EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 3436-3463 ISSN 1307-5543 – ejpam.com Published by New York Business Global Employing a Generalization of Open Sets Defined by Ideals to Initiate Novel Rough Approximation Spaces with a Chemical Application M. Hosny1,2, Tareq M. Al-shami3,4,∗ 1 Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia 2 Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo, Egypt 3 Department of Engineering Mathematics & Physics, Faculty of Engineering & Technology, Future University, New Cairo, Egypt 4 Jadara University Research Center, Jadara University, Jordan Abstract. A close similarity and analogy between rough set theory and topology is attributed to the corresponding behavior of lower and upper rough approximations with interior and closure topological operators, respectively. This relation motivates joint studies between topology and this theory. We endeavor by rough set theory to enlarge the knowledge we obtain from the information systems, for this reason, we apply the abstract concept of ideal structures to build new general- ized approximation spaces with less vagueness. In the present work, we employ a novel type of nearly open sets in topology so-called “L-θβλ-open” with an ideal structure to introduce novel approximation spaces satisfying the desired properties concerning shrinking the boundary region of uncertainty and expanding the domain of confirmed information. We set up the fundamentals of the proposed rough paradigms and demonstrate their superiority over the preceding paradigms induced by some nearly open sets. Two algorithms are furnished to illustrate the way of specifying the family of L-θβλ-open sets and exploring whether a subset is L-θβλ-definable or L-θβλ-rough. Then, we put forward the concepts of rough membership relations and functions and uncover their core characterizations. Finally, we examine the proposed models to model a real situation in the Chemistry field and clarify how our models improve the outcomes of generalized approximation spaces over the previous models. 2020 Mathematics Subject Classifications: 03E99, 54A05, 54E99 Key Words and Phrases: Rough set, topology, ideal, L-θβλO-open set ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5392 Email addresses: Maly@kku.edu.sa (M. Hosny), tareqalshami83@gmail.com (T.M. Al-shami) https://www.ejpam.com 3436 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3437 1. Introduction Nowadays, one can see a rapid growth of interest in the theory of rough sets and its ap- plications, evident from the number of international conferences and workshops dedicated to investigating the progression of rough set theory, as well as the high-quality papers pub- lished as a result of this attention. This theory was initiated by Pawlak [40, 41] in the early 1980s as a non-statistical technique to analyze data tables acquired from human experts or measurements. The philosophy of rough set theory in addressing the complex problems individuals face in practical life is based on dividing a set of data containing uncertainty into three regions. The first region includes the confirmed information extracted from this set, terminologically known as the lower approximation. The second region represents the information for which we cannot determine its belonging or non-belonging to the set, known as the upper approximation. The third region, known as the boundary region, is defined as the difference between the upper approximation and the lower approximation. Rough set theory begins with the concept of an equivalence relationship, which is a strict term when modeling many realistic problems. This strictness prompted many re- searchers and authors to search for alternative methods to the equivalence classes, leading to the development of the neighborhood idea. Initially defined by Yao, he [50, 51] formu- lated the concepts of right neighborhoods and left neighborhoods as the equivalents of the equivalence classes derived from Pawlak’s original model. Over time, with the desire to increase the confirmed information, other models were proposed to improve the approxi- mation operators and accuracy measures. For instance, rough set paradigms introduced by using minimal neighborhoods [3], containment neighborhoods [5], maximal neighborhoods [8, 16], subset neighborhoods [10, 52], adhesion neighborhoods [36], etcetera. Attention was paid early by [48] to the similarity between the behaviors of lower and upper rough approximations and interior and closure topological operators. There- fore, topological structures have been proposed to study information systems and ap- ply topological operators as alternative tools for these approximations; see, for instance [4, 17, 22, 34, 43, 46, 49, 53]. Diverse techniques have been introduced to create topolog- ical spaces utilizing neighborhood systems. For example, one can take the neighborhood of each point as a subbase of a topology [33] or initiate the topology using the following formula: ϑλ = {V ⊆ X : ∀y ∈ V,Gλ(y) ⊆ V } [45]. To develop decision-making methods for information systems from a topological standpoint, several authors have employed ab- stract topological principles and their generalizations, such as nearly open sets [1, 2, 6, 7], supra topology [9], minimal structures [18], infra topology [13], and bitopology [44]. The authors of [32, 47] put forward the notion of ideal over a set X as a nonempty subcollection of the power set of X which is closed under finite union and hereditary property. In [28], new topologies are derived from an old one using ideals. With a strong desire to increase the amount of confirmed information, which gives the decision-maker a greater opportunity to make more accurate decisions, the ideal structure was integrated into generalized approximation spaces. The concept was first employed by Kandil et al. [29]. This concept was later exploited by researchers in the study of information systems, explaining the advantages of this tool in various ways, including topological approaches, M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3438 as illustrated in many published manuscripts [11, 12, 14, 21, 23, 24, 27, 39]. Researchers have the freedom to choose the tool that is most efficient for addressing the problem and achieving the greatest possible amount of desired characteristics of Pawlak paradigms. Michael [38] came up with a brilliant idea to enlarge a family of semi-open sets using ideals, then some authors [25, 26] followed this technique to aggrandize the classes of α-open, β-open, and pre-open sets. This work deals with generalized approximation spaces using a topological approach and enhances the prominence of using ideals via rough set theory studies, as a tool to de- mystify the data. We suggest a broader general framework of topological approximation spaces via ideals, satisfying the desirable characteristics of original models and enhanc- ing decision reliability. The presentation of this article is organized as follows: Section 2 covers the fundamentals required to make the paper self-contained. Then, in Section 3, we define a new class of nearly open sets, namely, L-θβλ-open sets, which is strictly stronger than the class of L-βλ-open sets. We draw the main properties of this class and articulate its relationships with the preceding ones with the aid of examples. Section 4 is devoted to constructing rough set models utilizing the class of L-θβλ-open sets. We com- pare the approximation operators, boundary regions, and accuracy values of the proposed paradigms with those presented in other studies. In Section 5, we display a new type of rough membership functions and apply to describe the main concepts of the proposed rough set models. We provide a practical example in Section 6 to illustrate the superiority of the current models over the former models and their applicability in addressing realistic problems. Lastly, we draw conclusions from the present study and summarize its most important findings in Section 7. The presentation of this article is organized as follows: Section 2 covers the fundamentals required to make the paper self-contained. Then, in Section 3, we define a new class of nearly open sets, namely, L-θβλ-open sets, which is strictly stronger than the class of L-βλ-open sets. We draw the main properties of this class and articulate its relationships with the preceding ones with the aid of examples. Section 4 focuses on developing rough set models using the L-θβλ-open sets. We compare the approximation operators, boundary regions, and accuracy values of the proposed models with those in existing studies. In Section 5, we introduce a new type of rough member- ship functions and apply them to explain the central concepts of the proposed rough set models. Section 6 provides a practical example that demonstrates the advantages of the current models over previous ones and their effectiveness in solving real-world problems. Finally, in Section 7, we conclude the study by summarizing its key findings. 2. Preliminaries In this segment, we cover the main contributions via topological (generalized) approx- imation spaces that are required to understand the main contributions and significance of this manuscript. Definition 1. [32, 47] An ideal L over the universe X ̸= ∅ is a subfamily of the power set of X satisfying the below terms. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3439 (i) V ∈ L and Z ∈ L ⇒ V ∪ Z ∈ L. (ii) V ∈ L and Z ⊆ V ⇒ Z ∈ L. Through this content, X indicates for a finite nonempty set. Definition 2. [9] Let R be a binary relation on X. Then, the λ-neighborhood of an element y in X, symbolized by Gλ(y), λ ∈ {a,b, â, b̂, i,u, î, û}, is given by: (i) Ga(y) = {x ∈ X : yRx}. (ii) Gb(y) = {x ∈ X : xRy}. (iii) Gâ(y) = ∩y∈Ga(x)Ga(x), or Gâ(y) = ∅ when there does not exists Ga(x) containing y. (iv) Gb̂(y) = ∩y∈Gb(x)Gb(x), or Gb̂(y) = ∅ when there does not exists Gb(x) containing y. (v) Gi(y) = Ga(y) ∩Gb(y). (vi) Gu(y) = Ga(y) ∪Gb(y). (vii) Gî(y) = Gâ(y) ∩Gb̂(y). (viii) Gû(y) = Gâ(y) ∪Gb̂(y). Moving forward, we utilize this symbol λ throughout this manuscript to refer to the types of neighbourhoods of {a,b, â, b̂, i,u, î, û}. Definition 3. [45] If Ξλ : X → P (X) is a mapping that assigns for each y in X a Gλ in P (X), then we called a 3-tuple (X,R,Ξλ) a Gλ-space. Theorem 1. [30, 31, 45] It may generate a topology ϑλ on X using Gλ-neighbourhoods by the next formula ϑλ = {V ⊆ X : ∀y ∈ V,Gλ(y) ⊆ V } Every member of ϑλ is named a λ-open set and we call a subset a λ-closed set if its complement is a λ-open set. The class of Γλ is given by Γλ = {F ⊆ X : F ′ ∈ ϑλ}, where F ′ is the complement of F . Definition 4. [45] The λ-lower and λ-upper approximations, λ-boundary region and λ- accuracy of V ⊆ X, inspired by the topological space (X,ϑλ) given in above theorem, are respectively formulated by the subsequent formulas: Rλ(V ) is the union of all λ-open sets which are contained in V ; that is V = intλ(V ), where intλ is the topological λ-interior operator. Rλ(V ) is the intersection of all λ-closed sets containing V ; that is, V = clλ(V ), where clλ is the topological λ-closure operator. BNDλ(V ) = Rλ(V )−Rλ(V ). ACCλ(V ) = |Rλ(V )| |Rλ(V )| , for each subset V ̸= ∅. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3440 Remember that a subset V is named λ-exact if Rλ(V ) = Rλ(V ). Otherwise, V is λ-rough. In what follows, we recall some definitions of λ-nearly open sets. Definition 5. [15, 20] Let (X,R,Ξλ) be a Gλ-space. V ⊆ X is said to be (i) λ-preopen (Pλ-open), if intλ(clλ(V )) ⊇ V . (ii) λ-semiopen (Sλ-open), if clλ(intλ(V )) ⊇ V . (iii) αλ-open, if V ⊆ intλ[clλ(intλ(V ))]. (iv) βλ-open (semi preopen), if V ⊆ clλ[intλ(clλ(V ))]. (v) δβλ-open, if V ⊆ clλ[intλ(cl δ λ(V ))], where clδλ(V ) = {y ∈ X : V ∩ intλ(clλ(G)) ̸= ∅, G ∈ ϑλ and y ∈ G}. (vi) ∧ βλ -set if V = ∧ βλ (V ), where ∧ βλ (V ) = ∩{G : V ⊆ G,G ∈ βλO(X)}. The families of λ-nearly open subsets of X are assigned by ηλO(X), where η ∈ {α, P, S, β, δβ, ∧ β}. The complements of the λ-nearly open sets are known as λ-nearly closed sets and denoted by ηλC(X). Henceforth, we mean by η the elements of the set {P, S, α, β, δβ, ∧ β}, unless otherwise stated. Definition 6. [15, 20] Let (X,R,Ξλ) be a Gλ-space and V ⊆ X. The ηλ-lower and ηλ-upper approximations, ηλ-boundary regions and ηλ-accuracy of V are respectively given by: Rη λ(V ) = ∪{G ∈ ηλO(X) : G ⊆ V } = ηλ-interior of V . Rη λ(V ) ∩ {H ∈ ηλC(X) : V ⊆ H} = ηλ-closure of V . BNDη λ(V ) = Rη λ(V )−Rη λ(V ). ACCηλ(V ) = |Rη λ(V )| |Rη λ(V )| , where |Rη λ(V )| ≠ 0, |Rη λ(V )| denotes the cardinality of Rη λ(V ). Definition 7. [20] A subset V of a Gλ-space (X,R,Ξλ) is called: (i) δβλ-definable (δβλ-exact) if Rδβ λ (V ) = Rδβ λ (V ) or BNDδβ λ (V ) = ∅. (ii) δβλ-rough if Rδβ λ (V ) ̸= Rδβ λ (V ) or BNDδβ λ (V ) ̸= ∅. (iii) ∧ βλ -definable ( ∧ βλ -exact) if R ∧ β λ (V ) = R ∧ β λ (V ) or BND ∧ β λ (V ) = ∅. (iv) ∧ βλ -rough if R ∧ β λ (V ) ̸= R ∧ β λ (V ) or BND ∧ β λ (V ) ̸= ∅. Definition 8. [22, 23] Let L be an ideal on X. We call a subset V of a Gλ-space (X,R,Ξλ): M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3441 (i) L-αλ-open providing that there exists G ∈ ϑλ s.t. (V −intλ(clλ((G)) ∈ L and (G−V ) ∈ L. (ii) L-Pλ-open providing that there exists G ∈ ϑλ s.t. (V −G) ∈ L and (G− clλ(V )) ∈ L. (iii) L-Sλ-open providing that there exists G ∈ ϑλ s.t. (V −clλ(G)) ∈ L and (G−V ) ∈ L. (iv) L-βλ-open providing that there exists G ∈ ϑλ s.t. (V −clλ(G)) ∈ L and (G−clλ(V )) ∈ L. (v) L-δβλ-open providing that there exists G ∈ ϑλ s.t. (V −clλ(G)) ∈ L and (G−clδλ(V )) ∈ L. (vi) L- ∧ βλ -set, if V = L − ∧ βλ (V ), where L- ∧ βλ (V ) = ∩{G : V ⊆ G,G ∈ L-βλO(X)}. These sets are called L-λ-nearly open sets, the complement of the L-λ-nearly open sets is called L-λ-nearly closed sets, the families of L-λ-nearly open sets of X denoted by L-ηλO(X) and the families of L-λ-nearly closed sets of X denoted by L-ηλC(X). Proposition 1. [23] (i) Every δβλ-open is L-δβλ-open. (ii) Every ∧ βλ -set is L- ∧ βλ -set. Proposition 2. [23] The next implications hold true: ϑλ(Γλ) ⇒ L-αλO(L-αλC) L-PλO(L-PλC) ⇓ ⇓ L-SλO(L-SλC) ⇒ L-βλO(L-βλC) ⇒ L-δβλO(L-δβλC). ϑλ(Γλ) ⇒ L-αλO(L-αλC) L-PλO(L-PλC) ⇓ ⇓ L-SλO(L-SλC) ⇒ L-βλO(L-βλC) ⇒ L- ∧ βλO (L- ∧ βλC ). ϑλ(Γλ) ⇒ αλO(αλC) PλO(PλC) ⇓ ⇓ SλO(SλC) ⇒ βλO(βλC) ⇒ δβλO(δβλC). ϑλ(Γλ) ⇒ αλO(αλC) PλO(PλC) ⇓ ⇓ SλO(SλC) ⇒ βλO(βλC) ⇒ ∧ βλO ( ∧ βλC ). Definition 9. [22, 23] The L-ηλ-lower and L-ηλ-upper approximations, L-ηλ-boundary regions and L-ηλ-accuracy of V are respectively given by: RL−η λ (V ) = ∪{G ∈ L-ηλO(X) : G ⊆ V } = L-ηλ-interior of V . RL−η λ (V ) = ∩{H ∈ L-ηλC(X) : V ⊆ H} = L-ηλ-closure of V . BNDL−η λ (V ) = RL−η λ (V )−RL−η λ (V ). ACCL−η λ (V ) = |RL−η λ (V )| |RL−η λ (V )| , where |RL−η λ (V )| ≠ 0. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3442 Remember that a subset V is called an L-ηλ-definable (L-ηλ-exact) set if R L−η λ (V ) = RL−η λ (V ). Otherwise, V is an L-ηλ-rough set. Theorem 2. [23] For a subset V of a Gλ-space (X,R,Ξλ), we have: (i) Rα λ(V ) ⊆ Rp λ(V ) ⊆ Rγ λ(V ) ⊆ Rβ λ(V ) ⊆ Rδβ λ (V ) ⊆ RL−αβ λ (V ). (ii) Rα λ(V ) ⊆ Rs λ(V ) ⊆ Rγ λ(V ) ⊆ Rβ λ(V ) ⊆ Rδβ λ (V ) ⊆ RL−αβ λ (V ). (iii) Rλ(V ) ⊆ Rδβ λ(V ) ⊆ RL−αβ λ (V ). (iv) Rα λ(V ) ⊆ Rp λ(V ) ⊆ Rγ λ(V ) ⊆ Rβ λ(V ) ⊆ R ∧ β λ (V ) ⊆ RL− ∧ βλ (V ). (v) Rα λ(V ) ⊆ Rs λ(V ) ⊆ Rγ λ(V ) ⊆ Rβ λ(V ) ⊆ R ∧ β λ (V ) ⊆ RL− ∧ βλ (V ). (vi) Rλ(V ) ⊆ R ∧ β λ (V ) ⊆ RL− ∧ βλ (V ). (vii) RL−δβ λ (V ) ⊆ Rδβ λ (V ) ⊆ Rβ λ(V ) ⊆ Rγ λ(V ) ⊆ Rp λ(V ) ⊆ Rα λ(V ). (viii) RL−δβ λ (V ) ⊆ Rδβ λ (V ) ⊆ Rβ λ(V ) ⊆ Rγ λ(V ) ⊆ Rs λ(V ) ⊆ Rα λ(V ). (ix) RL−δβ λ (V ) ⊆ Rδβ λ(V ) ⊆ Rλ(V ). (x) RL− ∧ β λ (V ) ⊆ R ∧ β λ (V ) ⊆ Rβ λ(V ) ⊆ Rγ λ(V ) ⊆ Rp λ(V ) ⊆ Rα λ(V ). (xi) RL− ∧ β λ (V ) ⊆ R ∧ β λ (V ) ⊆ Rβ λ(V ) ⊆ Rγ λ(V ) ⊆ Rs λ(V ) ⊆ Rα λ(V ). (xii) RL− ∧ β λ (V ) ⊆ R ∧ β λ (V ) ⊆ Rλ(V ). When we combine an ideal L with a Gλ-space (X,R,Ξλ), we write the quadruple (X,R,Ξλ,L); this quadruple is symbolized by L −Gλ-space. Proposition 3. [23] For a subset V of an L −Gλ-space (X,R,Ξλ,L), we have: (i) RL−P λ (V ) ⊆ RL−β λ (V ) ⊆ RL−δβλ(V ). (ii) RL−α λ (V ) ⊆ RL−S λ (V ) ⊆ RL−β λ (V ) ⊆ RL−δβ λ (V ). (iii) RL−δβ λ (V ) ⊆ RL−β λ (V ) ⊆ RL−P λ (V ). (iv) RL−δβ λ (V ) ⊆ RL−β λ (V ) ⊆ RL−S λ (V ) ⊆ RL−α λ (V ). (v) RL−P λ (V ) ⊆ RL−β λ (V ) ⊆ RL− ∧ βλ(V ). (vi) RL−α λ (V ) ⊆ RL−S λ (V ) ⊆ RL−β λ (V ) ⊆ RL− ∧ βλ(V ). (vii) RL− ∧ βλ(V ) ⊆ RL−β λ (V ) ⊆ RL−P λ (V ). M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3443 (viii) RL− ∧ βλ(V ) ⊆ RL−β λ (V ) ⊆ RL−S λ (V ) ⊆ RL−α λ (V ). Definition 10. Let (X,R,Ξλ) be a Gλ-space and V ⊆ X. The θλ-closure is given by clθλ(V ) = {y ∈ X : V ∩ clλ(G) ̸= ∅, G ∈ ϑλ and y ∈ G}. Definition 11. [42] The rough membership function of a subset V of X is defined, under an equivalence relation R on X, as µV : X → [0, 1], where µV (y) = |[y]R ∩V | |[y]R| , y ∈ X. [y]R denotes to an equivalence classes. Definition 12. [23] The λ-rough membership functions of a subset V of X is given by µλV → [0, 1], where µλV (y) = |{∩Gλ(y)}∩V | |∩Gλ(y)| . Definition 13. [35] The λ-rough nearly membership function of a subset V of X is defined by µηλV → [0, 1] as follows µηλV (y) = { 1 : 1 ∈ ψηλV (y) min(ψηλV (y)) : otherwise where ψηλV (y) = { |ηλ(y)∩V | |ηλ(y)| : y ∈ ηλ(y) and ηλ(y) ∈ ηλO(X)}, η ∈ {α, P, S, β}. Definition 14. [22, 23] The L − λ-nearly rough membership functions of a subset V of X is defined by µL−ηλV → [0, 1], as follows µL−ηλV (y) = { 1 : 1 ∈ ψL−ηλ V (y) min(ψL−ηλ V (y))) : otherwise where ψL−ηλ V (y) = { |L−ηλ(y)∩V | |L−ηλ(y)| : y ∈ L − ηλ(y) and L − ηλ(y) ∈ L-ηλO(X)}. Lemma 1. [23] Let V be a subset of an L −Gλ-space (X,R,Ξλ,L). Then (i) µλV (y) = 1 ⇒ µηλV (y) = 1 ⇒ µL−ηλV (y) = 1, ∀ y ∈ X. (ii) µλV (y) = 0 ⇒ µηλV (y) = 0 ⇒ µL−ηλV (y) = 0, ∀ y ∈ X. Definition 15. [45] Let (X,R,Ξλ) be a Gλ-space, y ∈ X and V ⊆ X: (i) If y ∈ Rλ(V ), then y λ-certainly belongs to V , denoted by y ∈λV . (ii) If y ∈ Rλ(V ), then y λ-probably belongs to V , denoted by y ∈λV . (iii) If y ∈ Rη λ(V ), then y λ-nearly certainly (ηλ-certainly) belongs to V , denoted by y ∈ηλV, η ∈ {α, P, S, β}. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3444 (iv) If y ∈ Rη λ(V ), then y λ-nearly probably (ηλ-probably) belongs to V , denoted by y ∈ηλV, η ∈ {α, P, S, β}. Definition 16. [23] Let (X,R,Ξλ) be a Gλ-space, y ∈ X and V ⊆ X: (i) If y ∈ RL−η λ (V ), then y is λ-nearly certainly with respect to L ( L − ηλ-certainly) belongs to V , denoted by y ∈L−η λ A. (ii) If y ∈ RL−η λ (V ), then y is λ-nearly probably with respect to L (briefly L−ηλ-probably) belongs to V , denoted by y ∈L−η λ A. Proposition 4. [23] The subsequent properties hold true for each subset V . (i) if y ∈λA⇒ y ∈ηλA⇒ y ∈L−η λ A. (ii) if y ∈L−η λ A⇒ y ∈ηλA⇒ y ∈λA. 3. L-θβλ-open sets This section aims to adopt a fresh class of nearly open sets called L-θβλ-open sets, serving as an introduction to building rough set paradigms. This type of nearly open sets is established by replacing the empty difference of θβ-open sets with the belonging of difference to the ideal, which enlarges the class of θβ-open sets. We conclude the core characterizations of this class and elucidate its relationship with the forgoing classes. Definition 17. A subset V of an L−Gλ-space (X,R,Ξλ,L) is called L-θβλ-open providing that ∃ G ∈ ϑλ s.t. (V − clλ(G)) ∈ L and (G − clθλ(V )) ∈ L. We call a complement of a L-θβλ-open set an L-θβλ-closed set. The classes of all L-θβλ-open and L-θβλ-closed are respectively symbolized by L-θβλO(X) and L-θβλC(X). Example 1. Let X = {y1, y2, y3, y4, y5},L = {∅, {y3}}, and R = {(y1, y1), (y1, y2), (y2, y2), (y3, y3), (y3, y4), (y4, y3), (y4, y4), (y5, y2), (y5, y3), (y5, y4)}. Then, the topology generated by a relation R in the case of λ = a is ϑa = {X, ∅, {y2}, {y1, y2}, {y3, y4}, {y2, y3, y4}, {y, y2, y3, y4}, {y2, y3, y4, y5}} and L-θβaO(X) is the power set of X. We demonstrate in the next result that the class of L-θβλ-open sets is wider than the classes of L-δβλ-open sets, L- ∧ βλ -sets. Proposition 5. (i) Every L-δβλ-open set is L-θβλ-open set. (ii) Every L- ∧ βλ -set is L-θβλ-open set. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3445 Proof. It is evident by Definitions 8 [23] and 17. Remark 1. Example 1 yields an evidence that the converse of Proposition 5 fails. By this example, we remark that L-θβaO(X) = P (X), L-δβaO(X) = P (X)−{{y5}}, and L-∧ βa O(X) = {X, ∅, {y2}, {y3}, {y4}, {y5}, {y1, y2}, {y2, y3}, {y2, y4}, {y2, y5}, {y3, y4}, {y3, y5}, {y4, y5}, {y, y2, y3}, {y1, y2, y5}, {y1, y2, y4}, {y2, y3, y4}, {y2, y3, y5}, {y2, y4, y5}, {y3, y4, y5} , {y1, y2, y3, y4}, {y1, y2, y3, y5}, {y1, y2, y4, y5},{y2, y3, y4, y5}}. Now, {y5} is an L-θβaO(X)-open set, but it is neither an L-δβaO(X)-open set nor an L- ∧ βa -set. Also, the next result clarifies that the class of L-θβλ-open sets is wider than the classes of δβλ-open sets and ∧ βλ -sets. Proposition 6. (i) Every δβλ-open set is L-θβλ-open set. (ii) Every ∧ βλ -set is L-θβλ-open set. Proof. By using Propositions 1 [23] and 5. Example 2. Let X = {y1, y2, y3, y4},L = {∅, {y3}}, and R = {(y1, y1), (y1, y2), (y2, y1), (y2, y2), (y3, y3), (y4, y3), (y4, y4))}. Then, the topology generated by a relation R in the case of λ = a is ϑa = {X, ∅, {y3}, {y1, y2}, {y1, y2, y3}}. Now, L-θβaO(X) is the power set of X and δβaO(X) is P (X)\{{y4}}. One can check that {y4} is an L-θβaO(X)-open set, but it is not δβaO(X)-open. Example 3. Let X = {y1, y2, y3, y4},L = {∅, {y3}} and R = {(y1, y1), (y1, y3), (y2, y1), (y2, y3), (y3, y3), (y4, y4)} Then, the topology generated by a relation R in the case of λ = a is ϑa = {X, ∅, {y3}, {y4}, {y1, y3}, {y3, y4}, {y1, y3, y4}}. Note that L-θβaO(X) = P (X) and ∧ βa O(X) = {X, ∅, {y2}, {y3}, {y4}, {y1, y3}, {y2, y3}, {y2, y4}, {y3, y4}, {y1, y2, y3}, {y1, y3, y4}, {y2, y3, y4}}. The next proposition elucidates that the the class of L-θβλ-open sets is proper wider than the class of L-δβλ-open sets. Therefore, the class of L-θβλ-open sets is also wider than the classes of all L-λ-near open sets introduced in Definition 8 [21], i.e., L-βλ-open, L-Pλ-open, L-Sλ-open and L-αλ-open sets. Moreover, it is wider than the classes of δβλ-open sets. Hence, it is also wider than all classes of λ-near open sets introduced in Definition 5 [15], i.e., βλ-open, Pλ-open, Sλ-open and αλ-open sets. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3446 Proposition 7. The next implications hold true: ϑλ(Γλ) ⇒ L-αλO(L-αλC) L-PλO(L-PλC) ⇓ ⇓ L-SλO(L-SλC) ⇒ L-βλO(L-βλC) ⇒ L-δβλO(L-δβλC) ⇓ L-θβλO(L-θβλC). ϑλ(Γλ) ⇒ L-αλO(L-αλC) L-PλO(L-PλC) ⇓ ⇓ L-SλO(L-SλC) ⇒ L-βλO(L-βλC) ⇒ L- ∧ βλO (L- ∧ βλC ) ⇓ L-θβλO(L-θβλC). ϑλ(Γλ) ⇒ αλO(αλC) PλO(PλC) ⇓ ⇓ SλO(SλC) ⇒ βλO(βλC) ⇒ δβλO(δβλC) ⇓ L-θβλO(L-θβλC). ϑλ(Γλ) ⇒ αλO(αλC) PλO(PλC) ⇓ ⇓ SλO(SλC) ⇒ βλO(βλC) ⇒ ∧ βλO ( ∧ βλC ) ⇓ L-θβλO(L-θβλC). Proof. By Propositions 2 [23], 5, 6 the proof is obvious. Theorem 3. The union of two L-θβλ-open subsets is L-θβλ-open. That is, the family of L-θβλ-open subsets is closed under finite union. Proof. Take arbitrary two L-θβλ-open subsets V and W . Then, there are open sets G and H s.t. the four sets (V \ clλ(G)), (G \ clθλ(V )), (W \ clλ(H)) and (W − clθλ(B)) belong to L. Since (G\clθλ(V ∪W )) ⊆ (G\clθλ(V )) ∈ L, (H \clθλ(V ∪W )) ⊆ (H \clθλ(W )) ∈ L, we have (G\clθλ(V ∪W ))∪ (H \clθλ(V ∪W )) ∈ L. Let Z = G∪H, then (Z−clθλ(V ∪W )) ∈ L. Also, (V \ clλ(Z)) ⊆ (V \ clλ(G)) ∈ L and (W \ clλ(Z)) ⊆ (W \ clλ(H)) ∈ L. Then, (V \ clλ(Z))∪ (W \ clλ(Z)) ⊆ (V \ clλ(G))∪ (W \ clλ(H)) ∈ L and so ((V ∪W ) \ clλ(Z)) ⊆ (V \ clλ(G)) ∪ (W \ clλ(H)) ∈ L. Hence, V ∪W is an L-θβλ-open subset. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3447 In Algorithm 1, we present the steps to calculate the family of L-θβλ-open subsets. Input : The universal set X, a relation R, and an ideal L under consideration. Output: The family of L-θβλ-open subsets. 1 Ask the the expert(s) to give a relation L over X; 2 Choose a λ type; 3 for every y ∈ X do 4 compute Gλ(y) 5 end 6 Construct a topology ϑλ = {V ⊆ X : ∀y ∈ V,Gλ(y) ⊆ V } on X; 7 Initiate C1 = {clλ(V ) : V ∈ ϑλ}; 8 Construct a θ-topology ϑθλ = {V ∈ ϑλ : intθλ(V ) = V }; 9 Define F = P (X) \ ϑλ; 10 Initiate C2 = {clθλ(W ) :W ∈ F}; 11 Ask the the expert(s) to give an ideal L over X; 12 for every W ∈ F do 13 if ∃ V ∈ ϑλ s.t. (W \ clλ(V )) ∈ L and (V \ clθλ(W )) ∈ L then 14 W is an L-θβλ-open set; 15 W ∈ F∗ 16 else 17 W is not an L-θβλ-open set 18 end 19 end 20 L-θβλO(X) = ϑλ ∪ F∗. Algorithm 1: Determination of the family of L-θβλ-open subsets 4. Approximations spaces by using L-θβλ-open sets Herein, we establish novel rough paradigms inspired by the family of L-θβλ-open sets. We focus on the role of the proposed rough paradigms in developing decision-making meth- ods through the preservation of most properties of the standard model given by Pawlak and heighten the accuracy measures of extracted knowledge compared to paradigms stud- ied in the literature. Additionally, we make comparisons between the proposed models for all cases of λ with the assistance of counterexamples. Definition 18. Let V be a subset of an L − Gλ-space (X,R,Ξλ,L). We respectively define the L-θβλ-lower, L-θβλ-upper approximations, L-θβλ-boundary regions and L-θβλ- accuracy of V as follows: RL−θβ λ (V ) = ∪{G ∈ L-θβλO(X) : G ⊆ A} = L-θβλ-interior of V . RL−θβ λ (V ) = ∩{H ∈ L-θβλC(X) : V ⊆ H} = L-θβλ-closure of V . BNDL−θβ λ (V ) = RL−θβ λ (V )−RL−θβ λ (V ). ACCL−θβ λ (V ) = |RL−θβ λ (V )| |RL−θβ λ (V )| , where |RL−θβ λ (V )| ≠ 0. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3448 Proposition 8. Let V,W be subsets of an L −Gλ-space (X,R,Ξλ,L). Then, (i) RL−θβ λ (V ) ⊆ V ⊆ RL−θβ λ (V ) equality hold if V = ∅ or X. (ii) V ⊆W ⇒ RL−θβ λ (V ) ⊆ RL−θβ λ (W ). (iii) V ⊆W ⇒ RL−θβ λ (V ) ⊆ RL−θβ λ (W ). (iv) RL−θβ λ (V ∩W ) ⊆ RL−θβ λ (V ) ∩RL−θβ λ (W ). (v) RL−θβ λ (V ∪W ) ⊇ RL−θβ λ (V ) ∪RL−θβ λ (W ). (vi) RL−θβ λ (V ∪W ) ⊇ RL−θβ λ (V ) ∪RL−θβ λ (W ). (vii) RL−θβ λ (V ∩W ) ⊆ RL−θβ λ (V ) ∩RL−θβ λ (W ). (viii) RL−θβ λ (V ) = (RL−θβ λ (V ′ )) ′ , RL−θβ λ (V ) = (RL−θβ λ (V ′ )) ′ . (ix) RL−θβ λ (RL−θβ λ (V )) = RL−θβ λ (V ). (x) RL−θβ λ (RL−θβ λ (V )) = RL−θβ λ (V ). (xi) RL−θβ λ (RL−θβ λ (V )) ⊆ RL−θβ λ (RL−θβ λ (V )). (xii) RL−θβ λ (RL−θβ λ (V )) ⊆ RL−θβ λ (RL−θβ λ (V )). Proof. The proof is warranted by using the properties of L-θβλ-interior and L-θβλ- closure operators. Definition 19. A subset V of an L−Gλ-space (X,R,Ξλ,L) is named an L-θβλ-definable (an L-θβλ-exact) set if RL−θβ λ (V ) = RL−θβ λ (V ). Otherwise, V is an L-θβλ-rough set. In Example 1 V = {y3} is L-θβa-exact. To articulate the relationships between the present rough paradigms (Definition 18) and those given in Definition 9 [22, 23], we provide the next two results. Theorem 4. Let V be a subset of an L −Gλ-space (X,R,Ξλ,L). Then: (i) RL−P λ (V ) ⊆ RL−β λ (V ) ⊆ RL−δβ λ (V ) ⊆ RL−θβ λ (V ). (ii) RL−α λ (V ) ⊆ RL−S λ (V ) ⊆ RL−β λ (V ) ⊆ RL−δβ λ (V ) ⊆ RL−θβ λ (V ). (iii) RL−P λ (V ) ⊆ RL−β λ (V ) ⊆ RL− ∧ βλ(V ) ⊆ RL−θβλ(V ). (iv) RL−α λ (V ) ⊆ RL−S λ (V ) ⊆ RL−β λ (V ) ⊆ RL− ∧ βλ(V ) ⊆ RL−θβλ(V ). (v) RL−θβ λ (V ) ⊆ RL−δβ λ (V ) ⊆ RL−β λ (V ) ⊆ RL−P λ (V ). M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3449 (vi) RL−θβ λ (V ) ⊆ RL−δβ λ (V ) ⊆ RL−β λ (V ) ⊆ RL−S λ (V ) ⊆ RL−α λ (V ). (vii) RL−θβλ(V ) ⊆ RL− ∧ βλ(V ) ⊆ RL−β λ (V ) ⊆ RL−P λ (V ). (viii) RL−θβλ(V ) ⊆ RL− ∧ βλ(V ) ⊆ RL−β λ (V ) ⊆ RL−S λ (V ) ⊆ RL−α λ (V ). Proof. It is warranted by Proposition 3. Corollary 1. Let V be a subset of an L −Gλ-space (X,R,Ξλ,L). Then: (i) BNDL−θβ λ (V ) ⊆ BNDL−δβ λ (V ) ⊆ BNDL−β λ (V ) ⊆ BNDL−P λ (V ). (ii) BNDL−θβ λ (V ) ⊆ BNDL−δβ λ (V ) ⊆ BNDL−β λ (V ) ⊆ BNDL−S λ (V ) ⊆ BNDL−α λ (V ). (iii) BNDL−θβλ(V ) ⊆ BNDL− ∧ βλ(V ) ⊆ BNDL−β λ (V ) ⊆ BNDL−P λ (V ). (iv) BNDL−θβλ(V ) ⊆ BNDL− ∧ βλ(V ) ⊆ BNDL−β λ (V ) ⊆ BNDL−S λ (V ) ⊆ BNDL−α λ (V ). (v) ACCL−P λ (V ) ⩽ ACCL−β λ (V ) ⩽ ACCL−δβ λ (V ) ⩽ ACCL−θβ λ (V ). (vi) ACCL−α λ (V ) ⩽ ACCL−S λ (V ) ⩽ ACCL−β λ (V ) ⩽ ACCL−δβ λ (V ) ⩽ ACCL−θβ λ (V ). (vii) ACCL−P λ (V ) ⩽ ACCL−β λ (V ) ⩽ ACCL− ∧ βλ(V ) ⩽ ACCL−θβλ(V ). (viii) ACCL−α λ (V ) ⩽ ACCL−S λ (V ) ⩽ ACCL−β λ (V ) ⩽ ACCL− ∧ βλ(V ) ⩽ ACCL−θβλ(V ). Remark 2. By Example 1, we will illustrate that the converse of the implications in Theorem 4 and Corollary 1 is not always true as follows. (i) If V = {y5}, then RL−θβ a (V ) = A,RL−θβ a (V ) = A, BNDL−θβ a (V ) = ∅, ACCL−θβ a (V ) = 1, and RL−δβ a (V ) = ∅,RL−δβ a (V ) = A, BNDL−δβ a (V ) = A,ACCL−δβ a (V ) = 0. (ii) If V = {y}, then RL−θβ a (V ) = A,RL−θβ a (V ) = A,BNDL−θβ a (V ) = ∅, ACCL−θβ a (V ) = 1, and RL− ∧ βa(V ) = ∅,RL− ∧ βa(V ) = A,BNDL− ∧ βa(V ) = A,ACCL− ∧ βa(V ) = 0. Corollary 2. Let V be a subset of an L −Gλ-space (X,R,Ξλ,L). Then: (i) V is L-αλ-exact ⇒ V is L-Sλ-exact ⇒ V is L-βλ-exact ⇒ V is L-δβλ-exact ⇒ V is L-θβλ-exact. (ii) V is L-Pλ-exact ⇒ V is L-βλ-exact ⇒ V is L-δβλ-exact ⇒ V is L-θβλ-exact. (iii) V is λ-exact ⇒ V is L-αλ-exact ⇒ V is L-Sλ-exact ⇒ V is L-βλ-exact ⇒ V is L- ∧ βλ -exact ⇒ V is L-θβλ-exact. (iv) V is L-Pλ-exact ⇒ V is L-βλ-exact ⇒ V is L- ∧ βλ -exact ⇒ V is L-θβλ-exact. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3450 (v) V is L-θβλ-rough ⇒ V is L-δβλ-rough ⇒ V is L-βλ-rough ⇒ V is L-Sλ-rough ⇒ V is L-αλ-rough. (vi) V is L-θβλ-rough ⇒ V is L-δβλ-rough ⇒ V is L-βλ-rough ⇒ V is L-Pλ-rough. (vii) V is L-θβλ-rough ⇒ V is L- ∧ βλ -rough ⇒ V is L-βλ-rough ⇒ V is L-Sλ-rough ⇒ V is L-αλ-rough. (vii) V is L-θβλ-rough ⇒ V is L- ∧ βλ -rough ⇒ V is L-βλ-rough ⇒ V is L-Pλ-rough. Remark 3. By Example 1, we will illustrate that the converse of the implications in Corollary 2 fails. (i) If V = {y5}, then it is L-θβa-exact, but it is not L-δβa-exact and consequently, not L-βa-exact, not L-Sa-exact, not L-αa-exact and not L-Pa-exact. (ii) If V = {y}, then it is L-θβa-exact, but it is not L- ∧ βa-exact and consequently, not L-βa-exact, not L-Sa-exact, not L-αa-exact and not L-Pa-exact. We elucidate the interrelations between the present rough paradigms (Definition 18) and the those displayed in Definition 4 [45] and Definition 6 [15, 20]. Theorem 5. Let V be a subset of an L −Gλ-space (X,R,Ξλ,L). Then: (i) Rα λ(V ) ⊆ Rp λ(V ) ⊆ Rγ λ(V ) ⊆ Rβ λ(V ) ⊆ Rδβ λ (V ) ⊆ RL−θβ λ (V ). (ii) Rα λ(V ) ⊆ Rs λ(V ) ⊆ Rγ λ(V ) ⊆ Rβ λ(V ) ⊆ Rδβ λ (V ) ⊆ RL−θβ λ (V ). (iii) Rα λ(V ) ⊆ Rp λ(V ) ⊆ Rγ λ(V ) ⊆ Rβ λ(V ) ⊆ R ∧ β λ (V ) ⊆ RL−θβλ(V ). (iv) Rα λ(V ) ⊆ Rs λ(V ) ⊆ Rγ λ(V ) ⊆ Rβ λ(V ) ⊆ R ∧ βλ (V ) ⊆ RL−θβλ(V ). (v) Rλ(V ) ⊆ RL−θβ λ (V ). (vi) RL−θβ λ (V ) ⊆ Rδβ λ (V ) ⊆ Rβ λ(V ) ⊆ Rγ λ(V ) ⊆ Rp λ(V ) ⊆ Rα λ(V ). (vii) RL−θβ λ (V ) ⊆ Rδβ λ (V ) ⊆ Rβ λ(V ) ⊆ Rγ λ(V ) ⊆ Rs λ(V ) ⊆ Rα λ(V ). (viii) RL−θβλ(V ) ⊆ R ∧ β λ (V ) ⊆ Rβ λ(V ) ⊆ Rγ λ(V ) ⊆ Rp λ(V ) ⊆ Rα λ(V ). (ix) RL−θβλ(V )) ⊆ R ∧ β λ (V ) ⊆ Rβ λ(V ) ⊆ Rγ λ(V ) ⊆ Rs λ(V ) ⊆ Rα λ(V ). (x) RL−θβ λ (V ) ⊆ Rλ(V ). Proof. (i) By Theorem 2 [23], Rα λ(V ) ⊆ Rp λ(V ) ⊆ Rγ λ(V ) ⊆ Rβ λ(V ) ⊆ Rδβ λ (V ) and Rδβ λ (V )) = ∪{G ∈ δβλO(X) : G ⊆ A} ⊆ ∪{G ∈ L-θβλO(X) : G ⊆ A} = RL−θβ λ (V ) (by Proposition 6). M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3451 (ii)–(iv) It is similar to (i). (v) By Theorem 2 [23], Rλ(V ) ⊆ Rδβλ(V ), and by (1) Rδβλ(V ) ⊆ RL−θβ λ (V ). Hence, Rλ(V ) ⊆ RL−θβ λ (V ). (vi)–(x) They are similar to (i)–(v). The subsequent corollary points out that the greater the size of the boundary region, the lower the accuracy measures. Corollary 3. If V is a subset of an L−Gλ-space (X,R,Ξλ,L), then the next properties are satisfied. (i) BNDL−θβλ(V ) ⊆ BNDδβ λ (V ) ⊆ BNDβ λ(V ) ⊆ BNDγ λ(V ) ⊆ BNDp λ(V ) ⊆ BNDα λ(V ). (ii) BNDL−θβλ(V ) ⊆ BNDδβ λ (V ) ⊆ BNDβ λ(V ) ⊆ BNDγ λ(V ) ⊆ BNDs λ(V ) ⊆ BNDα λ(V ). (iii) BNDL−θβλ(V ) ⊆ BND ∧ β λ (V ) ⊆ BNDβ λ(V ) ⊆ BNDγ λ(V ) ⊆ BNDp λ(V ) ⊆ BNDα λ(V ). (iv) BNDL−θβλ(V ) ⊆ BND ∧ β λ (V ) ⊆ BNDβ λ(V ) ⊆ BNDγ λ(V ) ⊆ BNDs λ(V ) ⊆ BNDα λ(V ). (v) BNDL−θβ λ (V ) ⊆ BNDλ(V ). (vi) ACCαλ(V ) ⩽ ACCpλ(V ) ⩽ ACCγλ(V ) ⩽ ACCβλ(V ) ⩽ ACCδβλ (V ) ⩽ ACCL−θβ λ (V ). (vii) ACCαλ(V ) ⩽ ACCsλ(V ) ⩽ ACCγλ(V ) ⩽ ACCβλ(V ) ⩽ ACCδβλ (V ) ⩽ ACCL−θβ λ (V ). (viii) ACCαλ(V ) ⩽ ACCpλ(V ) ⩽ ACCγλ(V ) ⩽ ACCβλ(V ) ⩽ ACC ∧ β λ (V ) ⩽ ACC L−θβ λ (V ). (ix) ACCαλ(V ) ⩽ ACCsλ(V ) ⩽ ACCγλ(V ) ⩽ ACCβλ(V ) ⩽ ACC ∧ β λ (V ) ⩽ ACC L−θβ λ (V ). (x) ACCλ(V ) ⩽ ACCL−θβ λ (V ). Remark 4. The converse of the implications in Theorem 5 and Corollary 3 is not true in general as shown in (i) Example 2, if V = {y4}, then RL−θβ a (V ) = A,RL−θβ a (V ) = A, BNDL−θβ a (V ) = ∅, ACCL−θβ a(V ) = 1, and Rδβ a(V ) = ∅, Rδβ a(V ) = A and BNDδβ a(V ) = A,ACCδβa(V ) = 0. (ii) Example 3, if V = {y}, then RL−θβ a (V ) = A,RL−θβ a (V ) = A,BNDL−θβ a (V ) = ∅, ACCL−θβ a (V ) = 1, and RL− ∧ βa(V ) = ∅,RL− ∧ βa(V ) = A,BNDL− ∧ βa(V ) = A, ACCL− ∧ βa(V ) = 0. Corollary 4. For a subset V of an L−Gλ-space (X,R,Ξλ,L), we have the next results. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3452 (i) V is αλ-exact ⇒ V is Sλ-exact ⇒ V is βλ-exact ⇒ δβλ-exact ⇒ V is L-θβλ-exact. (ii) V is Pλ-exact ⇒ V is βλ-exact ⇒ V is δβλ-exact ⇒ V is L-θβλ-exact. (iii) V is αλ-exact ⇒ V is Sλ-exact ⇒ V is βλ-exact ⇒ V is ∧ βλ -exact ⇒ V is L-θβλ- exact. (iv) V is Pλ-exact ⇒ V is βλ-exact ⇒ V is ∧ βλ -exact ⇒ V is L-θβλ-exact. (v) V is λ-exact ⇒ V is L-θβλ-exact. (vi) V is L-θβλ-rough ⇒ V is δβλ-rough ⇒ V is βλ-rough ⇒ V is Sλ-rough ⇒ V is αλ-rough. (vii) V is L-θβλ-rough ⇒ V is δβλ-rough ⇒ V is βλ-rough ⇒ V is Pλ-rough. (viii) V is L-θβλ-rough ⇒ V is ∧ βλ -rough ⇒ V is βλ-rough ⇒ V is Sλ-rough ⇒ V is αλ-rough. (ix) V is L-θβλ-rough ⇒ V is ∧ βλ -rough ⇒ V is βλ-rough ⇒ V is Pλ-rough. (x) V is L-θβλ-rough ⇒ V is λ-rough. Remark 5. The converse of Corollary 4 is wrong in general. We demonstrate this claim in the following. (i) Example 2, if V = {y4}, then it is L-θβa-exact, but it is neither δβa-exact nor R-exact. (ii) Example 3, if V = {y}, then it is L-θβa-exact, but it is neither ∧ βa -exact nor a-exact. Remark 6. We can say that the present rough set models (Definition 18), with the com- parison of Abd El-Monsef et al.’s method 4 [45], Amer et al.’s method [15] and Hosny’s method 6 [20] and Hosny’s method 9 [22, 23], enlarge the confirmed knowledge by maxi- mizing the L-θβλ-lower approximations and minimizing the L-θβλ-upper approximations as illustrated in Theorems 4 and 5. That is, the present approach successfully shrinks the boundary region, which refer to size of ambiguity. Furthemore, Corollaries 3 and 1 con- firm that the our accuracy introduced in Definition 18 is greater than the previous ones in Definitions 4 [15], 6 [15, 20] and 9 [22, 23]. In Algorithm 2, we present the steps to calculate a subset’s boundary region and accuracy measure and determine whether an L-θβλ-definable set or an L-θβλ-rough set. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3453 Input : The universal set X, a relation R, and an ideal L under consideration. Output: Boundary region BNDL−θβ λ and accuracy measure ACCL−θβ λ of a subset. 1 Carry out steps 1–20 given in Algorithm 1; 2 Build L-θβλC(X) = {H ⊆ X : Hc ∈ L-θβλO(X)}; 3 for a subset E ⊆ X do 4 compute RL−θβ λ (E) = ∪{G ∈ L-θβλO(X) : G ⊆ E}; 5 compute RL−θβ λ (E) = ∩{H ∈ L-θβλC(X) : E ⊆ H}; 6 compute BNDL−θβ λ (E) = RL−θβ λ (E)−RL−θβ λ (E); 7 compute ACCL−θβ λ (E) = |RL−θβ λ (E)| |RL−θβ λ (E)| 8 end 9 Print BNDL−θβ λ (E); 10 Print ACCL−θβ λ (E); 11 if ACCL−θβ λ (E) = 1 then 12 Print E is an L-θβλ-definable set 13 else 14 Print E is an L-θβλ-rough set 15 end Algorithm 2: Calculate the boundary region and accuracy measure of a subset 5. L-θβλ-rough membership functions In this segment, we introduce the notion of L-θβλ-rough membership functions as a generalization of classical rough membership functions. We exploit this notion to describe the approximation operators given in the preceding section. Definition 20. Let (X,R,Ξλ,L) be an L −Gλ-space, y ∈ X, and V ⊆ X. (i) if y ∈ RL−θβ λ (V ), then y is λ-θβ-certainly with respect to L ( L−θβλ-certainly) belongs to V , denoted by y ∈L−θβλV . (ii) if y ∈ RL−θβ λ (V ), then y is λ-θβ-probably with respect to L (briefly L− θβλ-probably) belongs to V , denoted by y ∈L−θβλV . It is called λ-θβ-strong and λ-θβ-weak membership relations with respect to L respectively. Remark 7. According to Definition 18, the L-θβλ-lower and L-θβλ-upper approximations for any V ⊆ X can be written as: (i) RL−θβ λ (V ) = {y ∈ X : y ∈L−θβλV }. (ii) RL−θβ λ (V ) = {y ∈ X : y ∈L−θβλV }. M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3454 Lemma 2. Let (X,R,Ξλ,L) be an L −Gλ-space and V ⊆ X. Then (i) if y ∈L−θβλV , then y ∈ V . (ii) if y ∈ V , then y ∈L−θβλV . Proof. Straightforward. Proposition 9. Let (X,R,Ξλ,L) be an L −Gλ-space and V ⊆ X. Then (i) if y ∈λA⇒ y ∈ηλA⇒ y ∈L−η λ V ⇒ y ∈L−θβλV . (ii) if y ∈L−θβλA⇒ y ∈L−η λ A⇒ y ∈ηλA⇒ y ∈λV . Proof. We prove (i) and the other similarly. y ∈λA ⇒ y ∈ηλA ⇒ y ∈L−η λ V by Propo- sition 4. Let y ∈L−η λ V . Then, y ∈ RL−η λ (V ) ⇒ y ∈ RL−θβ λ (V )( by Proposition 4) ⇒ y ∈L−θβλV . Remark 8. The converse of Proposition 9 is not true in general, as it is shown in Example 1 (i) if V = {y5}, then y2 ∈L−θβaV , but y2 ∈L−δβaV . (ii) if V = {y1}, then y2 ∈L−θβaV , but y2 ∈L− ∧ βaV . Definition 21. Let (X,R,Ξλ) be a Gλ-space, L be an ideal on X,V ⊆ X and y ∈ X. The L − θβλ-rough membership functions of V are defined by µ L−θβλ V → [0, 1], where µ L−θβλ V (y) = {1 if 1∈ψL−θβλ V (y). min(ψ L−θβλ V (y)) otherwise. }. and ψ L−θβλ V (y) = |L−θβλ(y)∩V | |L−θβλ(y)| , y ∈ L − θβλ(y), L − θβλ(y) ∈ L-θβλO(X). Remark 9. The L − θβλ-rough membership functions are used to define the L-θβλ-lower and L-θβλ-upper approximations as follows: (i) RL−θβ λ (V ) = {y ∈ X : µ L−θβλ V (y) = 1}. (ii) RL−θβ λ (V ) = {y ∈ X : µ L−θβλ V (y) > 0}. (iii) BNDL−θβ λ (V ) = {y ∈ X : 0 < µ L−θβλ V (y) < 1}. Proposition 10. Let (X,R,Ξλ,L) be an L −Gλ-space and V,W ⊆ X. Then (i) if µ L−θβλ V (y) = 1 ⇔ y ∈L−θβλV . (ii) if µ L−θβλ V (y) = 0 ⇔ y ∈ X −RL−θβ λ (V ). (iii) if 0 < µ L−θβλ V (y) < 1 ⇔ y ∈ BNDL−θβ λ (V ). M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3455 (iv) if µ L−θβλ A′ (y) = 1− µ L−θβλ V (y), ∀ y ∈ X. (v) if µ L−θβλ V ∪B (y) ≥ max(µ L−θβλ V (y), µ L−θβλ B (y)), ∀ y ∈ X. (vi) if µ L−θβλ V ∩B (y) ≤ min(µ L−θβλ V (y), µ L−θβλ B (y)), ∀ y ∈ X. Proof. We prove (i), and the others similarly. y ∈L−θβλV ⇔ y ∈ RL−θβ λ (V ). Since RL−θβ λ (V ) is L− θβλ-open set contained in V , thus |RL−θβ λ (V )∩V | |RL−θβ λ (V )| = |RL−θβ λ (V )| |RL−θβ λ (V )| = 1. Then, 1 ∈ ψ L−θβλ V (y) and accordingly µ L−θβλ V (y) = 1. In the next, we prove an important result showing the interrelations between the relations of λ-rough membership [35] 12, λ-nearly rough membership [45] 13, λ-nearly rough membership w.r.t L [22, 23] 14, and L-θβλ-rough membership functions. Lemma 3. Let (X,R,Ξλ,L) be an L −Gλ-space and V ⊆ X. Then (i) µλV (y) = 1 ⇒ µηλV (y) = 1 ⇒ µL−ηλV (y) = 1 ⇒ µ L−θβλ V (y) = 1, ∀ y ∈ X. (ii) µλV (y) = 0 ⇒ µηλV (y) = 0 ⇒ µL−ηλV (y) = 0 ⇒ µ L−θβλ V (y) = 0, ∀ y ∈ X. Proof. (i) µλV (y) = 1 ⇒ µηλV (y) = 1 ⇒ µL−ηλV (y) = 1 directly from Lemma 1. Let µL−ηλV (y) = 1, then y ∈ RL−η λ (V ) ⇒ y ∈ RL−θβ λ (V ) ⇒ µ L−θβλ V (y) = 1,∀ y ∈ X. (ii) µλV (y) = 0 ⇒ µηλV (y) = 0 ⇒ µL−ηλV (y) = 0 directly from Lemma 1. Let µL−ηλV (y) = 0, then y ∈ X −RL−η λ (V ) ⇒ y ∈ X −RL−θβ λ (V ) ⇒ µ L−θβλ V (y) = 0, ∀ y ∈ X. Remark 10. By Example 1, one can see that the converse of Lemma 3 fails. Remark 11. According to Lemma 3, the current Definition 21 is also generalization of the approaches in [35] and 11 [42]. 6. Practical application We allocated this part to examine the proposed models to cope with a real situation in the field of Chemistry. We explain how our models improve the outcomes of generalized approximation spaces over the previous models displayed in [15, 22, 23, 45]. The authors of [19] presented information systems of amino acids (AAs) with some characterizations. To facilitate the mathematical computations, we shall select a sample of that information system as given in Table 1; that is, we choose data of five AAs, say, C = {y1, y2, y3, y4, y5} described by five attributes as follows ν1 is PIE, ν2 is surface area (SAC), ν3 is molecular refractivity (MR), ν4 is side chain polarity (LAM), and ν5 is molecular volume (Vol). M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3456 Table 1: Quantitative attributes of five amino acids. ν1 ν2 ν3 ν4 ν5 y1 0.23 254.2 2.126 -0.02 82.2 y2 -0.48 303.6 2.994 -1.24 112.3 y3 -0.61 287.9 2.994 -1.08 103.7 y4 0.45 282.9 2.933 -0.11 99.1 y5 -0.11 335.0 3.458 -0.19 127.5 Table 2: Ga of each element of C inspired by each relation Rk. G1a(yi) G2a(yi) G3a(yi) G4a(yi) G5a(yi) y1 {y1, y4} C C {y1, y4, y5} C y2 C {y2, y5} {y2, y3, y4, y5} C {y2, y5} y3 C {y2, y3, y4, y5} {y2, y3, y4, y5} C {y2, y3, y4, y5} y4 {y4} {y2, y3, y4, y5} {y2, y3, y4, y5} {y1, y4, y5} {y2, y3, y4, y5} y5 {y1, y4, y5} {y5} {y5} {y1, y4, y5} {y3, y5} Let us take relations on C as: Rk = {(yi, yj) : yi(νk) − yj(νk) < σyk 2 } for i, j, k = 1, 2, 3, 4, 5 s.t. σyk is the standard deviation of the quantitative attributes. The right neighbourhood Gka of each element of C generated by each one of these relations Rk is presented in Table 2. Now, we associate each element of C with all its Ga by the following relation Ha(yi) = 5⋂ k=1 Gka(yi). For the sake of brevity, we conduct the computation for four AAs, say, Y = C \ {y5} = {y1, y2, y3, y4}. Therefore, we first reduce Table 2 to Table 3. Table 3: Ga of each element of Y inspired by each relation Rk. G1a(yi) G2a(yi) G3a(yi) G4a(yi) G5a(yi) y1 {y1, y4} Y Y {y1, y4} Y y2 Y {y2} {y2, y3, y4} Y {y2} y3 Y {y2, y3, y4} {y2, y3, y4} Y {y2, y3, y4} y4 {y4} {y2, y3, y4} {y2, y3, y4} {y1, y4} {y2, y3, y4} Now, we associate each element of Y with all its Ga by the following relation Ha(yi) = 4⋂ k=1 Gka(yi). Accordingly, we obtain the following neighbourhoods: • Ha(y1) = {y1, y4}, M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3457 • Ha(y2) = {y2}, • Ha(y3) = {y2, y3, y4}, and • Ha(y4) = {y4}. Thus, the topology initiated by these neighbourhoods (using the formula ϑa = {V ⊆ Y : ∀y ∈ V,H(y) ⊆ V }) is: ϑa = {∅,Y, {y2}, {y4}, {y2, y4}, {y1, y4}, {y1, y2, y4}, {y2, y3, y4}}. The family of all β-open, δ-open and θ-open subsets of this topology respectively are: βaO(Y) = {∅,Y, {y2}, {y4}, {y2, y4}, {y1, y4}, {y2, y3}, {y3, y4}, {y1, y2, y4}, {y1, y3, y4}, {y2, y3, y4}}, δaO(Y) = {∅,Y, {y2}, {y1, y4}, {y1, y2, y4}}, and θaO(Y) = {∅,Y}. If we take L = {∅, {y1}} as an ideal structure on Y, then we find the following: • L-βaO(Y) = {∅,Y, {y2}, {y4}, {y1, y2}, {y2, y4}, {y1, y4}, {y2, y3}, {y3, y4}, {y1, y2, y4}, {y1, y3, y4}, {y2, y3, y4}, {y1, y2, y3}} = βaO(Y) ∪ {{y1, y2}, {y1, y2, y3}}, • L-δβaO(Y) = {∅,Y, {y1}, {y2}, {y4}, {y1, y2}, {y2, y4}, {y1, y4}, {y2, y3}, {y3, y4}, {y1, y2, y4}, {y1, y3, y4}, {y2, y3, y4}, {y1, y2, y3}} = L-βaO(Y) ∪ {{y1}, {y1, y3}}, and • L-θβaO(Y) = P (Y). M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3458 T ab le 4 : B o u n d ar y re g io n s an d ac cu ra cy m ea su re s ca lc u la te d w it h re sp ec t to A m er et al . [1 5 ] m et h o d , H o sn y [2 3 ] m et h o d , an d th e pr es en t m et h o d . M et h o d s A m er et al . m et h o d β a O (Y ) H os n y m et h o d s L -β a O (Y ) H os n y m et h o d s L -δ β a O (Y ) T h e p re se n t m et h o d L -θ β a O (Y ) V ⊆ Y B N D β a (V ) A C C β a (V ) B N D L − β a (V ) A C C L − β a (V ) B N D L − δ β a (V ) A C C L − δ β a (V ) B N D L − θ β a (V ) A C C L − θ β a (V ) {y 1 } {y 1 } 0 {y 1 } 0 ∅ 1 ∅ 1 {y 2 } ∅ 1 ∅ 1 ∅ 1 ∅ 1 {y 3 } {y 3 } 0 {y 3 } 0 {y 3 } 0 ∅ 1 {y 4 } {y 1 } 1/ 2 ∅ 1 ∅ 1 ∅ 1 {y 1 ,y 2 } {y 1 } 1/ 2 ∅ 1 ∅ 1 ∅ 1 {y 1 ,y 3 } {y 1 ,y 3 } 0 {y 1 ,y 3 } 0 {y 2 } 2/ 3 ∅ 1 {y 1 ,y 4 } ∅ 1 ∅ 1 ∅ 1 ∅ 1 {y 2 ,y 3 } ∅ 1 ∅ 1 ∅ 1 ∅ 1 {y 2 ,y 4 } {y 1 ,y 3 } 1/ 4 {y 1 ,y 3 } 1 /2 ∅ 1 ∅ 1 {y 3 ,y 4 } {y 1 } 1/ 3 ∅ 1 ∅ 1 ∅ 1 {y 1 ,y 2 ,y 3 } {y 1 } 2/ 3 ∅ 1 ∅ 1 ∅ 1 {y 1 ,y 2 ,y 4 } {y 3 } 3/ 4 {y 3 } 3 /4 {y 3 } 3 /4 ∅ 1 {y 1 ,y 3 ,y 4 } ∅ 1 ∅ 1 ∅ 1 ∅ 1 {y 2 ,y 3 ,y 4 } {y 1 } 3/ 4 {y 1 } 3 /4 ∅ 1 ∅ 1 M. Hosny, T.M. Al-shami / Eur. J. Pure Appl. Math, 17 (4) (2024), 3436-3463 3459 According to the computations of boundary regions and accuracy measures of subsets displayed in Table 4, we remark the following points: there are different techniques intro- duced in the literature to approximate subsets using some forms of subsets of topological spaces. Our rough approximation space minimizes the upper approximation and maxi- mizes the lower approximation, which leads to downsizing (or removing) the boundary regions. As a result, it outperforms other rough models given in the published litera- ture like [15, 22, 23, 45], which makes it the most refined technique. For instance, the above table shows that a subset {y3} is considered a rough set according to the models of [15, 22, 23, 45], whereas this subset and all other subsets are exact according to the model investigated herein. This observation confirms that the current model is more beneficial for coping with real-life scenarios since it extracts a greater amount of information and reduces data ambiguity. Furthermore, the proposed paradigm adheres to most properties of Pawlak’s model without any restrictions, as demonstrated in Proposition 8. In this regard, We empha- size that the methodology of using nearly open sets in topology can achieve some or all properties of the Pawlak model, depending on the frameworks these families of subsets form, whether they are topology, supra topology, infra topology, or minimal structures. To elucidate this point, we note that the family of α-open sets constitutes a topology. Thus, rough set models inspired by this family will fulfill all the properties of the Pawlak model. In contrast, the family of semi-open sets constitutes a supra topology. Therefore, rough set models inspired by this family lose some properties of the Pawlak model related to the distribution of the union and intersection operators to the upper and lower approx- imations, respectively. On the other hand, we find that families that do not achieve all the properties of the Pawlak model expand the confirmed knowledge and produce a greater accuracy measure than those families that achieve all the properties of the Pawlak model. 7. Conclusions The notion of rough neighborhoods was introduced in the literature with the aim of removing the strict term of an equivalence relation that limited the application of the classical rough set models. Such rough neighborhoods have shown to be useful in several applications. Some formulas have been proposed to institute a topology from these neigh- borhoods making topological spaces a vital instrument to represent rough approximation operators and analyze information systems. One of the important topological tools to reduce the vagueness of knowledge is nearly open sets. Despite this tool being applied by many researchers, there remain other types that should be investigated. This work goes along with this line of research. We have studied generalized ap- proximation spaces using the ideas of L-θβλ-open sets and ideal structures. We have explored their structural properties and pointed out the importance of the present models in maximizing the domain of confirmed information and minimizing the boundary region of uncertainty. Therefore, this work is a foundation for handling complicated paradigms in decision-making. We also showed the superiority of the proposed rough paradigms over different kinds of preceding paradigms induced by some nearly open sets. To facilitate the REFERENCES 3460 way of specifying the family of L-θβλ-open sets and exploring whether a subset is L-θβλ- definable or L-θβλ-rough, we have initiated two algorithms. 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