Topological structures and their generalizations are of crucial importance in data analysis, which have manifested in different fields, for example, in physics [1], chemistry [2], medicine [3], soft set theory [4], etc. Lashin et al. [5] used topological notions to study different issues in rough set theory in order to generalize Pawlak’s concepts [6] in different applications and integrate the concepts of rough and fuzzy sets. Topological structures were applied in rough sets to improve evolutionary-based feature selection technique using the extension of knowledge [7], decision making of COVID-19 [8,9], and enhanced feature selection based on integration containment neighborhoods rough set approximations and binary honey badger optimization [10]. Several articles [11–19] extended the application fields of Pawlak’s model. Popa et al. introduced the minimal structure spaces [20], as a generalization of topological spaces to analyze information systems. Various consequences of minimal spaces can be viewed in [21]. EL-Sharkasy [22,23] studied several sorts of sets in minimal structure spaces with some of their characterizations. The notion of the ideal 𝒥 , which is a nonempty class of sets of Λ fulfills the finite additivity and the hereditary property, is famed in the study of topological problems and their other usages defined by Kuratowski [24]. The advantages of utilizing an ideal in rough set theory are reducing the boundary region and improving the accuracy degree. Accordingly, the study of this theory with ideals is an interesting subject that has delivered the attention of many researchers [25–27]. As an important base for modulation of knowledge extraction and processing, this work will combine the notions of minimal structure approximation spaces and ideals (𝒥MSAS) , where these concepts are not only in most fields of mathematics but also in several real-life problems.

The main contributions of this work are constructing ideal minimal structure approximation space 𝒥MSAS from minimal structure approximation space using the notion of ideal and applying them in the decision-making of the classification of amino acids. In addition, some sorts of near open and closed sets via 𝒥MSAS view are proposed and verified.

Herein, some vital concepts and results are introduced, which are helpful in the sequel.

Definition 1 [13] Consider the binary relation η on the nonempty finite set Λ (named a universe set). Thus, the pair (Λ,η) is indicated as a generalized approximation space (briefly, an approximation space).

Definition 2 [13] Suppose that (Λ,η) is an approximation space. Thus, the right neighborhood Nr(s) of a point s∈Λ is defined by Nr(s)={y∈Λ:sηy} . Moreover, the class of all right neighborhoods in (Λ,η) is given by Nr(Λ)={Nr(s):s∈Λ} .

Definition 3 [22,23] For any approximation space (Λ,η) , then:

(i) the minimal structure of (Λ,η) is the class (Λ) = {∅,Λ}∪Nr(Λ) . Accordingly, the triple (Λ,η,MS(Λ)) is titled a minimal structure approximation space (briefly, MSAS ).

(ii) the elements of MS(Λ) are named MS(Λ) -open sets and their complements are MS(Λ) -closed sets.

(iii) the class of all MS(Λ) -closed sets, symbolized by (MS(Λ))c , is proposed by:

(MS(Λ))c={B:Bc∈MS(Λ)}, where Bc represents a complement of B in Λ.

Definition 4 [22,23] Let B be a subset of an MSAS (Λ,η,MS(Λ)) .Then:

(i) the minimal lower approximation of B is LMS(B)=∪{U∈MS(Λ):U⊆B} .

(ii) the minimal upper approximation of B is UMS(B)=∩{V∈(MS(Λ))c:B⊆V} .

(iii) the minimal accuracy of B is σr(B)=|LMS(B)||UMS(B)| , where |UMS(B)|≠0 .

Proposition 1 [22] If η is a dominance (reflexive and transitive) relation on a universe Λ , therefore MS( Λ ) represents a base for a topology on Λ .

Definition 5 [23] A subset B of an Λ is called:

(i) MS -regular open, if B = LMS(UMS(B)) (resp. MS -regular closed if B = UMS(LMS(B))) .

(ii) MS -semi open, if B ⊆ UMS(LMS(B)) (resp. MS -semi closed if LMS(UMS(B)) ⊆ B ).

(iii) MS -pre open, if B ⊆ LMS(UMS(B)) (resp. MS -pre closed if UMS(LMS(B)) ⊆ B ).

The sets of all MS -regular open, MS -regular closed, MS -semi open, MS -semi closed, MS -pre open and MS -pre closed sets of (Λ,η,MS(Λ)) are symbolized as MS - RO(Λ) , MS - RC(Λ) , MS - SO(Λ) , MS - SC(Λ) , MS - PO(Λ) and MS - PC(Λ) , respectively.

This section aims to introduce the ideal minimal structure approximation space 𝒥MSAS as a generalization of minimal structure approximation space MSAS via ideal. Some characteristics of the proposed method are obtained. Furthermore, the relations between the new approach 𝒥MSAS and the previous ones in [6,22,23] are studied.

Definition 6 Let (Λ,η,MS(Λ),𝒥) be an 𝒥MSAS . If B⊆Λ , then:

(i) a minimal lower approximation LMS𝒥(B) of B with respect to an ideal 𝒥 is

LMS𝒥(B) = η_r𝒥(B) ∩ B,  where η_r𝒥(B)=∪{U∈MS(Λ):U−B∈𝒥}.

(ii) a minimal upper approximation UMS𝒥(B) of B with respect to an ideal 𝒥 is

UMS𝒥(B)= η¯r𝒥(B) ∪ B, where η¯r𝒥(B) = ∩{V∈(MS(Λ))c:B−V∈𝒥}.

(iii) a minimal accuracy σr𝒥(B) of B with respect to an ideal 𝒥 is

σr𝒥(B)=|LMS𝒥(B)||UMS𝒥(B)|, where |UMS𝒥(B)|≠0. 

Remark 1 When 𝒥={∅} in Definition 6, the existing approximations are the same as the previous one in Definition 4.

To study the prime properties of LMS𝒥(.) , and UMS𝒥(.) approximations, firstly the properties of η_r𝒥(.) , and η¯r𝒥(.) will be investigated.

Proposition 2 For an 𝒥MSAS (Λ,η,MS(Λ),𝒥) . If B,B`⊆Λ , then the following conditions are satisfied:

(i) η_r𝒥(B)=η¯r𝒥(Bc))c , and η¯r𝒥(B)=(η_r𝒥(Bc))c .

(ii) η_r𝒥(Λ)=Λ , and η¯r𝒥(∅)=∅ .

(iii) if B⊆B` , then η_r𝒥(B) ⊆ η_r𝒥(B`) , and η¯r𝒥(B) ⊆ η¯r𝒥(B`) .

(iv) η_r𝒥(B∩B`) ⊆ η_r𝒥(B)∩η_r𝒥(B`) , and η_r𝒥(B∪B`)⊇ η_r𝒥(B)∪η_r𝒥(B`) .

(v) η¯r𝒥(B∪B`) ⊇ η¯r𝒥(B) ∪ η¯r𝒥(B`) , and η¯r𝒥(B∩B`) ⊆ η¯r𝒥(B)∩η¯r𝒥(B`) .

Proof. The first item will be proved, and the others similarly.

(η¯r𝒥(Bc))c=(∩{V∈(MS(Λ))c:Bc−V∈𝒥})c=∪{Vc∈MS(Λ):Vc−B∈𝒥}=η_r𝒥(B).

Example 1 Let Λ={a,b,c,d} , and η={(a,a),(a,b),(b,b),(b,c),(c,d),(d,c)} be a binary relation on Λ . Then, Nr(a)={a,b} , Nr(b)={b,c} , Nr(c)={d} , and Nr(d)={c} .

Hence, MS(Λ)={∅,Λ,{c},{d},{a,b},{b,c}} , and MS(Λ)c={∅,Λ,{c,d},{a,d},{a,b,d},{a,b,c}} .

Remark 2 Example 1 represents the following:

(i) the inversion of statement (iii) from Proposition 2 is not true. Suppose 𝒥={∅,{b}} , and let B={b} , B`={a} . Accordingly, η_r𝒥(B)=∅ , η_r𝒥(B`)={a,b} , η¯r𝒥(B)=∅ , η¯r𝒥(B`)={a,b} . Hence, η_r𝒥(B) ⊆ η_r𝒥(B`) and η¯r𝒥(B) ⊆ η¯r𝒥(B`) although B ⊈ B` .

(ii) the inclusion of the first part of the statement (iv) of Proposition 2 cannot be exchanged by an equality symbol. Suppose 𝒥={∅,{b}} , and let B={a} , B`={c} . Consequently, η_r𝒥(B)={a,b} , η_r𝒥(B`)={b,c} , and η_r𝒥(B∩B`)=∅ . Hence, η_r𝒥(B)∩η_r𝒥(B`) ⊈ η_r𝒥(B∩B`) .

(iii) the inclusion of the second part of the statement (iv) of Proposition 2 cannot be exchanged by equality symbol. Suppose 𝒥={∅,{b}} , and let B={a,b,d} , B`={b,c,d} . Thus, η¯r𝒥(B)={a,d} , η¯r𝒥(B`) = {c,d} , and η¯r𝒥(B∪B`)=Λ . Hence, η¯r𝒥(B∪B`) ⊈ η¯r𝒥(B) ∪ η¯r𝒥(B`) .

(iv) the inclusion of the first part of the statement (v) of Proposition 2 cannot be exchanged by equality symbol. Consider 𝒥={∅,{c}} , and let B={a} , B`={b} . Therefore, η_r𝒥(B)={c} , η_r𝒥(B`)={b,c} , and η_r𝒥(B∪B`) = {a,b,c} . So, η_r𝒥(B∪B`) ⊈  η_r𝒥(B)∪η_r𝒥(B`) .

(v) the inclusion of the second part of the statement (v) of Proposition 2 cannot be exchanged by equality symbol. Consider 𝒥 = {∅,{c}} , and let B={a,c,d} , B`={b,c,d} . As a result, η¯r𝒥(B)={a,d} , η¯r𝒥(B`)={a,b,d} , and η¯r𝒥(B∩B`)={d} . Hence, η¯r𝒥(B∩B`) ⊉ η¯r𝒥(B)∩η¯r𝒥(B`) .

The succeeding proposition is realized depending on Proposition 1.

Proposition 3 Let η , and MS(Λ) be a dominance relation, and a minimal structure on Λ , respectively. If B,B`⊆Λ , then for any ideal 𝒥 the following results hold: η_r𝒥(B∩B`) = η_r𝒥(B)∩η_r𝒥(B`) , and η¯r𝒥(B∪B`) = η¯r𝒥(B) ∪ η¯r𝒥(B`) .

The next remark is devoted to clarifying the differences between the existing approximations and the preceding one of Propositions 2 & 3 in [22].

Remark 3 Let B , B` be subsets of an 𝒥MSAS (Λ,η,MS(Λ),𝒥) . Then Example 1 with 𝒥={∅,{c}} confirms the following:

(i) η_r𝒥(∅) ≠ ∅ . As η_r𝒥(∅)={c} .

(ii) η¯r𝒥(Λ) ≠ Λ . As η¯r𝒥(Λ)={a,b,d} .

(iii) η_r𝒥(B) ⊈ B . As B={a} , η_r𝒥(B)={c} .

(iv) B ⊈ η¯r𝒥(B) . As B={b,c,d} , η¯r𝒥(B)={a,b,d} .

Proposition 4 Let 𝒥,𝒥∙ be two ideals on an 𝒥MSAS (Λ,η,MS(Λ)) , and B ⊆ Λ . Then, the next results hold:

(i) if 𝒥⊆𝒥∙ , then η_r𝒥(B) ⊆ η_r𝒥∙(B) .

(ii) if 𝒥⊆𝒥∙ , then η¯r𝒥∙(B) ⊆ η¯r𝒥(B) .

(iii) η¯r(𝒥∪𝒥∙)(B)=η¯r𝒥(B) ∪ η¯r𝒥∙(B) and η¯r(𝒥∩𝒥∙)(B)=η¯r𝒥(B) ∩ η¯r𝒥∙(B) .

(iv) η_r𝒥(B) ∪ η_r𝒥∙(B)=η_r(𝒥∪𝒥∙)(B) and η_r(𝒥∩𝒥∙)(B) = η_r𝒥(B) ∩ η_r𝒥∙(B) .

(v) B∈𝒥 iff η¯r𝒥(B)=∅ .

(vi) Bc∈𝒥 iff η_r𝒥(B)=Λ .

Proof. The first item will be proved, and the others similarly.

Let w∈η_r𝒥(B) , and then there exists U∈MS(Λ) such that w∈U and U−B∈𝒥 . Since 𝒥⊆𝒥∙ , then w∈η_r𝒥∙(B) and so η_r𝒥(B) ⊆ η_r𝒥∙(B) .

Example 2 Let η be a binary relation on Λ={a,b,c} such that Nr(a)={a} , Nr(b)=Λ , and Nr(c)={a,c} . Then MS(Λ)={∅,Λ,{a},{a,c}} , and MS(Λ)c={∅,Λ,{b},{b,c}} .

Remark 4 Suppose 𝒥={∅,{a}} , and 𝒥∙={∅,{b}} in Example 2. Then,

(i) the inversion of statement (i) from Proposition 4 is false. Suppose B={b} , then η_r𝒥∙(B)=∅ , η_r𝒥(B)={a} . Hence, η_r𝒥∙(B) ⊆ η_r𝒥(B) , while 𝒥∙ ⊈ 𝒥 .

(ii) the inversion of statement (ii) from Proposition 4 is not true. Suppose B = {a,b} , then η¯r𝒥∙(B)=Λ , η¯r𝒥(B) = {b} . Hence, η¯r𝒥(B) ⊆ η¯r𝒥∙(B) , while 𝒥∙ ⊈ 𝒥 .

Consequently, for any relation, the new kind of rough approximations 𝒥MSAS is more accurate than the preceding ones [6,22,23].

Remark 5 Table 1 compare between lower approximations, upper approximations, and the accuracy and Table 2 compare between boundary, internal edge, and external edge defined in Definitions 4 and 6 which are given by the relation η of Example 1 and ideal 𝒥={∅,{c}} .

The next result exhibits the connections among the lower, and upper approximations and the degree of accuracy that were offered in both Definitions 4, and 6.

Theorem 1 Let (Λ,η,MS(Λ),𝒥) be an 𝒥MSAS . If B⊆Λ . Then, the next properties are held:

(i) LMS(B) ⊆ LMS𝒥(B) .

(ii) UMS𝒥(B) ⊆ UMS(B) .

(iii) σr(B) < σr𝒥(B) .

Proof. The first item will be proved, and the others similarly.

Let w∈LMS(B) , and then there exists U∈MS(Λ) such that w∈U and U⊆B . Hence, U−B=∅∈𝒥 . i.e., w∈η_r𝒥(B) . So, LMS(B) ⊆ η_r𝒥(B) . Since LMS(B) ⊆ B , therefore LMS(B) ⊆ LMS𝒥(B) .

Remark 6 It must be perceived that the suggested approximations LMS𝒥(.) , and UMS𝒥(.) in Definition 6 have equal characteristics of η_r𝒥(.) and η¯r𝒥(.) identified in Propositions 1 and 3. Moreover, it fulfills the next properties:

(i) LMS𝒥(∅)=∅ and UMS𝒥(Λ)=Λ .

(ii) LMS𝒥(B) ⊆ B ⊆ UMS𝒥(B) .

Definition 7 For an 𝒥MSAS (Λ,η,MS(Λ),𝒥) . If B⊆Λ , then

(i) a minimal positive of B with respect to an ideal 𝒥 is MS - Por𝒥(B)=LMS𝒥(B) ,

(ii) a minimal exterior (negative) of B with respect to an ideal 𝒥 is MS - Exr𝒥(B)=Λ−UMS𝒥(B) ,

(iii) a minimal boundary of B with respect to ideal 𝒥 is MS - br𝒥(B)=UMS𝒥(B)−LMS𝒥(B) ,

(iv) a minimal internal edge of B with respect to an ideal 𝒥 is LMS𝒥 -edg (B)=B−LMS𝒥(B) , and

(v) a minimal external edge of B with respect to an ideal 𝒥 is UMS𝒥 -edg (B)=UMS𝒥(B)−B .

Remark 7 According to Example 1 with 𝒥={∅,{c}} , Table 2 illustrates the next relationships for B⊆Λ :

(i) MS - br𝒥(B) ⊆ MS - br(B) ,

(ii) LMS𝒥 -edg (B) ⊆ LMS -edg (B) , and

(iii) UMS𝒥 -edg (B) ⊆ UMS -edg (B) .

Definition 8 For an 𝒥MSAS (Λ,η,MS(Λ),𝒥) , a subset B of Λ is categorized as:

(i) MS -definable (exact) with respect to an ideal 𝒥 , if MS - br𝒥(B)=∅ . Otherwise, a subset B is called MS -undefinable (rough) with respect to the 𝒥 .

(ii) MS -internally definable with respect to an ideal 𝒥 , if LMS𝒥(B)=B .

(iii) MS -externally definable with respect to an ideal 𝒥 , if UMS𝒥(B)=B .

Remark 8 According to Table 1 and Example 1 with 𝒥={∅,{c}} , it is noticed that:

(i) the sets {c},{d},{a,b},{c,d},{a,b,c},{a,b,d} , and Λ are MS -definable with respect to an ideal 𝒥 .

(ii) the sets {b},{b,c},{b,d} , and {b,c,d} are MS -internally definable with respect to an ideal 𝒥 .

(iii) the sets {a},{a,c},{a,d} , and {a,c,d} are MS -externally definable with respect to an ideal 𝒥 .

Corollary 1 For a 𝒥MSAS (Λ,η,MS(Λ),𝒥) . Then,

(i) every MS -definable is MS -definable with respect to an ideal 𝒥 .

(ii) every MS -undefinable with respect to an ideal 𝒥 is MS -undefinable.

This section aims to introduce and investigate some sorts of near open (resp. closed) sets via the viewpoint of an 𝒥MSAS .

Definition 9 A subset B of an 𝒥MSAS (Λ,η,MS(Λ),𝒥) is called:

(i) MS𝒥 -regular open if B = LMS𝒥(UMS𝒥(B))

(ii) MS𝒥 -semi open if B ⊆ UMS𝒥(LMS𝒥(B)) .

(iii) MS𝒥 -pre open if B ⊆ LMS𝒥(UMS𝒥(B)) .

Remark 9

(i) The complement of an MS𝒥 -regular open (resp. MS𝒥 -semi open, and MS𝒥 -pre open) set is known MS𝒥 -regular closed (resp. MS𝒥 -semi closed, and MS𝒥 -pre closed) set.

(ii) The set of all MS𝒥 -regular open (resp. MS𝒥 -regular closed, MS𝒥 -semi open, MS𝒥 -semi closed, MS𝒥 -pre open and MS𝒥 -pre closed) sets of (Λ,η,MS(Λ)) is denoted by MS𝒥 - RO(Λ) (resp. MS𝒥 - RC(Λ) , MS𝒥 - SO(Λ) , MS𝒥 - SC(Λ) , MS𝒥 - PO(Λ) and MS𝒥 - PC(Λ) ).

Remark 10 In a topological space, the class of semi (resp. pre-) open sets is contained in the class of ideal semi (resp. pre-) open sets. While this fact is discussed for minimal structures, surprisingly it is incorrect i.e., MS - SO(Λ) and MS𝒥 - SO(Λ) (resp. MS - PO(Λ) and MS𝒥 - PO(Λ) ) are incomparable, as shown in Examples 3, and 4.

Example 3 (Continued from Example 2) Consider 𝒥={∅,{a},{b},{a,b}} is an ideal on (Λ,η,MS(Λ)) , that will result in.

(i) MS - RO(Λ)={∅,Λ} .

(ii) MS - SO(Λ)={∅,{a},{a,c},{a,b},Λ} .

(iii) MS - PO(Λ)={∅,{a},{a,c},{a,b},Λ} .

(iv) MS𝒥 - RO(Λ)={∅,{a},{b,c},Λ} .

(v) MS𝒥 - SO(Λ)={∅,{a},{c},{a,c},{b,c},Λ} .

(vi) MS𝒥 - PO(Λ)={∅,{a},{c},{a,c},{b,c},Λ} .

Example 4 Consider Λ={a,b,c,d} , η is a binary relation on Λ , and Nr(a)={a} , Nr(b)={a,b} , Nr(c)={c} , and Nr(d)={b,c,d} . Accordingly, MS(Λ)={∅,Λ,{a},{c},{b,c,d},{a,b}} and

MS(Λ)c={∅,Λ,{a},{c,d},{a,b,d},{b,c,d}} . If 𝒥={∅,{a},{d},{a,d}} , and L={∅,{b},{d},{b,d}} be ideals on (Λ,η,MS(Λ)) , then

(i) MS - RO(Λ)={∅,Λ,{a},{c},{b,c,d},{a,b}} .

(ii) MS - SO(Λ)={∅,Λ,{a},{c},{a,b},{a,c},{c,d},{a,b,c},{a,b,d},{b,c,d}} .

(iii) MS - PO(Λ)={∅,Λ,{a},{c},{a,b},{a,c},{b,c},{a,b,c},{a,c,d},{b,c,d}} .