Rough sets theory via new topological notions based on ideals and applications

There is a close analogy and similarity between topology and rough set theory. As, the leading idea of this theory is depended on two approximations, namely lower and upper approximations, which correspond to the interior and closure operators in topology, respectively. So, the joined study of this theory and topology becomes fundamental. This theory mainly propose to enlarge the lower approximations by adding new elements to it, which is an equivalent goal for canceling elements from the upper approximations. For this intention, one of the primary motivation of this paper is the desire of improving the accuracy measure and reducing the boundary region. This aim can be achieved easily by utilizing ideal in the construction of the approximations as it plays an important role in removing the vagueness of concept. The emergence of ideal in this theory leads to increase the lower approximations and decrease the upper approximations. Consequently, it minimizes the boundary and makes the accuracy higher than the previous. Therefore, this work expresses the set of approximations by using new topological notions relies on ideals namely I-δβJ-open sets and I∧ βJ-sets. Moreover, these notions are also utilized to extend the definitions of the rough membership relations and functions. The essential properties of the suggested approximations, relations and functions are studied. Comparisons between the current and previous studies are presented and turned out to be more precise and general. The brilliant idea of these results is increased in importance by applying it in the chemical field as it is shown in the end of this paper. Additionally, a practical example induced from an information system is introduced to elucidate that the current rough membership functions is better than the former ones in the other studies.


Introduction
Rough set [26,27] is one of a nonstatistical technique to deal with the problems of uncertainty in data and incompleteness of knowledge. The rationale of this set is depended on that the human knowledge is categorized into three fundamental regions, inside, outside and boundary. Therefore, the essential idea of this set focuses on the lower and upper approximations which are used to define the boundary region and accuracy measure. In the classical rough set model approximations are based on the equivalence relations, but this condition does not always hold in many practical problems and also this restriction limits the wide applications of this set. In the recent times, lots of researchers are interested to generalize this set in many fields of applications [9,15,16,23]. It was also generalized by the topological point of view [20,21,29,31] by replacing the equivalence relations in the lower and upper approximations by the open and closed sets, respectively. In the past few years mathematicians turned their attention towards to near (or nearly) open concept as generalization of open sets to topological spaces [1, 19,24,25,30]. In this direction, numerous generalizations of the rough set were offered using the nearly open concepts instead of open sets [4][5][6]32]. In 2017, Amer et al. [8] utilized coincided with Hosny's definitions [11]. So, Hosny's definitions [11] are special case of the current definitions. The main object of Sections 4 and 6 is to propose two different and independent of new approximations. These approximations are based on I-δβ J -open sets and Iβ -sets. The properties of the present approximations and the connections among them are established and constructed in these sections. They are compared to the prior ones [2,8,11,12] and shown that the accuracy measure which deduced by the current approximations is the best. The goal of Section 7 is to define new kind of the rough membership functions via ideal namely, I-δβ J -rough membership functions and Iβ Jrough membership functions. It is proved that these functions are better than the previous ones such as Abd El-Monsef et al. [3], Hosny [12], Lin [22], Pawlak and Skowron [28] (see Lemmas 7.2,7.3 and Remark 7.8). Section 8 demonstrates the importance of this paper by some real life applications. Finally, Section 9 aims to outline the essential findings and a plan for the future work.

Preliminaries
Definition 2.1. [17] Let X be a non-empty set. I φ, I ∈ P(X) is an ideal on X, if (i) A ∈ I and B ∈ I ⇒ A ∪ B ∈ I.  (iii) < R >-nd: n <R> (x) = ∩ x∈n R (y) n R (y).
(vii) < I >-nd: n <I> (x) = n <R> (x) ∩ n <I> (x). Proposition 2.3. [12] Let (X, R, Ξ J ) be a J-ndS and I be an ideal on X. Then, the following implications hold: . Definition 2.22. [12] Let (X, R, Ξ J ) be a J-ndS, I be an ideal on X, A ⊆ X and x ∈ X. The I − J-nearly rough membership functions of A are defined by }.
. Lemma 2.1. [12] Let (X, R, Ξ J ) be a J-ndS, I be an ideal on X and A ⊆ X. Then Definition 2.23. [3] Let (X, R, Ξ J ) be a J-ndS, x ∈ X and A ⊆ X: , then x is J-surely belongs to A, denoted by x ∈ J A.
, then x is J-nearly surely (η J -surely) belongs to A, denoted by x ∈ η J A.
, then x is J-nearly possibly (η J -possibly) belongs to A, denoted by x ∈ η J A. It is called J-(nearly) strong and J-(nearly) weak membership relations respectively. Definition 2.24. [12] Let (X, R, Ξ J ) be a J-ndS, I be an ideal on X, x ∈ X and A ⊆ X : , then x is J-nearly surely with respect to I ( I − η J -surely) belongs to A, denoted by , then x is J-nearly possibly with respect to I (briefly I − η J -possibly) belongs to A, denoted by x ∈ It is called J-nearly strong and J-nearly weak membership relations with respect to I respectively.
Proposition 2.6. [12] Let (X, R, Ξ J ) be a J-ndS, I be an ideal on X and A ⊆ X. Then

I-δβ J -open sets
In this section, the concept of I-δβ J -open sets is presented as generalization of the J-nearly open sets in Definitions 2.6 [8], 2.9 [11] and also generalization of the I-J-nearly open sets in Definition 2.16 [12]. This concept is based on the notions of ideals. Moreover, the principle properties of this concept is studied and compared to the previous concepts.  The following proposition shows that the concept of I-δβ J -open sets is an extension of the concept of δβ J -open sets in Definition 2.9 [11]. (ii) According to Remark 2.1 [8] and Propositions 2.1 [11], 3.1, the current Definition 3.1 is also a generalization of Definition 2.6 [8].
The following theorem shows that Hosny's Definition 2.9 [11] is a special case of the current definition.  It should be noted that, Proposition 3.2 shows that, every I-β J -open is I-δβ J -open, but the converse is not necessarily true as shown in the following example.  Proof. By Propositions 2.4 and 3.2 [12], the proof is obvious.
It is clear that

Approximations spaces by using I-δβ J -open sets
The purpose of this section is to generalize the previous approximations in Definitions 2.4 [2], 2.7 [8], 2.10 [11] and 2.17 [12]. The current approximations are depended on the I-δβ J -open sets. The fundamental properties of these approximations are obtained. Furthermore, the current findings are compared to the previous approaches.
The proof of this proposition is simple using the I-δβ J -interior and I-δβ J -closure, so I omit it.
(b) the converse of parts (ii) and (iii) is not necessarily true: Remark 4.4. The converse of parts of Corollary 4.2 is not necessarily true as in Example 3.1: The following proposition and corollary are introduced the relationships between the current approximations in Definition 4.1 and the previous one in Definition 2.17 [12]. Table 1.
Comparison between the boundary and accuracy by using the current approximations in Definition 4.1 and the previous one in Definition 2.17 [12].

A
The previous one in Definition 2.17 [12] The current method in Definition 4.1 For example, take A = {a, b}, then the boundary and accuracy by the present method in Definition 4.1 are φ and 1 respectively. Whereas, the boundary and accuracy by using Hosny's method 2.17 [12] are {a, b} and 0 respectively. Remark 4.5. Example 3.2 shows that the converse of the implications in Corollary 4.4 is not true in general. For example, if take A = {a}, then it is I-δβ R -exact, but it is not I-β R -exact and consequently, not I-S R -exact, not I-α R -exact and not I- In the following lemma I summarize the fundamental properties of the subset Iβ J .
Lemma 5.1. For subsets A, B and A α (α ∈ ∆) of a J-ndS (X, R, Ξ J ), the following implications hold: Proof. I prove only (v) and (vi) since the other are consequences of Definition 5.1.
Therefore, x Iβ J (∩{A α : α ∈ ∆}). The following proposition shows that the concept of Iβ J -sets is an extension of the concept of Proof. This follows from Lemma 5.1. Remark 5.6. Let (X, R, Ξ J ) be a J-ndS, I be an ideal on X and A ⊆ X. Then the following statements are not true in general: So, the relationships among Iβ J -sets are not comparable as in Example 3.3:       It is clear that • In a similar way, I can add examples to show that, τ

Approximations spaces by using Iβ J -sets
The aim of this section is to present a new technique to define the approximations of rough sets by using the notion of Iβ J -sets. Some important significant properties of these approximations are investigated and compared to the previous approximations in Definitions 2.4 [2], 2.7 [8], 2.14 [11] and 2.17 [12]. The techniques in this section and Section 4 are different and independent.
The following proposition studies the main properties of the current Iβ J -lower and Iβ J -upper approximations.
Proposition 6.1. Let (X, R, Ξ J ) be a J-ndS, I be an ideal on X and A, B ⊆ X. Then, The proof of this proposition is simple using Iβ J -interior and Iβ J -closure, so I omit it.
Remark 6.1. Example 3.3 shows that (a) The inclusion in Proposition 6.1 parts (i), (iv), (v), (xi) and (xii) can not be replaced by equality relation: The converse of parts (ii) and (iii) is not necessarily true: Definition 6.2. Let (X, R, Ξ J ) be a J-ndS, I be an ideal on X and A ⊆ X. A is an Iβ J -definable The converse of parts of Corollary 6.2 is not necessarily true as in Example 5.2: (ii) If A = {b}, then it is R-rough and β R -rough, but it is not I-β R -rough.
The following proposition and corollary are introduced the relationships between the current approximations in Definition 6.1 and the previous one in Definition 2.17 [12].
In Table 2, the lower, upper approximations, boundary region and accuracy are calculated by using Hosny's method 2.17 [12] and the current approximations in Definition 6.1 by using Example 3.3. Table 2. Comparison between the boundary and accuracy by Hosny's method 2.17 [12] and the current approximations in Definition 6.1.

A
Hosny's method 2.17 [12] The current method in Definition 6.1 For example, take A = {e}, then the boundary and accuracy by the present method in Definition 6.1 are φ and 1 respectively. Whereas, the boundary and accuracy by using Hosny's method 2.17 [12] are {e} and 0 respectively. Remark 6.4. It should be noted that the Iβ J -approximations in this section and the I-δβ J -approximations in Section 4 are different and independent. As, the concepts of I-δβ J -open sets and Iβ J -sets are different and independent as shown in Remark 5.5.
The following results give the fundamental properties of the I − δβ J -rough membership functions.
Proposition 7.2. Let (X, R, Ξ J ) be a J-ndS, I be an ideal on X and A, B ⊆ X. Then otherwise.

}.
and ψ Remark 7.6. The I-J-nearly rough membership functions are used to define the Iβ J -lower and Iβ J -upper approximations as follows: The following results give the fundamental properties of the I − β J -rough membership functions.
Proposition 7.3. Let (X, R, Ξ J ) be a J-ndS, I be an ideal on X and A, B ⊆ X. Then Proof. It is similar to Proposition 7.2.
Lemma 7.3. Let (X, R, Ξ J ) be a J-ndS, I be an ideal on X and A ⊆ X. Then Proof. It isimilar to Lemma 7.2.
Remark 7.7. The converse of Lemma 7.3 is not true in general, as it is shown in Example 3.3.
Remark 7.8. According to Lemmas 7.2 and 7.3, the current Definitions 7.2 and 7.3 are also generalization of the approaches in [22] and 2.19 [28].

Applications
Finally in this section, an applied example in Chemistry field is introduced by applying the present Definition 3.1 and the previous one 2.6 in [8]. Furthermore, a practical example uses an equivalence relation that induced from an information system is introduced to compare between the current approach in Definition 7.2 and the previous approach for Pawlak and Skoworn 2.19 [28].
I consider the relations on X defined as: .., 5} where σ k represents the standard deviation of the quantitative attributes.
The right neighborhoods ∀x ∈ X with respect to the relations are shown in Table 4.  Table 4. Right neighborhood of seven reflexive relations.
The intersection of all right neighborhoods ∀x ∈ X is:  (iii) Similarly, it is easy to calculate I-β R O(X), τ and their approximations by the same manner in Tables 1 and 2. This also shows that the present methods is better than the previous ones in [2,8,11,12].
Example 8.2. Consider the following information system as in Table 5. The data about six students is given as shown below:  Table 5: (i) The set of universe:   [28] are computed as follows: µ A (x 1 ) = µ A (x 3 ) = µ A (x 6 ) = 1, µ A (x 2 ) = 1 2 . Obviously, the current Definition 7.2 is accurate more than the Definition of Pawlak and Skowron 2.19 [28].
Remark 8.1. It should be noted that for some elements that have decision (Reject) such that x 5 (i) The rough membership function with respect to the Definition of Pawlak and Skowron 2.19 [28] is µ A (x 5 ) = 1 2 . This means that x 5 may belong to the set A (Decision: Accept), A = {x 1 , x 2 , x 3 , x 6 } and this contradicts to Table 5.

Conclusions
Rough set theory is a vast area that has varied inventions, applications and interactions with many other branches of mathematical sciences. Deriving rough sets from topology is one such interaction. There is a close homogeneity between rough set theory and general topology. Topology is a rich source for constructs that can be helpful to enrich the original model of approximation spaces. Ideal is a fundamental concept in topological spaces and played an important role in the study of a generalization of rough set. Since the advent of the ideals, several research papers with interesting results in different respects came to existence. In the current results, ideals were very helpful for increasing the current lower approximations and decreasing the current upper approximations. Consequently, they reduced the boundary region and increased the accuracy measure. So, they removed the vagueness of a concept that is an essential goal for the rough set. The properties of the proposed concepts and methods were studied. It should be noted that the two methods in this paper were different and independent as it was shown. I gave not only their characterizations but also discussed the relationships among them and between the previous ones and shown to be more general. The present accuracy measures were more accurate and higher than the previous ones. Since, the boundary regions were decreased (or empty) by increasing the lower approximations and decreasing the upper approximations. Further, two kind of the rough membership functions with respect to ideals were introduced as extension of the former functions. Moreover, an applied example in chemical field was suggested by applying the current methods to illustrate the concepts in a friendly way. Finally, a particle example was provided to clarify the technique of the present rough membership functions and demonstrate their utility and efficiency. I hope the beauty of this work can pave way to many other research fields such as:  This is a part of the future research.