Processing math: 100%
Research article

θβ-ideal approximation spaces and their applications

  • Commentary on: AIMS Mathematics 7: 2479-2497
  • Received: 16 July 2021 Accepted: 28 October 2021 Published: 12 November 2021
  • MSC : 54A05, 54C10

  • The essential aim of the current work is to enhance the application aspects of Pawlak rough sets. Using the notion of a j-neighborhood space and the related concept of θβ-open sets, different methods for generalizing Pawlak rough sets are proposed and their characteristics will be examined. Moreover, in the context of ideal notion, novel generalizations of Pawlak's models and some of their generalizations are presented. Comparisons between the suggested methods and the previous approximations are calculated. Finally, an application from real-life problems is proposed to explain the importance of our decision-making methods.

    Citation: Ashraf S. Nawar, Mostafa A. El-Gayar, Mostafa K. El-Bably, Rodyna A. Hosny. θβ-ideal approximation spaces and their applications[J]. AIMS Mathematics, 2022, 7(2): 2479-2497. doi: 10.3934/math.2022139

    Related Papers:

    [1] İbrahim Aktaş . On some geometric properties and Hardy class of q-Bessel functions. AIMS Mathematics, 2020, 5(4): 3156-3168. doi: 10.3934/math.2020203
    [2] Orhan Göçür . Amply soft set and its topologies: AS and PAS topologies. AIMS Mathematics, 2021, 6(4): 3121-3141. doi: 10.3934/math.2021189
    [3] Mohammad H. M. Rashid, Feras Bani-Ahmad . An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality. AIMS Mathematics, 2023, 8(11): 26384-26405. doi: 10.3934/math.20231347
    [4] Raziuddin Siddiqui . Geometry of configurations in tangent groups. AIMS Mathematics, 2020, 5(1): 522-545. doi: 10.3934/math.2020035
    [5] Jinsheng Guo, Shuang-Ming Wang . Threshold dynamics of a time-periodic two-strain SIRS epidemic model with distributed delay. AIMS Mathematics, 2022, 7(4): 6331-6355. doi: 10.3934/math.2022352
    [6] Chen Lang, Hongyan Guan . Common fixed point and coincidence point results for generalized α-φE-Geraghty contraction mappings in b-metric spaces. AIMS Mathematics, 2022, 7(8): 14513-14531. doi: 10.3934/math.2022800
    [7] Mona Hosny . Rough sets theory via new topological notions based on ideals and applications. AIMS Mathematics, 2022, 7(1): 869-902. doi: 10.3934/math.2022052
    [8] Chenghua Gao, Maojun Ran . Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition. AIMS Mathematics, 2020, 5(2): 904-922. doi: 10.3934/math.2020062
    [9] Shoubin Sun, Lingqiang Li, Kai Hu, A. A. Ramadan . L-fuzzy upper approximation operators associated with L-generalized fuzzy remote neighborhood systems of L-fuzzy points. AIMS Mathematics, 2020, 5(6): 5639-5653. doi: 10.3934/math.2020360
    [10] Vediyappan Govindan, Inho Hwang, Choonkil Park . Hyers-Ulam stability of an n-variable quartic functional equation. AIMS Mathematics, 2021, 6(2): 1452-1469. doi: 10.3934/math.2021089
  • The essential aim of the current work is to enhance the application aspects of Pawlak rough sets. Using the notion of a j-neighborhood space and the related concept of θβ-open sets, different methods for generalizing Pawlak rough sets are proposed and their characteristics will be examined. Moreover, in the context of ideal notion, novel generalizations of Pawlak's models and some of their generalizations are presented. Comparisons between the suggested methods and the previous approximations are calculated. Finally, an application from real-life problems is proposed to explain the importance of our decision-making methods.



    Rough set philosophy [1] deals with ambiguity. In fact, this methodology was based on an equivalence relation which limited the application fields. Accordingly, to extend the application scope of these approaches, the equivalence relation is generalized to a rough set model constructed by any binary relation. Numerous proposals established in this line [231]. Nowadays, there are many applications of topology and its extensions, for instance [5,11,13,17,32,33].

    Abd El-Monsef et al. [19] presented a structure to generalize the standard rough set concept. In fact, they presented the notion of a j-neighborhood space (briefly, j-NS) constructed by a binary relation. Moreover, they used a general topology generated from the binary relation to set up generalized rough sets. This methodology paved the way for extra topological presentations in the rough set contexts and helped to formalize various applications from daily-life problems. After then, Amer et al. [20] applied some near open sets in a j-NS, and thus they managed to in generating new generalized rough set approximations, namely j-near approximations. In 2018, Hosny [21] extended these approximations to different approximations using the concepts of δβ-open and β-open sets. Using the interesting notion "Ideal", Hosny [22] proposed another idea for generalizing Pawlak's theory. She has introduced the ideas of Ij-lower and Ij-upper approximations as extensions for Pawlak's approximations and several of their generalizations.

    Ideal represents an important notion in topological spaces and has a vital part in the study of topological problems. This concept is very interesting in rough set theory since it is considered as a bridge between rough sets and topological structures. Moreover, the ideal can be considered as a class of some objects in the information system that has some conditions and the expert wants to study and make a new granulation for the data collected from real-life problem according to this class. Few authors have used the term "ideal" to describe and produce non-granular rough approximations over general approximation spaces in recent years. These relation-based rough set studies have aimed to obtain fine properties analogous to those of classical rough approximations. The authors severely generalize these ideas in this research paper, and they investigate associated semantic features. Granules are used implicitly in the building of approximations, and thus the idea of θβj-ideal approximations are introduced. In other words, we suggested two different methods depending on topological concepts, and hence the significance of these approaches was that they were based on ideals, which were topological tools, and the two proposed methods represent two opinions rather than one. Therefore, these techniques open the way for more topological applications in the rough set context.

    The fundamental contributions of the present work are to introduce different extensions of Pawlak's rough context and its generalizations. In fact, we suggest two different types to extend and strengthen the already methods in the literature (namely, Abd El-Monsef et al. [19], Amer et al. [20], and Hosny [21,22] techniques). Moreover, we illustrate that the proposed methods have extended the application fields in real-life problems and help in extracting the data in the information system.

    This paper is organized into five different sections besides the introduction and the conclusion. Section 2 is devoted to state the main concepts, results, and methods that are used throughout the paper. Additionally, we give a summary of previous methods which are compared with our approaches. By using the new notion of θβj-open sets, we proposed a new technique to generalize Pawlak's rough models and some of its generalizations in section 3. The central properties of this method are examined and compared to Abd El-Monsef et al. [19], Amer et al. [20], and Hosny [21,22] methods. Additionally, many results with counterexamples are investigated to reason that the suggested method is stronger than the other methods. In setion 4, we present and examine the models of θβj-ideal lower and θβj-ideal upper approximations for any subset. Some properties of these operators will be examined and demonstrated that θβj-ideal approximations are more accurate than the other methods. Finally, in section 5, a practical example is presented to clarify the importance of the proposed techniques in decision-making. Besides, the proposed methods are compared with the previous methodologies. From real-life problems, we use an information system (with decision attributes). This system depends on a binary relation which means that Pawlak's rough sets can't be applied here. So, we apply our methods and will show that the recommended methods are stronger than the other methods. Therefore, we can say that our methods extend the application fields for rough sets from a topological point of view.

    In the next, we give the basic ideas and consequences used in current research.

    Definition 2.1. [34] A non-empty class I of subsets of a set X is called an ideal on X, if it satisfies the subsequent conditions

    1) If AI and BA, then BI (hereditary),

    2) If AI and BI, then ABI (finite additivity).

    Definition 2.2. [19] Suppose that R is an arbitrary binary relation on a non-empty finite set X. Therefore, the j-neighborhoods of xX (Symbolically, Nj(x), j{r,, r,,i,u,i,u}), are given by:

    (i) Right neighborhood (briefly, r-neighborhood):

    Nr(x)={yX:xRy}

    (ii) Left neighborhood (briefly, -neighborhood):

    N(x)={yX:yRx}

    (iii) Minimal right neighborhood (briefly, r-neighborhood):

    Nr(x)=xNr(y)Nr(y)

    (iv) Minimal left neighborhood (briefly, -neighborhood):

    N(x)=xN(y)N(y).

    (v) Intersection of right and left neighborhoods (briefly, i-neighborhood):

    Ni(x)=Nr(x)N(x).

    (vi) Union of right and left neighborhoods (briefly, u-neighborhood):

    Nu(x)=Nr(x)N(x).

    (vii) Intersection of minimal right and minimal left neighborhoods (briefly, i-neighborhood):

    Ni(x)=Nr(x)N(x).

    (viii) Union of minimal right and minimal left neighborhoods (briefly, u-neighborhood):

    Nu(x)=Nr(x)N(x).

    Definition 2.3. [19] Suppose that R is an arbitrary binary relation on a non-empty finite set X and ψj:XP(X) is a mapping which assigns an j-neighborhood for each xX in P(X). The triple (X,R,ψj) is named a j-neighborhood space (in briefly, j-NS).

    Theorem 2.1. [19] If (X,R,ψj) is a j-NS, then j{r,,r,,u,i,u,i} the family τj={AX:pA,Nj(p)A}forms a topology on X.

    Definition 2.4. [19] Let (X,R,ψj) be a j-NS. A subset AX is said to be a j-open set if Aτj, the complement of a j-open set is called a j-closed set. A class Γj of all j-closed sets is given by Γj={FX:Fcτj}, such that Fc represents a complement of F.

    Definition 2.5. [19] Let (X,R,ψj) be a j-NS, and AX. Then, j{r,,r,,u,i,u,i}, we define the j-lower and the j-upper approximations, the j-boundary regions and the j-accuracy of approximations of A, respectively, by:

    R_j(A)={Gτj:GA}=intj(A), where intj(A) represents j-interior of A.

    ¯Rj(A)={FΓj:FA}=clj(A), where clj(A) represents j-closure of A.

    Bj(A)=¯Rj(A)R_j(A).

    σj(A)=|R_j(A)||¯Rj(A)|, where |¯Rj(A)|0.

    Definition 2.6. [19] Suppose that (X,R,ψj) is a j-NS, and AX, then for each j{r,,r,,u,i,u,i}, the subset Ais named a j-exact set if R_j(A)=¯Rj(A)=A. Else, Ais called a j-rough set.

    Definition 2.7. [20] Suppose that (X,R,ψj) is a j-NS. A subset AX is named

    (1) j-regular open (Rj-open), if A=intj(clj(A));

    (2) j-preopen (Pj-open), if Aintj(clj(A));

    (3) j-semi open (Sj-open), if Aclj(intj(A));

    (4) γj-open (bj-open), if Aintj(clj(A))clj(intj(A));

    (5) αj-open, if Aintj[clj(intj(A))];

    (6) βj-open (semi preopen), if Aclj[intj(clj(A))];

    (7) δβj-open, if Aclj[intj(clδj(A))].

    Remark 2.1. [20]

    (i) The previous types of sets are called j-near open sets and the families of j-near open sets of X symbolized by KjO(X), K{R,P,S,γ,α,β,δβ}.

    (ii) The complements of the j-near open sets are called j-near closed sets and the families of j-near closed sets of X symbolized by KjC(X), K{R,P,S,γ,α,β,δβ}.

    Definition 2.8. [21] Let (X,R,ψj) be a j-NS, and AX. A subset βj(A) is assumed as follows: βj(A)={G:AG,GβjO(X)}. The complement of βj(A)-set is called βj(A)-set.

    Definition 2.9. [21] Let (X,R,ψj) be a j-NS, and AX. A subset A is said to be a βj-set if A=βj(A). The family of all βj-sets and βj-sets are symbolized by βjO(X) and βjC(X), respectively.

    Definition 2.10. [20,21] Let (X,R,ψj) be a j-NS, and AX. Then, for each j{r,,r,,u,i,u,i} and K{R,P,S,γ,α,β,δβ,β}, the j-near lower, j-near upper approximations, j-near boundary regions and j-near accuracy of the approximations of A are assumed respectively by:

    R_Kj(A)={GKjO(X):GA}.
    ¯RKj(A)={FKjC(X):FA}.
    BKj(A)=¯RKj(A)R_Kj(A).
    σKj(A)=|R_Kj(A)||¯RKj(A)|,where|¯RKj(A)|0.

    Definition 2.11. [20,21] Let (X,R,ψj) be a j-NS, and AX. Then, for each j{r,,r,,u,i,u,i} and K{R,P,S,γ,α,β,δβ,β}, A is called a j-near definable (j-near exact) set if R_Kj(A)=¯RKj(A). Else, A is called a j-near rough set.

    Proposition 2.1. [20,21] If (X,R,ψj) is a j-NS, and AX, then for each j{r,,r,,u,i,u,i} and K{P,S,γ,α,β,δβ,β}, KR:

    R_j(A)R_Kj(A)A¯RKj(A)¯Rj(A).

    Proposition 2.2. [20,21] Consider(X,R,ψj) is a j-NS, and AX. Then, the following properties are satisfied:

    1)R_αj(A)R_Pj(A)R_γj(A)R_βj(A)R_δβj(A).2)R_αj(A)R_Pj(A)R_γj(A)R_βj(A)R_βj(A).3)R_αj(A)R_Sj(A)R_γj(A)R_βj(A)R_δβj(A).4)R_αj(A)R_Sj(A)R_γj(A)R_βj(A)R_βj(A).5)¯Rδβj(A)¯Rβj(A)¯Rγj(A)¯RPj(A)¯Rαj(A).6)¯Rβj(A)¯Rβj(A)¯Rγj(A)¯RPj(A)¯Rαj(A).7)¯Rδβj(A)¯Rβj(A)¯Rγj(A)¯RSj(A)¯Rαj(A).8)¯Rβj(A)¯Rβj(A)¯Rγj(A)¯RSj(A)¯Rαj(A).

    Proposition 2.3. [20,21] Let (X,R,ψj) be a j-NS, and AX. Then, the following statements are verified:

    1)Bδβj(A)Bβj(A)Bγj(A)BPj(A)Bαj(A).2)Bβj(A)Bβj(A)Bγj(A)BPj(A)Bαj(A).3)Bδβj(A)Bβj(A)Bγj(A)BSj(A)Bαj(A).4)Bβj(A)Bβj(A)Bγj(A)BSj(A)Bαj(A).5)σαj(A)σPj(A)σγj(A)σβj(A)σδβj(A).6)σαj(A)σPj(A)σγj(A)σβj(A)σβj(A).7)σαj(A)σSj(A)σγj(A)σβj(A)σδβj(A).8)σαj(A)σSj(A)σγj(A)σβj(A)σβj(A).

    Theorem 2.2. [22] Assume that (X,R,ψj) is a j-NS, AX and I is an ideal on X, then for each j{r,,r,,u,i,u,i} the collection τIj={AX:pA,Nj(p)AcI}is a topology on X.

    Definition 2.12. [22] Let (X,R,ψj) be a j-NS, and I be an ideal on X. A subset AX is named an Ij-open set if AτIj, the complement of an Ij-open set is named an Ij-closed set. A family TIj of all Ij-closed sets is given by TIj={FX:FcτIj}, such that Fc represents a complement of F.

    Definition 2.13. [22] Consider (X,R,ψj) is a j-NS, I is an ideal on X and AX. j{r,,r,,u,i,u,i}, the Ij-lower, Ij-upper approximations, Ij-boundary regions and Ij-accuracy of the approximations of A are assumed, respectively, by:

    R_Ij(A)={OτIj:OA}=intIj(A),whereintIj(A)representsIjinteriorofA.
    ¯RIj(A)={FTIj:FA}=clIj(A),whereclIj(A)representsIjclosureofA.
    BIj(A)=¯RIj(A)R_Ij(A).
    σIj(A)=|R_Ij(A)||¯RIj(A)|,where|¯RIj(A)|0.

    A new method for defining generalized rough sets using the idea of θβj-open sets is proposed. The current method's properties are investigated and compared to those of Abd El-Monsef et al. [19], Amer et al. [20], and Hosny [21].

    Definition 3.1. Consider (X,R,ψj) is a j-NS, AX and for each j{r,,r,,u,i,u,i}. The θj-closure of A is defined by clθj(A)={xX:Aclj(G),GτjandxG}. Moreover, A is called a θj-closed if A=clθj(A). The complement of a θj-closed set is θj-open. Note that: intθj(A)=Xclθj(XA).

    Definition 3.2. Consider (X,R,ψj) is aj-NS and AX. A subset A is called a θβj-open, if Aclj[intj(clθj(A))]. A is a θβj-closed set if its complement is a θβj-open set and the family of all θβj-open and θβj-closed sets are symbolized by θβjO(X) and θβjC(X), respectively.

    Example 3.1. Let X={a,b,c,d,e} and ={(a,a),(a,e),(b,c),(b,d),(b,e),(c,c),(c,d),(d,c), (d,d),(e.e)} be a binary relation given on X. Then, we get

    Nr(a)={a,e},Nr(b)={c,d,e},Nr(c)=Nr(d)={c,d},Nr(e)={e}.

    Therefore, the topology associated with this relation is τr={X,,{e},{a,e},{c,d},{c,d,e},{a,c,d,e},{b,c,d,e}, and hence we obtain θβrO(X)=P(X).

    The next results demonstrate that θβj-open sets are more accurate than δβj-open sets and βj-open sets. Therefore, θβj-open sets are stronger than other j-near open sets such as Rj -open, Pj-open, Sj-open, γj-open, αj-open and βj-open sets. Moreover, by appling the characteristics of the definitions ofj-interior, j-closure, δj-closure and θj-closure operators, it is easy to prove these results, so we omit the proof.

    Proposition 3.1. Each δβj-open set is θβj-open.

    Proposition 3.2. Each βj-open set is θβj-open.

    Remark 3.1. The reverse of Propositions 3.1 and 3.2 is not essentially correct as showing in Example 3.1.

    Definition 3.3. Let (X,R,ψj) be a j-NS, and AX. Then, the θβj-lower, θβj-upper approximations, θβj-boundary regions and θβj-accuracy of the approximations of A are defined, respectively, by:

    R_θβj(A)={GθβjO(X):GA}=θβjinteriorofA.
    ¯Rθβj(A)={FθβjC(X):FA}=θβjclosureofA.
    Bθβj(A)=¯Rθβj(A)R_θβj(A).
    σθβj(A)=|R_θβj(A)||¯Rθβj(A)|,where|¯Rθβj(A)|0.

    To illustrate the connections among the existing approaches in Definition 3.3 and the preceding one in Definitions 2.5 [19] and 2.10 [20,21], we present the following results.

    Theorem 3.1. Let (X,R,ψj) be a j-NS, and AX. Then, the next statements are verified:

    1)R_αj(A)R_Pj(A)R_γj(A)R_βj(A)R_δβj(A)R_θβj(A).2)R_αj(A)R_Pj(A)R_γj(A)R_βj(A)R_βj(A)R_θβj(A).3)R_αj(A)R_Sj(A)R_γj(A)R_βj(A)R_δβj(A)R_θβj(A).4)R_αj(A)R_Sj(A)R_γj(A)R_βj(A)R_βj(A)R_θβj(A).5)R_j(A)R_θβj(A).6)¯Rθβj(A)¯Rδβj(A)¯Rβj(A)¯Rγj(A)¯RPj(A)¯Rαj(A).7)¯Rθβj(A)¯Rβj(A)¯Rβj(A)¯Rγj(A)¯RPj(A)¯Rαj(A).8)¯Rθβj(A)¯Rδβj(A)¯Rβj(A)¯Rγj(A)¯RSj(A)¯Rαj(A).9)¯Rθβj(A)¯Rβj(A)¯Rβj(A)¯Rγj(A)¯RSj(A)¯Rαj(A).10)¯Rθβj(A)¯Rj(A).

    Proof.

    1) R_αj(A)R_Pj(A)R_γj(A)R_βj(A)R_δβj(A), by Proposition 2.2 (1) and R_δβj(A)={GδβjO(X):GA}{GθβjO(X):GA}=R_θβj(A) (Proposition 3.1).

    2) R_αj(A)R_Pj(A)R_γj(A)R_βj(A)R_βj(A), by Proposition 2.2 (2) and R_βj(A)={GβjO(X):GA}{GθβjO(X):GA}=R_θβj(A) (Proposition 3.2).

    3) Similar to (1).

    4) Similar to (2).

    5) By Proposition 2.1, R_jR_Kj(A), and K{P,S,γ,α,β,δβ,β}, such that KR, and from (1) R_Kj(A)R_θβj(A). Therefore, R_jR_θβj(A).

    (6)-(9) Similar to (1) and (2).

    10) By Proposition 2.1, ¯RKj(A)¯Rj(A), and K{P,S,γ,α,β,δβ,β}, KR, and by (6) ¯Rθβj(A)¯RKj(A). Therefore, ¯Rθβj(A)¯Rj(A).

    Corollary 3.1. If (X,R,ψj) is a j-NS, and AX, then the following statements are satisfied:

    1)Bθβj(A)Bδβj(A)Bβj(A)Bγj(A)BPj(A)Bαj(A).2)Bθβj(A)Bβj(A)Bβj(A)Bγj(A)BPj(A)Bαj(A).3)Bθβj(A)Bδβj(A)Bβj(A)Bγj(A)BSj(A)Bαj(A).4)Bθβj(A)Bβj(A)Bβj(A)Bγj(A)BSj(A)Bαj(A).5)Bθβj(A)Bj(A).6)σαj(A)σPj(A)σγj(A)σβj(A)σδβj(A)σθβj(A).7)σαj(A)σPj(A)σγj(A)σβj(A)σβj(A)σθβj(A).8)σαj(A)σSj(A)σγj(A)σβj(A)σδβj(A)σθβj(A).9)σαj(A)σSj(A)σγj(A)σβj(A)σβj(A)σθβj(A).10)σj(A)σθβj(A).

    Remark 3.2. According to Theorem 3.1, we noted that the proposed technique decreases the boundary region by enhancing the θβj-lower approximation and reducing the θβj-upper approximation comparing them with the methods of Abd El-Monsef et al. (in Definition 2.5), Amer et al. and Hosny (in Definition 2.10). Furthermore, the accuracy that given by Definition 3.3 is higher than the other accuracies in Definitions 2.5 and 2.10. In Example 3.1 as demonstrated by Corollary 3.1. To this end, we compute the approximations, boundary regions, and the accuracy measure using the proposed method in Definition 3.3 and compare them with the previous techniques "Abd El-Monsef et al., Amer et al., Hosny" as exposed in Table 1.

    Table 1.  Comparison of the boundary and accuracy methods of Abd El-Monsef et al. [19], Amer et al. [20], and Hosny [21], as well as the existing technique (Definition 3.3).
    AX Abd El Monsef et al. method Amer et al. method Hosny methods The current method
    Br(A) σr(A) Bβr(A) σβr(A) Bδβr(A) σδβr(A) Bβr(A) σβr(A) Bθβr(A) σθβr(A)
    {a} {a} 0 {a} 0 1 {a} 0 1
    {b} {b} 0 {b} 0 {b} 0 1 1
    {c} {b,c,d} 0 1 1 1 1
    {d} {b,c,d} 0 1 1 1 1
    {e} {a,b} 1/3 {a} 1/2 1 {a} 1/2 1
    {a,b} {a,b} 0 {a,b} 0 1 {a} 1/2 1
    {a,c} {a,b,c,d} 0 {a} 1/2 1 {a} 1/2 1
    {a,d} {a,b,c,d} 0 {a} 1/2 1 {a} 1/2 1
    {a,e} {b} 2/3 1 1 1 1
    {b,c} {b,c,d} 0 1 1 1 1
    {b,d} {b,c,d} 0 1 1 1 1
    {b,e} {a,b} 1/3 {a} 2/3 1 {a} 2/3 1
    {c,d} {b} 2/3 1 1 1 1
    {c,e} {a,b,c,d} 1/5 {a} 2/3 1 {a} 2/3 1
    {d,e} {a,b,c,d} 1/5 {a} 2/3 1 {a} 2/3 1
    {a,b,c} {a,b,c,d} 0 {a} 2/3 1 {a} 2/3 1
    {a,b,d} {a,b,c,d} 0 {a} 2/3 1 {a} 2/3 1
    {a,b,e} {b} 2/3 1 1 1 1
    {a,c,d} {a,b} 1/2 {a} 2/3 1 {a} 2/3 1
    {a,c,e} {b,c,d} 2/5 1 1 1 1
    {a,d,e} {b,c,d} 2/5 1 1 1 1
    {b,c,d} {b} 2/3 1 1 1 1
    {b,c,e} {a,b,c,d} 1/5 {a} 3/4 1 {a} 3/4 1
    {b,d,e} {a,b,c,d} 1/5 {a} 3/4 1 {a} 3/4 1
    {c,d,e} {a,b} 3/5 {a,b} 3/5 1 {a} 3/4 1
    {a,b,c,d} {a,b} 1/2 {a} 3/4 1 {a} 3/4 1
    {a,b,c,e} {b,c,d} 2/5 1 1 1 1
    {a,b,d,e} {b,c,d} 2/5 1 1 1 1
    {a,c,d,e} {b} 4/5 {b} 4/5 {b} 4/5 1 1
    {b,c,d,e} {a} 4/5 {a} 4/5 1 {a} 4/5 1
    U 1 1 1 1 1

     | Show Table
    DownLoad: CSV

    We present the ideas of θβj-ideal lower and θβj-ideal upper approximations for any subset in this section. Some of the related characteristics of them will be examined; demonstrating that the θβj-ideal approximations represent the finest ones and precise than the other approaches.

    Definition 4.1. If (X,R,ψj) is aj-NS, I be an ideal on X. Then a subset A in X is called an I-θβj-open, if Aclj[intj(clθj(A))]. A complement of an I-θβj-open set is an I-θβj-closed. A family of all I-θβj-open and I-θβjclosed sets are denoted by IθβjO(X) and IθβjC(X), respectively.

    Note that: clθj(A)=AAθj, where Aθj={xX:Aclj(G)I,GτjandxG}.

    Definition 4.2. Consider (X,R,ψj) is a j-NS, I is an ideal on X and AX. For all j{r,,r,,u,i,u,i}, I-θβj-lower, I-θβj-upper approximations, I-θβj-boundary regions and I-θβj-accuracy of the approximations of A are defined respectively, by:

    R_Iθβj(A)={GIθβjO(X):GA}=intIθβj(A),whereintIθβj(A)representsI-θβjinteriorofA.
    ¯RIθβj(A)={FIθβjC(X):FA}=clIθβj(A),whereclIθβj(A)representsIθβjclosureofA.
    BIθβj(A)=¯RIθβj(A)R_Iθβj(A).
    σIθβj(A)=|R_Iθβj(A)||¯RIθβj(A)|,where|¯RIθβj(A)|0.

    The following proposition proposes the essential properties of the existing I-θβj-lower and I-θβj-upper approximations.

    Proposition 4.1. Let (X,R,ψj) be a j-NS, I be an ideal on X and A,BX. Then:

    (i)R_Iθβj(A)A¯RIθβj(A).(ii)R_Iθβj()=¯RIθβj()=andR_Iθβj(X)=¯RIθβj(X)=X(iii)IfAB,thenR_Iθβj(A)R_Iθβj(B)and¯RIθβj(A)¯RIθβj(B)(iv)R_Iθβj(A)R_Iθβj(B)R_Iθβj(AB)(v)¯RIθβj(A)¯RIθβj(B)¯RIθβj(AB)(vi)R_Iθβj(AB)R_Iθβj(A)R_Iθβj(B)(vii)¯RIθβj(AB)¯RIθβj(A)¯RIθβj(B)(viii)R_Iθβj(Ac)=(¯RIθβj(A))c(ix)¯RIθβj(Ac)=(R_Iθβj(A))c(x)R_Iθβj(R_Iθβj(A))=R_Iθβj(A)(xi)¯RIθβj(¯RIθβj(A))=¯RIθβj(A)(xii)R_Iθβj(R_Iθβj(A))¯RIθβj(R_Iθβj(A))(xiii)R_Iθβj(¯RIθβj(A))¯RIθβj(¯RIθβj(A))

    The proof of this proposition is simple using the properties of I-θβj-interior and I-θβj-closure operators, so we omit it.

    Remark 4.1. In the next example, we explain that the relation of implication in parts (i), (iv), (v), (vi), (vii), (xii), and (xiii) of Proposition 4.1 cannot be replaced by equality relation:

    Example 4.1. Let X={a,b,c,d,e}, R={(a,a),(a,e),(b,a),(b,c),(b,d),(b,e), (c,c),(c,d), (d,c),(d,d),(e.e)} and I={,{a},{c},{a,c}}.

    Thus Nr(a)={a,e}, Nr(b)={a,c,d,e}, Nr(c)=Nr(d)={c,d}, Nr(e)={e}.

    Then, I-θβrO(X)={X,,{b},{d},{e},{a,b},{a,e},{b,c},{b,d},{b,e},{c,d},{d,e},{a,b,c}, {a,b,d},{a,b,e},{a,d,e},{b,c,d},{b,c,e},{b,d,e},{c,d,e},{a,b,c,d},{a,b,c,e},{a,b,d,e}, {a,c,d,e},{b,c,d,e}}.

    Thus we have

    (1) For part (i), assume that A={a}. Therfore, R_Iθβr(A)= and thus AR_Iθβr(A). Also, if A={b,d}, then ¯RIθβr(A)={b,c,d} and hence ¯RIθβr(A)A.

    (2) For part (iv), assume that A={a} and B={b}. Therfore, R_Iθβr(A)=, R_Iθβr(B)={b}, and R_Iθβj(AB)={a,b}.

    (3) For part (v), assume that A={b} and B={d}. Therfore, ¯RIθβr(A)={b}, ¯RIθβr(B)={d}, and ¯RIθβr(AB)={b,c,d}.

    (4) For part (vi), if A={a,b} and B={a,e}, then R_Iθβr(A)={a,b}, R_Iθβr(B)={a,e}, and R_Iθβj(AB)=.

    (5) For part (vii), if A={b,c} and B={b,d}, then ¯RIθβr(A)={b,c}, ¯RIθβr(B)={b,c,d}, and ¯RIθβr(AB)={b}.

    (6) For part (xii), if A={b,d}, then R_Iθβr(R_Iθβr(A))={b,d} and ¯RIθβj(R_Iθβj(A))={b,c,d}, and therefore ¯RIθβj(R_Iθβj(A))R_Iθβj(R_Iθβj(A)).

    (7) For part (xiii), if A={c}, then ¯RIθβj(¯RIθβj(A))={c} and R_Iθβj(¯RIθβj(A))=, and therefore ¯RIθβj(¯RIθβj(A))R_Iθβj(¯RIθβj(A)).

    Definition 4.3. Let (X,R,ψj) be a j-NS, I be an ideal on X. The subset Ain X is named an I-θβj-definable (I-θβj-exact) set if ¯RIθβj(A)=R_Iθβj(A). Else, A is an I-θβj-rough set.

    Note that: In Example 4.1, A={b} is an I-θβr-exact set, while B={a} is an I-θβr-rough set.

    Remark 4.2. Consider (X,R,ψj) is a j-NS, I is an ideal on X, and AX. Then a finite intersection of two I-θβr-rough sets essentially not to be an I-θβr-rough set as in Example 4.1 {a,b} and {a,e} are I-θβr-rough sets, {a,b}{a,e}={a} is not an I-θβr-rough set.

    The subsequent theorem and its corollaries explain the relations amongst the present approximations in Definition 4.2 and the others in Definitions 2.5 [21] and 2.13 [22].

    Theorem 4.1. Consider(X,R,ψj) is a j-NS, I is an ideal on X, and AX. Then:

    (i)R_j(A)R_Ij(A)R_Iθβj(A).
    (ii)¯RIθβj(A)¯RIj(A)¯Rj(A).

    Proof:

    (i)R_j(A)={Gτj:GA}{GτIj:GA}=R_Ij(A){GIθβjO(X):GA}=R_Iθβj(A).

    (ii) Similar to (i).

    Corollary 4.1. Assume that (X,R,ψj) is a j-NS, I is an ideal on X, and AX. Then:

    (i)BIθβj(A)BIj(A)Bj(A).
    (ii)σj(A)σIj(A)σIθβj(A).

    Corollary 4.2. Suppose that (X,R,ψj) is a j-NS, I is an ideal on X, and AX. Then:

    (1) Each j-exact subset in X is I-θβj-exact.

    (2) Each I-j-exact subset in X is I-θβj-exact.

    (3) Each I-θβj-rough subset in X is j-rough.

    (4) Each I-θβj-rough subset in X is I-j-rough.

    Remark 4.3. Example 4.1 confirms that the opposite of parts of Corollary 4.2 is not necessarily true.

    (1) if A={b}, then it is I-θβr-exact, but it is not r-exact.

    (2) if A={d}, then it is I-θβr-exact, but it is not I-r-exact.

    (3) if A={e}, then it is r- rough, but it is not I-θβr- rough.

    (4) if A={a,b}, then it is I-r-rough, but it is not I-θβr- rough.

    For example, take A={d}: By using the present technique in Definition 4.2, the boundary and accuracy of A are and 1 respectively. Whereas, the boundary and accuracy by using Abd El-Monsef et al.'s method in Definition 2.5 are {b,c,d} and 0 respectively, and by using Hosny method in Definition 2.13 are {b,c} and 1/3 respectively.

    The next propositions and corollaries show some the relationships among the I-θβj-lower, I-θβj-upper approximations, I-θβj-boundary regions and I-θβj-accuracy.

    Proposition 4.2. Assume that(X,R,ψj) is a j-NS, I is an ideal on X, and AX. Then, the following properties are true.

    (i)R_Iθβu(A)R_Iθβr(A)R_Iθβi(A).
    (ii)R_Iθβu(A)R_Iθβl(A)R_Iθβi(A).
    (iii)R_Iθβ<u>(A)R_Iθβ<r>(A)R_Iθβ<i>(A).
    (iv)R_Iθβ<u>(A)R_Iθβ<l>(A)R_Iθβ<i>(A).

    Now, using Example 4.1, Table 2 represents a comparison between the boundary regions and the accuracy of the approximations using different methods.

    Table 2.  A comparison between the boundary regions and the accuracy of approximations, in Example 4.1, using different methods.
    AX Abd El-Monsef method Hosny method Current method
    Br(A) σr(A) BIr(A) σIr(A) BIθβr(A) σIθβr(A)
    {a} {a,b} 0 {a} 0 {a} 0
    {b} {b} 0 {b} 0 1
    {c} {b,c,d} 0 {c} 0 {c} 0
    {d} {b,c,d} 0 {b,c} 1/3 1
    {e} {a,b} 1/3 {a,b} 1/3 1
    {a,b} {a,b} 0 {a,b} 0 1
    {a,c} {a,b,c,d} 0 {a,c} 0 {a,c} 0
    {a,d} {a,b,c,d} 0 {a,b,c} 1/4 {a} 1/2
    {a,e} {b} 2/3 {b} 2/3 1
    {b,c} {b,c,d} 0 {b,c} 0 1
    {b,d} {b,c,d} 0 {b,c} 1/3 {c} 2/3
    {b,e} {a,b} 1/3 {a,b} 1/3 {a} 2/3
    {c,d} {b} 2/3 {b} 2/3 1
    {c,e} {a,b,c,d} 1/5 {a,b,c} 1/4 {c} 1/2
    {d,e} {a,b,c,d} 1/5 {a,b,c} 2/5 1
    {a,b,c} {a,b,c,d} 0 {a,b,c} 0 1
    {a,b,d} {a,b,c,d} 0 {a,b,c} 1/4 {c} 3/4
    {a,b,e} {b} 2/3 {b} 2/3 1
    {a,c,d} {a,b} 1/2 {a,b} 1/2 {a} 2/3
    {a,c,e} {b,c,d} 2/5 {b,c} 1/2 {c} 2/3
    {a,d,e} {b,c,d} 2/5 {b,c} 3/5 1
    {b,c,d} {b} 2/3 {b} 2/3 1
    {b,c,e} {a,b,c,d} 1/5 {a,b,c} 1/4 {a} 3/4
    {b,d,e} {a,b,c,d} 1/5 {a,c} 3/5 {a,c} 3/5
    {c,d,e} {a,b} 3/5 {a,b} 3/5 1
    {a,b,c,d} {a,b} 1/2 {a,b} 1/2 1
    {a,b,c,e} {b,c,d} 2/5 {b,c} 1/2 1
    {a,b,d,e} {b,c,d} 2/5 {c} 4/5 {c} 4/5
    {a,c,d,e} {b} 4/5 {b} 4/5 1
    {b,c,d,e} {a,b} 3/5 {a} 4/5 {a} 4/5
    X 1 1 1

     | Show Table
    DownLoad: CSV

    Proposition 4.3. Suppose that (X,R,ψj) is a j-NS, I is an ideal on X, and AX. Then, j{r,,r,,u,i,u,i}, the following statements are valid.

    (i)¯RIθβi(A)¯RIθβr(A)¯RIθβu(A).
    (ii)¯RIθβi(A)¯RIθβl(A)¯RIθβu(A).
    (iii)¯RIθβ<i>(A)¯RIθβ<r>(A)¯RIθβ<u>(A).
    (iv)¯RIθβ<i>(A)¯RIθβ<l>(A)¯RIθβ<u>(A).

    The proof of Propositions 4.2 and 4.3 was omitted since it is obvious.

    Corollary 4.3. Let (X,R,ψj) be a j-NS, I be an ideal on X, and AX. Then, j{r,,r,,u,i,u,i}, we have

    (i)BIθβi(A)BIθβr(A)BIθβu(A).
    (ii)BIθβi(A)BIθβl(A)BIθβu(A).
    (iii)BIθβ<i>(A)BIθβ<r>(A)BIθβ<u>(A).
    (iv)BIθβ<i>(A)BIθβ<l>(A)BIθβ<u>(A).

    Corollary 4.4. Consider (X,R,ψj) is a j-NS, I is an ideal on X, and AX. Then, j{r,,r,,u,i,u,i}, we have

    (i)σIθβu(A)σIθβr(A)σIθβi(A).
    (ii)σIθβu(A)σIθβl(A)σIθβi(A).
    (iii)σIθβ<u>(A)σIθβ<r>(A)σIθβ<i>(A).
    (iv)σIθβ<u>(A)σIθβ<l>(A)σIθβ<i>(A).

    Here, a practical example in the arena of chemistry is provided, using the current method in "Definition 4.2" to explain the concepts with an illustrative manner. The main goal is to present the benefits of the proposed methodologies for improving the accuracy measure. This is evident from the comparisons provided between our approaches and previous methods such as those presented via Abd El-Monsef et al. and M. Hosny.

    Example 5.1. [35,36] Suppose that H={c1,c2,c3,c4,c5} represents five amino acids (AAs). The (AAs) are described in terms of five attributes: a1 = PIE, a2 = SAC = surface area, a3 = MR = molecular refractivity, a4 = LAM = the side chain polarity and a5 = Vol = molecular volume as showed in Table 3.

    Table 3.  Lists all of the quantitative characteristics of the five AAs.
    a1 a2 a3 a4 a5
    c1 0.23 254.2 2.126 0.02 82.2
    c2 0.48 303.6 2.994 1.24 112.3
    c3 0.61 287.9 2.994 1.08 103.7
    c4 0.45 282.9 2.993 0.11 99.1
    c5 0.11 335.0 3.458 0.19 127.5

     | Show Table
    DownLoad: CSV

    Assume the following five reflexive relations on H are defined:

    Rs={(cu,cv)H×H:cu(as)cv(as)<σs2,u,v,s=1,2,,5}.

    Where σs is the standard deviation of the quantitative attributes as, and s=1,2,,5. The right neighbourhoods for all elements cu of H where u=1,2,,5, with respect to the relations Rs shown in Table 4.

    Table 4.  Right Neighborhood of five relations.
    cuR1 cuR2 cuR3 cuR4 cuR5
    c1 {c1,c4} H H {c1,c4,c5} H
    c2 H {c2,c5} {c2,c3,c4,c5} H {c2,c5}
    c3 H {c2,c3,c4,c5} {c2,c3,c4,c5} H {c2,c3,c4,c5}
    c4 {c4} {c2,c3,c4,c5} {c2,c3,c4,c5} {c1,c4,c5} {c2,c3,c4,c5}
    c5 {c1,c4,c5} {c5} {c5} {c1,c4,c5} {c3,c5}

     | Show Table
    DownLoad: CSV

    As a result, in order to demonstrate the set of all condition attributes, we compute the intersections of all right neighborhoods for each element of H as follows:

    c1R=5s=1c1Rs={c1,c4}, c2R=5s=1c2Rs={c2,c5}, c3R=5s=1c3Rs={c2,c3,c4,c5}, c4R=5s=1c4Rs={c4} and c4R=5s=1c4Rs={c5}.

    Hence, the topology generated by these right neighborhoods is given by:

    τr={H,,{c4},{c5},{c1,c4},{c2,c5},{c4,c5},{c1,c4,c5},{c2,c4,c5},{c1,c2,c4,c5},{c2,c3,c4,c5}}.

    By Chemistry's expert, if I={,{c1},{c4},{c1,c4}} is the selected ideal, then the topology generated by this ideal is:

    τIr={H,,{c1},{c4},{c5},{c1,c4},{c1,c5},{c2,c5},{c4,c5},{c1,c2,c5},{c1,c4,c5},{c2,c3,c5},
    {c2,c4,c5},{c1,c2,c3,c5},{c1,c2,c4,c5},{c2,c3,c4,c5}}.

    Therefore, I-θβrO(X)={H,,{c2},{c3},{c4},{c5},{c1,c3}{c1,c4},{c2,c3},{c2,c4},{c2,c5},{c3,c4},{c3,c5},{c4,c5},{c1,c2,c3},{c1,c2,c4},{c1,c3,c4},{c1,c3,c5},{c1,c4,c5},{c2,c3,c4},{c2,c3,c5},{c2,c4,c5},{c3,c4,c5},{c1,c2,c3,c4},{c1,c2,c3,c5},{c1,c2,c4,c5},{c1,c3,c4,c5},{c2,c3,c4,c5}}.

    Now, Table 5 presents a comparison between the boundary (resp. accuracy) using the current technique in Definition 4.2 and the other approaches.

    Table 5.  A comparison between the boundary and accuracy using the techniques of Abd El-Monsef et al. [19], Hosny [22] and the proposed technique in Definition 4.2.
    AH Abd El-Monsef method Hosny method Current method
    Br(A) σr(A) BIr(A) σIr(A) BIθβr(A) σIθβr(A)
    {c1} {c1} 0 1 {c1} 0
    {c2} {c2,c3} 0 {c2,c3} 0 1
    {c3} {c3} 0 {c3} 0 1
    {c4} {c1,c3} 1/3 1 1
    {c5} {c2,c3} 1/3 {c2,c3} 1/3 1
    {c1,c2} {c1,c2,c3} 0 {c2,c3} 1/3 {c1} 1/2
    {c1,c3} {c1,c3} 0 {c3} 1/2 1
    {c1,c4} {c3} 2/3 1 1
    {c1,c5} {c1,c2,c3} 1/4 {c2,c3} 1/2 {c1} 1/2
    {c2,c3} {c2,c3} 0 {c2,c3} 0 1
    {c2,c4} {c1,c2,c3} 1/4 {c2,c3} 1/3 1
    {c2,c5} {c3} 2/3 {c3} 2/3 1
    {c3,c4} {c1,c3} 1/3 {c3} 1/2 {c1} 2/3
    {c3,c5} {c2,c3} 1/3 {c2,c3} 1/3 1
    {c4,c5} {c1,c2,c3} 2/5 {c2,c3} 1/2 1
    {c1,c2,c3} {c1,c2,c3} 0 {c2,c3} 1/3 1
    {c1,c2,c4} {c2,c3} 1/2 {c2,c3} 1/2 1
    {c1,c2,c5} {c1,c3} 1/2 {c3} 3/4 {c1} 2/3
    {c1,c3,c4} {c3} 2/3 {c3} 2/3 1
    {c1,c3,c5} {c1,c2,c3} 1/4 {c2,c3} 1/2 1
    {c1,c4,c5} {c2,c3} 3/5 {c2,c3} 3/5 1
    {c2,c3,c4} {c1,c2,c3} 1/4 {c2,c3} 1/3 {c1} 3/4
    {c2,c3,c5} {c3} 2/3 1 1
    {c2,c4,c5} {c1,c3} 3/5 {c3} 3/4 1
    {c3,c4,c5} {c1,c2,c3} 2/5 {c2,c3} 1/2 {c1} 3/4
    {c1,c2,c3,c4} {c2,c3} 1/2 {c2,c3} 1/2 1
    {c1,c2,c3,c5} {c1,c3} 1/2 1 1
    {c1,c2,c4,c5} {c3} 4/5 {c3} 4/5 1
    {c1,c3,c4,c5} {c2,c3} 3/5 {c2,c3} 3/5 1
    {c2,c3,c4,c5} {c1} 4/5 1 {c1} 4/5

     | Show Table
    DownLoad: CSV

    Observation: From the previous comparisons in Table 5, we note the following:

    1)   There are several approaches to approximate the sets. The finest of these approaches is that there are assumed by using Iθβj-approximations of the current methods in Definition 4.2 for creating the rough approximations because the boundary regions in these cases are minimized (or removed) by maximizing the lower approximation and minimizing the upper approximation. Moreover, the accuracy degree, in these cases, is more accurate than the other types. For example, all proper subsets are rough in Abd El-Monsef's approaches. But there are many Iθβj- exact sets in the current methods.

    2)   The suggested method in Definition 4.2 is more accurate and stronger than M. Hosny's methods. Therefore, the suggested methodologies will be useful in decision-making for extracting the information and help in eliminating the ambiguity of the data in real-life problems.

    3)   The significance of the suggested approximations is not only that it is decreasing or deleting the boundary regions, but also, it's satisfying all characteristics of Pawlak's model without any restrictions as shown in Proposition 4.1.

    In our daily life, we often face some problems that necessitate complete decision-making. However, in the majority of these cases, we get perplexed as to the best solution. To find the most feasible solution to these problems, we must consider some solution-related parameters. One of the most important topics in rough sets is minimizing the boundary region, which aims to maximize the degree of decision-making accuracy. Topological structures formed by relations are one technique used to accomplish this goal. In this article, using the notion of a j-neighborhood space and the related concept of θβ-open sets, different methods for generalizing Pawlak rough sets and some of their enhancements have been proposed and their properties have been studied. Additionally, in the context of ideal notion, other generalizations to Pawlak's models and some of their enhancements such as (Abd El-Monsef et al. [19], Amer et al. [20], and Hosny [21,22] techniques) have been presented. Comparisons were made between the proposed methods and previous approaches published in the literature. Furthermore, numerous results have been proposed to explain why our approximations are more accurate and powerful than other methods.

    Finally, an application from Chemistry was proposed to demonstrate the significance of our decision-making methods. Moreover, it provides a comparison between the proposed methods with already existing in the literature. Also, this application proved that the suggested methods improve the accuracy measure which is useful in establishing an accurate decision. So, we can say that the proposed techniques may be useful in applications. In the future, we will apply the proposed techniques in more real-life applications.

    The authors would like to thank the referees and editor for their insightful comments and suggestions, which aided in the improvement of this paper. Furthermore, they would like to express their heartfelt gratitude to all Tanta Topological Seminar colleagues "Under the leadership of Prof. Dr. A. M. Kozae" for their interest, ongoing encouragement, and lively discussions. Furthermore, we would like to dedicate this work to the memory of Prof. Dr. M. E. Abd El-Monsef "God's mercy," who died in 2014.

    The authors declare that they have no conflicts of interest.



    [1] Z. Pawlak, Rough sets, Int. J. Inform. Comput. Sci., 11 (1982), 341-356. doi: 10.1007/BF01001956. doi: 10.1007/BF01001956
    [2] R. Slowinski, D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE T. Data En., 12 (2000), 331-336. doi: 10.1109/69.842271. doi: 10.1109/69.842271
    [3] E. A. Abo-Tabl, A comparison of two kinds of definitions of rough approximations based on a similarity relation, Inform. Sci., 181 (2011), 2587-2596. doi: 10.1016/j.ins.2011.01.007. doi: 10.1016/j.ins.2011.01.007
    [4] K. Y. Qin, J. L. Yang, Z. Pei, Generalized rough sets based on reflexive and transitive relations, Inform. Sci., 178 (2008), 4138-4141. doi: 10.1016/j.ins.2008.07.002. doi: 10.1016/j.ins.2008.07.002
    [5] M. Kondo, On the structure of generalized rough sets, Inform. Sci., 176 (2006), 589-600. doi: 10.1016/j.ins.2005.01.001. doi: 10.1016/j.ins.2005.01.001
    [6] Y. Y. Yao, Two views of the theory of rough sets in finite universes, Int. J. Approx. Reason., 15 (1996), 291-317. doi: 10.1016/S0888-613X(96)00071-0. doi: 10.1016/S0888-613X(96)00071-0
    [7] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for basic rough set concepts, In: International workshop on rough sets, fuzzy sets, data mining, and granular computing, Lecture Notes in Artificial Intelligence, Berlin, Heidelberg: Springer, 2005. doi: 10.1007/11548669_7.
    [8] M. K. El-Bably, T. M. Al-shami, Different kinds of generalized rough sets based on neighborhoods with a medical application, Int. J. Biomath., 14 (2021), 2150086. doi: 10.1142/S1793524521500868. doi: 10.1142/S1793524521500868
    [9] R. Abu-Gdairi, M. A. El-Gayar, M. K. El-Bably, K. K. Fleifel, Two different views for generalized rough sets with applications, Mathematics, 9 (2021), 2275. doi: 10.3390/math9182275. doi: 10.3390/math9182275
    [10] Z. M. Yu, X. L. Bai, Z. Q. Yun, A study of rough sets based on 1-neighborhood systems, Inform. Sci., 248 (2013), 103-113. doi: 10.1016/j.ins.2013.06.031. doi: 10.1016/j.ins.2013.06.031
    [11] M. K. El-Bably, E. A. Abo-Tabl, A topological reduction for predicting of a lung cancer disease based on generalized rough sets, J. Intell. Fuzzy Syst., 41 (2021), 3045-3060. doi: 10.3233/JIFS-210167. doi: 10.3233/JIFS-210167
    [12] Y. Y. Yao, Three-way decision and granular computing, Int. J. Approx. Reason., 103 (2018), 107-123. doi: 10.1016/j.ijar.2018.09.005. doi: 10.1016/j.ijar.2018.09.005
    [13] M. El Sayed, M. A. El Safety, M. K. El-Bably, Topological approach for decision-making of COVID-19 infection via a nano-topology model, AIMS Mathematics, 6 (2021), 7872-7894. doi: 10.3934/math.2021457. doi: 10.3934/math.2021457
    [14] M. E. Abd El-Monsef, M. A. EL-Gayar, R. M. Aqeel, On relationships between revised rough fuzzy approximation operators and fuzzy topological spaces, Int. J. Granul. Comput. Rough Sets Intell. Syst., 3 (2014), 257-271.
    [15] M. K. El-Bably, T. M. Al-shami, A. S. Nawar, A. Mhemdi, Corrigendum to "Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space", J. Intell. Fuzzy Syst., 2021, 1-9. doi: 10.3233/JIFS-211198. doi: 10.3233/JIFS-211198
    [16] M. K. El-Bably, K. K. Fleifel, O. A. Embaby, Topological approaches to rough approximations based on closure operators, Granul. Comput., 2021. doi: 10.1007/s41066-020-00247-x. doi: 10.1007/s41066-020-00247-x
    [17] B. K. Tripathy, A. Mitra, Some topological properties of rough sets and their applications, Int. J. Granul. Comput. Rough Sets Intell. Syst., 1 (2010), 355-369.
    [18] A. S. Nawar, Approximations of some near open sets in ideal topological spaces, J. Egypt. Math. Soc., 28 (2020), 5. doi: 10.1186/s42787-019-0067-0. doi: 10.1186/s42787-019-0067-0
    [19] M. E. Abd El-Monsef, O. A. Embaby, M. K. El-Bably, Comparison between rough set approximations based on different topologies, Int. J. Granul. Comput. Rough Sets Intell. Syst., 3 (2014), 292-305.
    [20] W. S. Amer, M. I. Abbas, M. K. El-Bably, On j-near concepts in rough sets with some applications, J. Intell. Fuzzy Syst., 32 (2017), 1089-1099. doi: 10.3233/JIFS-16169. doi: 10.3233/JIFS-16169
    [21] M. Hosny, On generalization of rough sets by using two different methods, J. Intell. Fuzzy Syst., 35 (2018), 979-993. doi: 10.3233/JIFS-172078. doi: 10.3233/JIFS-172078
    [22] M. Hosny, Idealization of j-approximation spaces, Filomat, 34 (2020), 287-301. doi: 10.2298/FIL2002287H. doi: 10.2298/FIL2002287H
    [23] M. E. Abd El-Monsef, M. A. EL-Gayar, R. M. Aqeel, A comparison of three types of rough fuzzy sets based on two universal sets, Int. J. Mach. Learn. Cyber., 8 (2017), 343-353. doi: 10.1007/s13042-015-0327-8. doi: 10.1007/s13042-015-0327-8
    [24] W. H. Xu, W. X. Zhang, Measuring roughness of generalized rough sets induced by a covering, Fuzzy Set. Syst., 158 (2007), 2443-2455. doi: 10.1016/j.fss.2007.03.018. doi: 10.1016/j.fss.2007.03.018
    [25] M. E. Abd El-Monsef, A. M. Kozae, M. K. El-Bably, On generalizing covering approximation space, J. Egypt. Math. Soc., 23 (2015), 535-545. doi: 10.1016/j.joems.2014.12.007. doi: 10.1016/j.joems.2014.12.007
    [26] A. S. Nawar, M. K. El-Bably, A. A. El-Atik, Certain types of coverings based rough sets with application, J. Intell. Fuzzy Syst., 39 (2020), 3085-3098. doi: 10.3233/JIFS-191542. doi: 10.3233/JIFS-191542
    [27] Y. R. Syau, E. B. Lin, Neighborhood systems and covering approximation spaces, Knowl.-Based Syst., 66 (2014), 61-67. doi: 10.1016/j.knosys.2014.04.017. doi: 10.1016/j.knosys.2014.04.017
    [28] F. F. Zhao, L. Q. Li, Axiomatization on generalized neighborhood system-based rough sets, Soft Comput., 22 (2018), 6099-6110. doi: 10.1007/s00500-017-2957-0. doi: 10.1007/s00500-017-2957-0
    [29] W. Yao, Y. H. She, L. X. Lu, Metric-based L-fuzzy rough sets: Approximation operators and definable sets, Knowl.-Based Syst., 163 (2019), 91-102. doi: 10.1016/j.knosys.2018.08.023. doi: 10.1016/j.knosys.2018.08.023
    [30] W. Yao, X. Q. Chen, Fuzzy partition and fuzzy rough approximation operators, J. Liaocheng Univ., 33 (2020), 1-4.
    [31] H. C. Lu, A. M. Khalil, W. Alharbi, M. A. El-Gayar, A new type of generalized picture fuzzy soft set and its application in decision making, J. Intell. Fuzzy Syst., 40 (2021), 12459-12475. doi: 10.3233/JIFS-201706. doi: 10.3233/JIFS-201706
    [32] H. M. Abu-Donia, A. S. Salama, Generalization of Pawlaks rough approximation spaces by using δβ-open sets, Int. J. Approx. Reason., 53 (2012), 1094-1105. doi: 10.1016/j.ijar.2012.05.001. doi: 10.1016/j.ijar.2012.05.001
    [33] T. M. Al-Shami, B. A. Asaad, M.A. El-Gayar, Various types of supra pre-compact and supra pre-Lindelöf spaces, Missouri J. Math. Sci., 32 (2020), 1-20. doi: 10.35834/2020/3201001. doi: 10.35834/2020/3201001
    [34] D. Jankovic, T. R. Hamlet, New topologies from old via ideals, Amer. Math. Monthly, 97 (1990), 295-310. doi: 10.1080/00029890.1990.11995593. doi: 10.1080/00029890.1990.11995593
    [35] N. E. Tayar, R. S. Tsai, P. A. Carrupt, B. Testa, Octan-1-ol-water partition coefficients of zwitterionic α-amino acids. Determination by centrifugal partition chromatography and factorization into steric/hydrophobic and polar components, J. Chem. Soc. Perkin Trans. 2, 1992, 79-84. doi: 10.1039/P29920000079. doi: 10.1039/P29920000079
    [36] B. Walczak, D. L. Massart, Rough sets theory, Chemometr. Intell. Lab. Syst., 47 (1999) 1-16. doi: 10.1016/S0169-7439(98)00200-7. doi: 10.1016/S0169-7439(98)00200-7
  • This article has been cited by:

    1. Tareq M. Al-shami, Topological approach to generate new rough set models, 2022, 8, 2199-4536, 4101, 10.1007/s40747-022-00704-x
    2. Mostafa A. El-Gayar, Abd El Fattah El Atik, Dan Selisteanu, Topological Models of Rough Sets and Decision Making of COVID-19, 2022, 2022, 1099-0526, 1, 10.1155/2022/2989236
    3. Tareq M. Al-shami, Amani Rawshdeh, Heyam H. Al-jarrah, Abdelwaheb Mhemdi, Connectedness and covering properties via infra topologies with application to fixed point theorem, 2023, 8, 2473-6988, 8928, 10.3934/math.2023447
    4. Mostafa K. El-Bably, Radwan Abu-Gdairi, Mostafa A. El-Gayar, Medical diagnosis for the problem of Chikungunya disease using soft rough sets, 2023, 8, 2473-6988, 9082, 10.3934/math.2023455
    5. El-Sayed A. Abo-Tabl, Mostafa K. El-Bably, Rough topological structure based on reflexivity with some applications, 2022, 7, 2473-6988, 9911, 10.3934/math.2022553
    6. Rukhsana Kausar, Shaista Tanveer, Muhammad Riaz, Dragan Pamucar, Cirovic Goran, Topological Data Analysis of m-Polar Spherical Fuzzy Information with LAM and SIR Models, 2022, 14, 2073-8994, 2216, 10.3390/sym14102216
    7. Mona Hosny, Generalization of rough sets using maximal right neighborhood systems and ideals with medical applications, 2022, 7, 2473-6988, 13104, 10.3934/math.2022724
    8. Esra Dalan Yildirim, A. Ghareeb, New Topological Approaches to Rough Sets via Subset Neighborhoods, 2022, 2022, 2314-4785, 1, 10.1155/2022/3942708
    9. Tareq M. Al-shami, Ibtesam Alshammari, Rough sets models inspired by supra-topology structures, 2022, 0269-2821, 10.1007/s10462-022-10346-7
    10. Rodyna A. Hosny, Radwan Abu-Gdairi, Mostafa K. El-Bably, Approximations by Ideal Minimal Structure with Chemical Application, 2023, 36, 1079-8587, 3073, 10.32604/iasc.2023.034234
    11. Zhaowen Li, Damei Luo, Guangji Yu, Reduction in a fuzzy probability information system based on incomplete set-valued data, 2023, 45, 10641246, 3749, 10.3233/JIFS-230865
    12. Oya Bedre Özbakir, Esra Dalan Yildirim, Aysegül Çaksu Güler, Different types of approximation operators on Gn-CAS via ideals, 2024, 38, 0354-5180, 727, 10.2298/FIL2402727B
    13. D. I. Taher, R. Abu-Gdairi, M. K. El-Bably, M. A. El-Gayar, Decision-making in diagnosing heart failure problems using basic rough sets, 2024, 9, 2473-6988, 21816, 10.3934/math.20241061
    14. R. Abu-Gdairi, A. A. El-Atik, M. K. El-Bably, Topological visualization and graph analysis of rough sets via neighborhoods: A medical application using human heart data, 2023, 8, 2473-6988, 26945, 10.3934/math.20231379
    15. Mostafa A. El-Gayar, Radwan Abu-Gdairi, Extension of topological structures using lattices and rough sets, 2024, 9, 2473-6988, 7552, 10.3934/math.2024366
    16. Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, K. A. Aldwoah, Ismail Ibedou, New soft rough approximations via ideals and its applications, 2024, 9, 2473-6988, 9884, 10.3934/math.2024484
    17. Rodyna A. Hosny, Radwan Abu-Gdairi, Mostafa K. El-Bably, Enhancing Dengue fever diagnosis with generalized rough sets: Utilizing initial-neighborhoods and ideals, 2024, 94, 11100168, 68, 10.1016/j.aej.2024.03.028
    18. Adeel Farooq, Musawwar Hussain, Muhammad Yousaf, Ahmad N. Al-Kenani, A new algorithm to compute fuzzy subgroups of a finite group, 2023, 8, 2473-6988, 20802, 10.3934/math.20231060
    19. Tareq M. Al-shami, Mohammed M. Ali Al-Shamiri, Murad Arar, Unavoidable corrections for θβ-ideal approximation spaces, 2024, 9, 2473-6988, 32399, 10.3934/math.20241553
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2416) PDF downloads(141) Cited by(19)

Figures and Tables

Tables(5)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog