×L MN r {π1, π2, π3}) Proposition 5. If (Π,×, ζι) forms a ι-NS, L denotes a grill on Π, and C,G ∈ Π. Sub- sequently, the ensuing assertions are accurate: (1) C ⊑ ×L MN ι (C); (2) ×L MN ι (ϕ) = ϕ; (3) ×L MN ι (C) = C; (4) If C ⊑ G, then ×L MN ι (C) ⊑ ×L MN ι (G); (5) ×L MN ι (C ⊔G) = ×L MN ι (C) ⊔ ×L MN ι (G); A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 15 of 22 (6) ×L MN ι (Cc) = (×L MN ι (C))c; (7) ×L MN ι (×L MN ι (C)) = ×L MN ι (C). Proof. The demonstration parallels that of Proposition 4.2. Corollary 4. If (Π,×, ζι) forms a ι-NS, L denotes a grill on Π, and C,G ∈ Π. Then, ×L MN ι (C ⊓G) ⊑ ×L MN ι (C) ⊓ ×L MN ι (G) Proof. It is directly deducible from Proposition 4.2 (4). Remark 5. In Example 3.10, the subsequent propositions are accurate: (1) ×L MN r ({π1 ⊓ π2}) = ϕ ⊑ {π2} = ×L MN r ({π1}) ⊓ ×L MN r ({π2)}); (2) ×L MN r ({π2}) = {π2} ⊑ {π2, π4} = ×L MN r ({π2, π4}); (3) {π1, π3} ⊑ ×L MN r ({π1, π3}) = {π1, π2, π3}. Definition 18. If (Π,×, ζι) forms a ι-NS, L denotes a grill on Π. The L-MN ι-accuracy is AL MN ι (C) = |×L MN ι (C)| |×L MN ι (C)| , where C ̸= ϕ, Proposition 4.9 ensures that when more data is added, the lo-app won’t get smaller. Similarly, there is no drop in the up-app. Thus, monotonicity is a characteristic of the suggested model. This characteristic guarantees that when additional information becomes available, the apps either increase in accuracy or remain constant, never declining. Proposition 6. Let (Π,×1, ζ1ι) and (Π,×2, ζ2ι) be two ι-NS, L represent a grill on Π, and ×. ⊑ ×... For all ι ∈ {r, l, i, u}, C is a subclass of Π, and the subsequent propositions are accurate: (1) ×L MN 1ι (C) ⊑ ×L MN 2ι (C); (2) ×L MN 2ι (C) ⊑ ×L MN 1ι (C); (3) AL MN 2ι (C) ≤ AL MN 1ι (C). Proof. (1) ×L MN 1ι (C) = ⊓{Y ∈ τLMN 1ι : C ⊑ Y } ⊑ ⊓{Y ∈ τLMN 2ι : C ⊑ Y } = ×L MN 2ι (C) (from proposition 3.18). So, ×L MN 1ι (C) ⊑ ×L MN 2ι (C). (2) Let ×L MN 2ι (C) = ⊔{G ∈ τLMN 2ι : G ⊑ C} ⊑ ⊔{G ∈ τLMN 1ι : G ⊑ C} = ×L MN 1ι (C) (from proposition 3.18). So, ×L MN 2ι (C) ⊑ ×L MN 1ι (C). A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 16 of 22 (3) AL MN 1ι (C) = |×L MN1ι (C)| |×L MN1ι (C)| ≤ |×L MN2ι (C)| |×L MN2ι (C)| = AL MN 2ι (C) Definition 19. If (Π,×, ζι) forms a ι-NS, L denotes a grill on Π. The L-MN ι-positive, L-MN ι-boundary, and L-MN ι-negative regions of C ⊑ Π are ×L+ MN ι (C) = ×L MN ι (C), ×L− MN ι (C) = Π\×L MN ι (C), BL MN ι (C) = ×L MN ι (C)\×L MN ι (C). Proposition 7. Let (Π,×1, ζ1ι) and (Π,×2, ζ2ι) be two ι-NS, L represent a grill on Π, and ×. ⊑ ×... For all ι ∈ {r, l, i, u}, C is a subclass of Π, and the subsequent propositions are accurate: (1) BL MN 1ι (C) ⊑ BL MN 2ι (C); (2) ×L− MN 2ι (C) ⊑ ×L− MN 1ι (C). Proof. (1) Let π1 ∈ BL MN 1ι (C). Then, π1 ∈ ×L MN 1ι (C)\×L MN 1ι (C) . Therefore, π1 ∈ ×L MN 1ι (C), and π1 ∈ (×L MN 1ι (C))c. So, π1 ∈ ×L MN 2ι (C), and π1 ∈ (×L MN 2ι (C))c. There- fore, π1 ∈ BL MN 2ι (C), BL MN 1ι (C) ⊑ BL MN 2ι (C). (2) This is derived from Proposition 4.9. Definition 20. If (Π,×, ζι) forms a ι-NS, L denotes a grill on Π. C ⊑ Π is L-MN ι-exact if ×L MN ι (C)) = ×L MN ι (C)) = C; otherwise, it is L-MN ι-rough. Proposition 8. If (Π,×, ζι) forms a ι-NS, L denotes a grill on Π. C ⊑ Π is L-MN ι-exact if and only if BL MN ι (C) = ϕ . Proof. Let C be any L-MN ι-exact. Then, BL MN ι (C) = ×L MN ι (C)\×L MN ι (C) = ϕ. Conversely, BL MN ι (C) = ϕ; hence, ×L MN ι (C)\×L MN ι (C) = ϕ, and so×L MN ι (C) ⊑ ×L MN ι (C). However, ×L MN ι (C) ⊑ ×L MN ι (C). Thus, ×L MN ι (C) = ×L MN ι (C), and C is L-MN ι-exact. A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 17 of 22 5. An Application of the Suggested Approach for Heart Failure This paragraph presents the experimental results of a preparation study conducted on five symptoms of heart disease, as outlined by Dickstein et al. [57], including seven pa- tients. The research was carried out in the cardiology department of Al-Azhar University [58]. The quantity of training data utilized Twenty-five records were analyzed,while the other data were forwarded to this institution displaying similar presented symptoms, com- prehensive history, physical examination, complete laboratory tests, resting electrocardio- gram, and conventional echocardiographic assessment were conducted. The information system contains data for just seven patients with similar characteristics, as discussed in Table 2, regarding the heart failure issue. The columns denote the symptoms, where ’E’ indicates the presence of symptoms and ’NE’ signifies their absence, pertaining to the diagnosis of heart failure [57] (condition characteristics). where BS is the breathlessness, OA is the orthopnea, PA is the paroxysmal nocturnal dyspnea, RE reduced exercise tol- erance, AG is the ankle swelling. The attribute D represents the determination of heart failure. The rows in Table 2, Π = {π1, π2, π3, π4, π5, π8, π9} represents the patients. Let us consider the expert of the system who has offered the subsequent relation × on the set of patients Π to delineate the connections among them based on their symptoms: πi×πj ⇔f(πi) ⊑ f(πj), where the function is defined by f(π1) = {BS,OA,PA,RE}, f(π2) = {RE,AG}, f(π3) = {BS,OA,PA,RE,AG}, f(π4) = {RE}, f(π5) = {BS,RE,AG}, f(π8) = {BS,OA,RE,AG}, f(π9) = {BS,PA,RE}. Then, × = {(π1, π1), (π1, π3), (π2, π2), (π2, π3), (π2, π5), (π2, π8), (π3, π3), (π4, π1) (π4, π2), (π4, π3), (π4, π4), (π4, π5), (π4, π8), (π4, π9), (π5, π3), (π5, π5), (π5, π8) (π8, π3), (π8, π8), (π9, π1), (π9, π3), (π9, π9)}. Hence, MN r(π1) = {π1, π3}, MN r(π2) = {π2, π3, π5, π8}, MN r(π3) = {π3}, MN r(π4) = Π, MN r(π5) = {π3, π5, π8}, MN r(π8) = {π3, π8}, and MN r(π9) = {π1, π3, π9}. Let L = {Π}, Then τLMN r = 2Π(the set of all subsets of Π). I1 = {π1, π3, π8, π9} are the afflicted patients, In contrast, the uninfected patients are represented by I2 = {π2, π4, π5}. Consequently, the lw-app, the up-app, and the accuracy of I1 are computed as Case(i) The the afflicted patients I1 = {π1, π3, π8, π9} (1) Ismail’s method [52] in Definition 17: ×MN r (I1) = I1; ×MN r(I1) = Π; AMN r(I1) = 4 7 ; BL MN r (I1) = {π2, π4, π5} A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 18 of 22 (2) The proposed Definitions 4.1, 4.8, and 4.10. ×L MN r (I1) = I1; ×L MN r (I1) = I1; AL MN r (I1) = 1; BL MN r (I2) = ϕ Case (ii) The uninfected patients I2 = {π2, π4, π5}.(1) Ismail’s method [52] in Defini- tion 17: ×MN r (I2) = ϕ; ×MN r(I2) = {π2, π4, π5}; AMN r(I1) = 0; BL MN r (I2) = {π2, π4, π5} (2) The proposed Definitions 4.1, 4.8, and 4.10. ×L MN r (I2) = I2; ×L MN r (I2) = I2; AL MN r (I2) = 1; BL MN r (I2) = ϕ. Consequently, Ismail’s boundaries [52] for infected and uninfected persons are {π2, π4, π5}, resulting in ambiguity and diminished decision precision. Conversely, the current method- ology produces a null border, so diminishing ambiguity and improving precision. 6. conclusion Rough set theory is a mathematical framework designed to address uncertainty. Grills can enhance this idea, serving as an effective instrument for diminishing ambiguity by enabling a wider approximation. A primary emphasis in the study of rough sets is the minimization of boundaries to improve precision. Grills is one of the most efficient meth- ods for accomplishing this. Consequently, numerous techniques employing grills for the construction of diverse topologies have been suggested. The previous topologies utilizing A. A. Azzam et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7160 19 of 22 Table 2: Information system for heart failure Patients BS OA PA RE AG D. π1 E E E E NE E π2 NE NE NE E E NE π3 E E E E E E π4 NE NE NE E NE NE π5 E NE NE E E NE π8 E E NE E E E π9 E NE E E NE E π10 NE NE NE E E NE grills were less refined than the current ones. The existing topologies are more extensive and provide a significant amount of information that is beneficial for the analysis of rough sets. We utilized the neighborhood relation with the grill, which is among the most effec- tive relations and has broadened the topological structures. This increases its relevance in domains necessitating extensive samples, such as worldwide epidemics. The characteris- tics of these topologies were examined, including comparative analyses among them. The smallest and largest topologies were determined, a task not accomplished in prior studies. Furthermore, novel approximations were proposed, employing these suggested topologies as an extension of the previous model. This increases its relevance in domains necessitat- ing extensive samples, such as worldwide epidemics. A promising avenue for future study will include the following: (1) Introducing two grills instead of one to generalize the current method; (2) Developing a soft-topology to enhance the existing work; (3) Extending the present paper to encompass fuzzy and picture sets; (4) Comparison between grills and ideals to generalize the prevailing methodology . Conflict of interest There are no conflicting interests, according to the authors. Acknowledgements The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2025/01/34111). References [1] Pawlak V., Rough sets. Int. J. Comput. Inf. Sci. 1982, 11, 341–356. [2] Pawlak V., Rough concept analysis. Bull. Pol. Acad. Sci. Math. 1985, 33, 495–498. [3] Ma X., Liu Q., Zhan J., A survey of decision-making methods based on certain hybrid soft set models. Artif. Intell. Rev. 2017, 47, 507–530. A. A. Azzam et al. / Eur. J. Pure Appl. 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