Processing math: 23%
Research article

Soft closure spaces via soft ideals

  • Received: 13 December 2023 Revised: 25 January 2024 Accepted: 29 January 2024 Published: 05 February 2024
  • MSC : 54A05, 54D10, 54D05, 54A10, 54C40

  • This paper was devoted to defining new soft closure operators via soft relations and soft ideals, and consequently new soft topologies. The resulting space is a soft ideal approximation. Many of the well known topological concepts were given in the soft set-topology. Particularly, it introduced the notations of soft accumulation points, soft continuous functions, soft separation axioms, and soft connectedness. Counterexamples were introduced to interpret the right implications. Also, a practical application of the new soft approximations was explained by an example of a real-life problem.

    Citation: Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, Ismail Ibedou. Soft closure spaces via soft ideals[J]. AIMS Mathematics, 2024, 9(3): 6379-6410. doi: 10.3934/math.2024311

    Related Papers:

    [1] Tareq M. Al-shami, Zanyar A. Ameen, A. A. Azzam, Mohammed E. El-Shafei . Soft separation axioms via soft topological operators. AIMS Mathematics, 2022, 7(8): 15107-15119. doi: 10.3934/math.2022828
    [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] G. Muhiuddin, Ahsan Mahboob . Int-soft ideals over the soft sets in ordered semigroups. AIMS Mathematics, 2020, 5(3): 2412-2423. doi: 10.3934/math.2020159
    [4] 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. AIMS Mathematics, 2024, 9(4): 9884-9910. doi: 10.3934/math.2024484
    [5] Tareq M. Al-shami, Salem Saleh, Alaa M. Abd El-latif, Abdelwaheb Mhemdi . Novel categories of spaces in the frame of fuzzy soft topologies. AIMS Mathematics, 2024, 9(3): 6305-6320. doi: 10.3934/math.2024307
    [6] Dina Abuzaid, Samer Al-Ghour . Supra soft Omega-open sets and supra soft Omega-regularity. AIMS Mathematics, 2025, 10(3): 6636-6651. doi: 10.3934/math.2025303
    [7] Samer Al-Ghour, Hanan Al-Saadi . Soft weakly connected sets and soft weakly connected components. AIMS Mathematics, 2024, 9(1): 1562-1575. doi: 10.3934/math.2024077
    [8] Tareq M. Al-shami, Abdelwaheb Mhemdi, Radwan Abu-Gdairi, Mohammed E. El-Shafei . Compactness and connectedness via the class of soft somewhat open sets. AIMS Mathematics, 2023, 8(1): 815-840. doi: 10.3934/math.2023040
    [9] Tareq M. Al-shami, El-Sayed A. Abo-Tabl . Soft $ \alpha $-separation axioms and $ \alpha $-fixed soft points. AIMS Mathematics, 2021, 6(6): 5675-5694. doi: 10.3934/math.2021335
    [10] Dina Abuzaid, Samer Al Ghour . Three new soft separation axioms in soft topological spaces. AIMS Mathematics, 2024, 9(2): 4632-4648. doi: 10.3934/math.2024223
  • This paper was devoted to defining new soft closure operators via soft relations and soft ideals, and consequently new soft topologies. The resulting space is a soft ideal approximation. Many of the well known topological concepts were given in the soft set-topology. Particularly, it introduced the notations of soft accumulation points, soft continuous functions, soft separation axioms, and soft connectedness. Counterexamples were introduced to interpret the right implications. Also, a practical application of the new soft approximations was explained by an example of a real-life problem.



    The paper [1] by Pawlak was the first article focused on the rough area between the interior set A and the closure set ¯A of a subset A in a universal set X. This idea led to many applications in decision theory. The theory of rough sets is constructed using the equivalence classes as its building blocks.

    The most efficacious tools to study the generalization of rough set theory are the neighborhood systems. The main idea in this theory is the upper and lower approximations that have been defined using different types of neighborhoods instead of equivalence classes such as left and right neighborhoods [2,3,4,5], minimal left neighborhoods [6] and minimal right neighborhoods [7], and the intersection of minimal left and right neighborhoods [8]. Afterwards, the approximations by minimal right neighborhoods which are determined by reflexive relations that form the base of the topological space defined in [9]. In 2018, Dai et al. [10] presented new kind of neighborhoods, namely the maximal right neighborhoods which were determined by similarity relations and have been used to propose three new kinds of approximations. Dai et al.'s approximations [10] differed from Abo-Tabl's approximations [9] in that the corresponding upper and lower approximations, boundary regions, accuracy measures, and roughness measures in two types of Dai et al.'s approximations [10] had a monotonicity. Later on, Al-shami [11] embraced a new type of neighborhood systems namely, the intersection of maximal right and left neighborhoods, and then used this type to present new approximations. These approximations improved the accuracy measures more than Dai et al.'s approximations [10]. Al-shami's [11] accuracy measures preserved the monotonic property under any arbitrary relation. The paper [12], by Molodtsov, was the first article that defined the notion "soft set", and it has many applications in uncertainty area or ambiguity decision. A theoretical research on soft set theory was given in [13] by Maji et al. The paper [14] by Ali et al. proposed many soft set-theoretical notions such as union, intersection, difference and complement. [15,16,17,18,19,20] objected to developing the theory and the applications of soft sets. In [21], the authors introduced the soft ideal notion. It is a completely new approach for modeling vagueness and uncertainty by reducing the boundary region and increasing the accuracy of a rough set which helped scholars to solve many real-life problems [4,22,23,24,25]. Recently, many extensions of the classical rough set approximations have been applied to provide new rough paradigms using certain topological structures and concepts like subset neighborhoods, containment neighborhoods, and maximal and minimal neighborhoods to deal with rough set notions and address some real-life problems [2,4,26,27,28]. Numerous researchers have recently examined some topological concepts, including continuity, separation axioms, closure spaces, and connectedness in ideal approximation spaces [29,30,31]. Ordinary rough sets were defined using an equivalence relation R on X, and produced two approximations, one is lower and second is upper. The space (X,R) is named approximation space. In the soft case, soft roughness used soft relations [32]. Some researchers transferred the common definitions in set-topology to soft set-topology, depending on that soft topology is an extension to the usual topology as explained in [15]. Many researchers objected to the basics of set-topology and subsequently the well-known embedding theorems but in point of view of soft set-topology with some real-life applications (see [33,34,35,36,37,38]). This paper used the notion of soft binary relations to ensure that the soft interior and soft closure in approximation spaces utilizing soft ideal to generate soft ideal approximation topological spaces based on soft minimal neighborhoods. We illustrated that soft rough approximations [17] are special cases of the current soft ideal approximations. Soft accumulation points, soft exterior sets, soft dense sets, and soft nowhere dense sets with respect to these spaces were defined and studied, and we gave some examples. We introduce and study soft ideal accumulation points in such spaces under a soft ideal defined on the given soft ideal. Soft separation axioms with respect to these soft ideal approximation spaces are reformulated via soft relational concepts and compared with examples to show their implications. In addition, we reformulate and study soft connectedness in these soft ideal approximation spaces. Finally, we defined soft boundary region and soft accuracy measure with respect to our soft ideal approximation spaces. We added two real life examples to illustrate the importance of the results obtained in this paper.

    This paper is divided into 6 sections beyond the introduction and the preliminaries. Section 3 defined the soft approximation spaces using a soft ideal. Section 4 is the main section of the manuscript and displays the properties of soft sets in the soft ideal approximation spaces. It has been generated using the concepts of R<x>R, soft neighborhoods and soft ideals. We study the main properties in soft ideal approximation spaces which are generalizations of the same properties of ideal approximation spaces given by Abbas et al. [31] and provide various illustrative examples. Section 5 introduced soft lower separation axioms via soft binary relations and soft ideal as a generalization of lower separation axioms given in [31]. We scrutinized its essential characterizations of some of its relationships associated with the soft ideal closure operators. Some illustrative examples are given. Section 6 reformulated and studied soft connectedness in [31] with respect to these soft ideal approximation spaces. Some examples are submitted to explain the definitions. Section 7 is devoted to comparing between the current purposed methods in Definitions 3.4–3.6 and to demonstrate that the method given in Definition 3.6 is the best in terms of developing the soft approximation operators and the values of soft accuracy. That is, the third approach in Definition 3.6 produces soft accuracy measures of soft subsets higher than their counterparts displayed in previous method 2.4 in [17]. Moreover, we applied these approaches to handle real-life problems. Section 8 is the conclusion.

    Through this paper, X stands for the universal set of objects, E denotes the set of parameters, LE denotes for a soft ideal, RE as a soft binary relation, P(X) represents all subsets of X, and SS(X) refers to the set of all soft subsets of X. All basic notions and notations of soft sets are found in [12,13,15,39,40].

    If (F,E) is a soft set of X and xX, then xˇ(F,E) whenever xF(e) for each eE. A soft set (F,E) of X with F(e)={x} for each eE is called a singleton soft set or a soft point and it is represented by xE or (x,E). Let (F1,E),(F2,E)ˇSS(X)E. Then, (F1,E) is a soft subset of (F2,E), represented by (F1,E)(F2,E), if F1(e)F2(e),eE. In that case, (F1,E) is called a soft subset of (F2,E) and (F2,E) is said to be a soft supset of (F1,E), (F2,E)(F1,E). Two soft subset (F1,E) and (F2,E) over X are called equal if (F1,E) is soft subset of (F2,E) and (F1,E) is soft supset of (F2,E). A soft set (F,E) over X is called a NULL soft set written as Φ if for each eE,F(e)=ϕ. Let A be a non-empty subset of X, then ˜AE or ˜A represents the absolute soft set (A,E) of X in which A(e)=A, for each eE. The soft intersection (resp. soft union) of (F1,E) and (F2,E) over X denoted by (F1F2,E) (resp. (F1F2,E)) and defined as (F1F2)(e)=F1(e)F2(e) (resp. (F1F2)(e)=F1(e)F2(e)) for each eE. Complementing a soft set (F,E) is represented by (F,E)c and it is defined as (F,E)c=(Fc,E) where Fc:EP(X) is a mapping defined by Fc(e)=XF(e) for all eE, and Fc is then a soft complement function of F.

    Definition 2.1. [32] Let (R,E)=RE be a soft set of X×X, that is R:EP(X×X). Then, RE is said to be a soft binary relation of X. RE is a collection of parameterized binary relations of X, from that R(e) is a binary relation on X for all parameters eE. The set of all soft binary relations of X is denoted by SBr(X).

    Definition 2.2. [15] Let ˜τ be a collection of soft sets over a universe X with a fixed set of parameters E. Then, ˜τSS(X)E is called a soft topology on X if

    (1)˜X,ΦEˇ˜τ,

    (2) the intersection of any two soft sets in ˜τ belongs to ˜τ,

    (2) the union of any number of soft sets in in ˜τ belongs to ˜τ.

    The triplet (X,˜τ,E) is called a soft topological space over X.

    Definition 2.3. [28] A mapping Cl:SS(X)ESS(X)E is called a soft closure operator on X if it satisfies these properties for every (F,E),(G,E)ˇSS(X)E:

    (1) Cl(Φ)=Φ,

    (2) (F,E)Cl(F,E),

    (3) Cl[(F,E)(G,E)]=Cl(F,E)Cl(G,E),

    (4) Cl(Cl(F,E))=Cl(F,E).

    Definition 2.4. [17] Let R:EP(X1×X2) and AX2. Then, the sets R_A(e),¯RA(e) could be defined by

    R_A(e)={xX1:ϕxR(e)A},¯RA(e)={xX1:xR(e)Aϕ}

    where xR(e)={yX2:(x,y)R(e)}. Moreover, R_:EP(X1) and ¯R:EP(X1) and we say (X1,X2,R) a generalized soft approximation space.

    Definition 2.5. [21] Let LE be a non-empty family of soft sets of X. Then, LESS(X)E is said to be a soft ideal on X if the following properties are fulfilled:

    (1)ΦˇLE,

    (2) (F,E)ˇLE and (G,E)(F,E) imply (G,E)ˇLE,

    (3) (F,E),(G,E)ˇLE imply (F,E)(G,E)ˇLE.

    In this section, we define the soft approximation spaces using soft ideals.

    Definition 3.1. Let RE be a soft binary relation of X and (x,y)X×X. Then, (x,y)ˇR whenever (x,y)R(e) for each eE.

    Definition 3.2. Let RE be a soft binary relation of X. Then, the soft afterset of xˇ˜X is xR={yˇ˜X:(x,y)ˇR}. Also, the soft foreset of xˇ˜X is Rx={yˇ˜X:(y,x)ˇR}.

    Definition 3.3. Let RE be a soft binary relation over X. Then, a soft set <x>R:EP(X) is defined by

    <x>R={xyR(yR)ify:xˇyR,Φo.w.

    Also, R<x>: EP(X) is the intersection of all foresets containing x, that is,

    R<x>={xyR(Ry)ify:xˇRy,Φo.w.

    Also, R<x>R=R<x><x>R.

    Lemma 3.1. Let RE be a soft binary relation over X. Then,

    (1) If xˇ<y>R, then <x>R⊑<y>R.

    (2) If xˇR<y>R, then R<x>RR<y>R.

    Proof. (1) Let zˇ<x>R=xˇwR(wR). Then, z is contained in any wR which contain x, and since x is contained in any uR which contains y, we have zˇ<y>R. Hence, <x>R⊑<y>R.

    (2) Straightforward from part (1).

    Definition 3.4. Let RE be a soft binary relation of X. For a soft set (F,E)ˇSS(X)E, the soft lower approximation Apr_1S(F,E) and the soft upper approximation ¯Apr1S(F,E) are defined by:

    Apr_1S(F,E)={xˇ(F,E): <x>R(F,E)}, (3.1)
    ¯Apr1S(F,E)=(F,E){xˇ˜X: <x>R(F,E)Φ}. (3.2)

    Theorem 3.1. Let (F,E),(G,E)ˇSS(X)E. The soft upper approximation defined by Eq (3.2) has the following properties:

    (1) ¯Apr1S(Φ)=Φ and ¯Apr1S(˜X)=˜X,

    (2) (F,E)¯Apr1S(F,E),

    (3) (F,E)(G,E)¯Apr1S(F,E)¯Apr1S(G,E),

    (4) ¯Apr1S[(F,E)(G,E)]¯Apr1S(F,E)¯Apr1S(G,E),

    (5) ¯Apr1S[(F,E)(G,E)]=¯Apr1S(F,E)¯Apr1S(G,E),

    (6) ¯Apr1S(¯Apr1S(F,E))=¯Apr1S(F,E),

    (7) ¯Apr1S(F,E)=[Apr_1S(F,E)c]c.

    Proof. (1),(2) It is clear from Definition 3.4.

    (3) Let xˇ¯Apr1S[(F,E). Then, <x>R(F,E)Φ. Since (F,E)(G,E), <x>R(G,E)Φ. Therefore, xˇ¯Apr1S(G,E). Hence, ¯Apr1S(F,E)¯Apr1S(G,E).

    (4) Immediately by part (3).

    (5) ¯Apr1S[(F,E)(G,E)]=[(F,E)(G,E)]{xˇ˜X: <x>R[(F,E)(G,E)]Φ}. Then,

    ¯Apr1S[(F,E)(G,E)]=[(F,E){xˇ˜X: <x>R(F,E)Φ}][(G,E){xˇ˜X: <x>R(G,E)Φ}]. Hence, ¯Apr1S[(F,E)(G,E)]=¯Apr1S((F,E))¯Apr1S((G,E)).

    (6) From part (2), we have ¯Apr1S(F,E)¯Apr1S(¯Apr1S(F,E)).

    Conversely, let xˇ¯Apr1S(¯Apr1S(F,E)). Then, <x>R¯Apr1S(F,E)Φ. Thus, there exists yˇ<x>R¯Apr1S(F,E). That means <y>R⊑<x>R (by Lemma 3.1 part (1)) and <x>R(F,E)Φ. Hence, xˇ¯Apr1S(F,E). This completes the proof.

    (7)

    [Apr_1S(F,E)c]c=[(F,E)c{xˇ˜X:<x>R(F,E)c}]c=(F,E){xˇ˜X:<x>R(F,E)Φ}=¯Apr1S(F,E).

    Example 3.1. Let X={a,b,c,d}, E={e1,e2} and

    RE={(e1,{(a,a),(a,b),(b,d),(c,d),(d,c),(d,d)),(e2,{(a,a),(a,b),(a,c),(b,d),(b,c),(c,d),(d,c),(d,d),(d,b))}}. Then, we have

    <a>R=<b>R={(e1,{a,b}),(e2,{a,b})},<c>R={(e1,{c,d}),(e2,{c,d})},

    <d>R={(e1,{d}),(e2,{d})}. Suppose (F1,E)={(e1,{a,c}),(e2,{a,c})} and

    (F2,E)={(e1,{a,d}),(e2,{a,d})}. Therefore,

    ¯Apr1S(F1,E)=(F,E){xˇ˜X:<x>R(F,E)Φ}={(e1,{a,b,c}),(e2,{a,b,c})}, ¯Apr1S(F2,E)=˜X and ¯Apr1S[(F1,E)(F2,E)]={(e1,{a,b}),(e2,{a,b})}. Hence, ¯Apr1S[(F1,E)(F2,E)]¯Apr1S(F1,E)¯Apr1S(F2,E).

    Corollary 3.1. Let RE be a soft binary relation of X. Then, the soft operator ¯Apr1S:SS(X)ESS(X)E is said to be a soft closure operator and (X,¯Apr1S) is standing for a soft closure space. Moreover, it induces a soft topology on X written as ˜τ1S and defined by ˜τ1S={(F,E)ˇSS(X)E:¯Apr1S(F,E)c=(F,E)c}.

    Theorem 3.2. Let (F,E),(G,E)ˇSS(X)E. The soft lower approximation defined by Eq (3.1) has the following properties:

    (1) Apr_1S(Φ)=Φ and Apr_1S(˜X)=˜X,

    (2) Apr_1S(F,E)(F,E),

    (3) (F,E)(G,E)Apr_1S(F,E)Apr_1S(G,E),

    (4) Apr_1S[(F,E)(G,E)]=Apr_1S(F,E)Apr_1S(G,E),

    (5) Apr_1S[(F,E)(G,E)]Apr_1S(F,E)Apr_1S(G,E),

    (6) Apr_1S(Apr_1S(F,E))=Apr_1S(F,E),

    (7) Apr_1S(F,E)=[¯Apr1S(F,E)c]c.

    Proof. It is the same as given in Theorem 3.1.

    Note that the equality in Theorem 3.2 part (5) did not hold in general (see Example 3.1).

    Take (F1,E)={(e1,{b,c}),(e2,{b,c})} and (F2,E)={(e1,{b,d}),(e2,{b,d})}. Then,

    Apr_1S(F1,E)={xˇ(F1,E):<x>R(F1,E)}=Φ, Apr_1S(F2,E)={(e1,{d}),(e2,{d})} and

    Apr_1S[(F1,E)(F2,E)]={(e1,{c,d}),(e2,{c,d})}, which means that

    Apr_1S[(F,E)(G,E)]Apr_1S(F,E)Apr_1S(G,E).

    Definition 3.5. Let RE be a soft binary relation over X and LE a soft ideal on X. For any soft set (F,E)ˇSS(X)E, the soft lower approximation and the soft upper approximation of (F,E) by LE, denoted by Apr_2S(F,E) and ¯Apr2S(F,E) are defined by:

    Apr_2S(F,E)={xˇ(F,E):<x>R(F,E)cˇLE}, (3.3)
    ¯Apr2S(F,E)=(F,E){xˇ˜X:<x>R(F,E)ˇLE}. (3.4)

    Theorem 3.3. Let (F,E),(G,E)ˇSS(X)E. The soft upper approximation defined by Eq (3.4) has the following properties:

    (1) ¯Apr2S(Φ)=Φ and ¯Apr2S(˜X)=˜X,

    (2) (F,E)¯Apr2S(F,E),

    (3) (F,E)(G,E)¯Apr2S(F,E)¯Apr2S(G,E),

    (4) ¯Apr2S[(F,E)(G,E)]¯Apr2S(F,E)¯Apr2S(G,E),

    (5) ¯Apr2S[(F,E)(G,E)]=¯Apr2S(F,E)¯Apr2S(G,E),

    (6) ¯Apr2S(¯Apr2S(F,E))=¯Apr2S(F,E),

    (7) ¯Apr2S(F,E)=[Apr_2S(F,E)c]c.

    Proof. (1),(2) Direct from Definition 3.5.

    (3) Let xˇ¯Apr2S[(F,E). Thus, <x>R(F,E)ˇLE. Since (F,E)(G,E) and LE is a soft ideal, <x>R(G,E)ˇLE. Therefore, xˇ¯Apr2S(G,E). Hence, ¯Apr2S(F,E)¯Apr2S(G,E).

    (4) Straightforward by part (3).

    (5) ¯Apr2S[(F,E)(G,E)]=[(F,E)(G,E)]{xˇ˜X:<x>R[(F,E)(G,E)]ˇLE}. Then, ¯Apr2S[(F,E)(G,E)]=[(F,E){xˇ˜X:<x>R(F,E)ˇLE}][(G,E){xˇ˜X:<x>R(G,E)ˇLE}]. Hence, ¯Apr2S[(F,E)(G,E)]=¯Apr2S((F,E))¯Apr2S((G,E)).

    (6) From part (2), we have ¯Apr2S(F,E)¯Apr2S(¯Apr2S(F,E)).

    Conversely, let xˇ¯Apr2S(¯Apr2S(F,E)). Then, <x>R¯Apr2S(F,E)ˇLE. Therefore, <x>R¯Apr1S(F,E)Φ. Thus, there exists yˇ<x>R¯Apr2S(F,E). That means <y>R⊑<x>R (by Lemma 3.1 part (1)) and <y>R(F,E)ˇLE. Then, <x>R(F,E)ˇLE. Hence, xˇ¯Apr2S(F,E). This completes the proof.

    (7)

    [Apr_2S(F,E)c]c=[(F,E)c{xˇ˜X:<x>R(F,E)ˇLE}]c=(F,E){xˇ˜X:<x>R(F,E)ˇLE}=¯Apr2S(F,E).

    Corollary 3.2. Let RE be a soft binary relation over X and LE be a soft ideal on X. Then, the soft operator ¯Apr2S:SS(X)ESS(X)E is said to be a soft closure operator and (X,¯Apr2S) is standing for a soft closure space. Moreover, it induces a soft topology on X written as ˜τ2S and defined by ˜τ2S={(F,E)ˇSS(X)E:¯Apr2S(F,E)c=(F,E)c}.

    Theorem 3.4. Let (F,E),(G,E)ˇSS(X)E. The soft lower approximation defined by Eq (3.3) has the following properties:

    (1) Apr_2S(Φ)=Φ and Apr_2S(˜X)=˜X,

    (2) Apr_2S(F,E)(F,E),

    (3) (F,E)(G,E)Apr_2S(F,E)Apr_2S(G,E),

    (4) Apr_2S[(F,E)(G,E)]=Apr_2S(F,E)Apr_2S(G,E),

    (5) Apr_2S[(F,E)(G,E)]Apr_2S(F,E)Apr_2S(G,E),

    (6) Apr_2S(Apr_2S(F,E))=Apr_2S(F,E),

    (7) Apr_2S(F,E)=[¯Apr2S(F,E)c]c.

    Proof. It is similar to that was given in Theorem 3.3.

    Definition 3.6. Let RE be a soft binary relation over X and LE be a soft ideal on X. For any soft set (F,E)ˇSS(X)E, the soft lower approximation and soft upper approximation of (F,E) by LE, denoted by Apr_3S(F,E) and ¯Apr3S(F,E) are defined by:

    Apr_3S(F,E)={xˇ(F,E):R<x>R(F,E)cˇLE}, (3.5)
    ¯Apr3S(F,E)=(F,E){xˇ˜X:R<x>R(F,E)ˇLE}. (3.6)

    Theorem 3.5. Let (F,E),(G,E)ˇSS(X)E. The soft upper approximation defined by Eq (3.6) has the following properties:

    (1) ¯Apr3S(Φ)=Φ and ¯Apr3S(˜X)=˜X,

    (2) (F,E)¯Apr3S(F,E),

    (3) (F,E)(G,E)¯Apr3S(F,E)¯Apr3S(G,E),

    (4) ¯Apr3S[(F,E)(G,E)]¯Apr3S(F,E)¯Apr3S(G,E),

    (5) ¯Apr3S[(F,E)(G,E)]=¯Apr3S(F,E)¯Apr3S(G,E),

    (6) ¯Apr3S(¯Apr3S(F,E))=¯Apr3S(F,E),

    (7) ¯Apr3S(F,E)=[Apr_3S(F,E)c]c.

    Proof. It is clear from Theorem 3.3.

    Corollary 3.3. Let RE be a soft binary relation over X and LE be a soft ideal on X. Then, the soft operator ¯Apr3S:SS(X)ESS(X)E is said to be a soft closure operator and (X,¯Apr3S) is standing for a soft closure space. In addition, (X,RE,LE) is said to be a soft ideal approximation space. Moreover, it induces a soft topology on X written as ˜τ3S and defined by ˜τ3S={(F,E)ˇSS(X)E:¯Apr3S(F,E)c=(F,E)c}. It is clear that ˜τ1S˜τ2S˜τ3S.

    Theorem 3.6. Let (F,E),(G,E)ˇSS(X)E. The soft lower approximation defined by Eq (3.5) has the following properties:

    (1) Apr_3S(Φ)=Φ and Apr_3S(˜X)=˜X,

    (2) Apr_3S(F,E)(F,E),

    (3) (F,E)(G,E)Apr_3S(F,E)Apr_3S(G,E),

    (4) Apr_3S[(F,E)(G,E)]=Apr_3S(F,E)Apr_3S(G,E),

    (5) Apr_3S[(F,E)(G,E)]Apr_3S(F,E)Apr_3S(G,E),

    (6) Apr_3S(Apr_3S(F,E))=Apr_3S(F,E),

    (7) Apr_3S(F,E)=[¯Apr3S(F,E)c]c.

    Corollary 3.4. Let RE be a soft binary relation over X, (F,E)ˇSS(X)E and LE be a soft ideal on X. Then,

    Apr_1S(F,E)Apr_2S(F,E)Apr_3S(F,E)(F,E)¯Apr3S(F,E)¯Apr2S(F,E)¯Apr1S(F,E).

    Proof. Direct from Definitions 3.4–3.6, using Lemma 3.1.

    We dedicate this is the main section of the manuscript to display the properties of soft sets in the soft ideal approximation spaces. It has been generated using the concepts of R<x>R, soft neighborhoods and soft ideals. We study the main properties in soft ideal approximation spaces which are generalizations of the same properties of ideal approximation spaces given by Abbas et al. in [31] and provide various illustrative examples.

    Lemma 4.1. Let (X,RE,LE) is be a soft ideal approximation space. Then,

    (1) Apr_1S(<x>R)=<x>R,

    (2) Apr_2S(<x>R)=<x>R,

    (3) Apr_3S(R<x>R)=R<x>R.

    Proof. We will ensure that item (1) and the other items will be similar. From Theorem 3.3 part (3), it is clear that Apr_2S(<x>R)⊑<x>R.

    Conversely, we will ensure that <x>RApr_2S(<x>R). Let yˇ<x>R. Then, by Lemma 3.1 part(1), <y>R⊑<x>R. Thus, <y>R(<x>R)c=Φ. So, <y>R(<x>R)cˇLE. Hence, yˇApr_2S(<x>R). Thus, <x>RApr_2S(<x>R).

    Proposition 4.1. Let (X,RE,LE) be a soft ideal approximation space. For xyˇ˜X,

    (1) xˇ¯Apr1S(yE) iff <x>RyEΦ and xˇ¯Apr1S(yE) iff <x>RyE=Φ,

    (2) xˇ¯Apr2S(yE) iff <x>RyEˇLE and xˇ¯Apr2S(yE) iff <x>RyEˇLE,

    (3) xˇ¯Apr3S(yE) iff R<x>RyEˇLE and xˇ¯Apr3S(yE) iff R<x>RyEˇLE.

    Proof. We will prove the second statement and the others will be similar. Let xˇ¯Apr2S(yE). Then,

    xˇ[yE{zˇ˜X: <z>RyEˇLE}]. Thus, <x>RyEˇLE. Conversely, let <x>RyEˇLE. Then, by Definition 3.6, xˇ¯Apr2S(yE).

    Proposition 4.2. Let (X,RE,LE) be a soft ideal approximation space and <x>RˇLE. Then,

    (1) Apr_1S(xE)=xE=¯Apr1S(xE),

    (2) Apr_2S(xE)=xE=¯Apr2S(xE),

    (3) Apr_3S(xE)=xE=¯Apr3S(xE).

    Proof. We will prove that the second statement and the others will be similar. Let <x>RˇLE. Then, <x>R[xE]cˇLE. Thus, xˇApr_2S(xE). So, Apr_2S(xE)=xE. Also, <x>RˇLE induces that <x>RyEˇLE for all yˇX. Hence, ¯Apr2S(xE)=xE.

    Theorem 4.1. Let (X,RE,LE) be a soft ideal approximation space and xˇ˜X,(F,E)ˇSS(X)E.

    If <x>R(F,E)ˇLE, then

    (1) <x>R¯Apr1S(F,E)=Φ,

    (2) <x>R¯Apr2S(F,E)ˇLE,

    (3) R<x>R¯Apr3S(F,E)ˇLE.

    Proof. We will prove the second part and the others will be similar. Suppose <x>R(F,E)ˇLE. It is clear that [<x>RxE](F,E)ˇLE. Then, xˇDS(F,E). Thus, <x>RDS(F,E)=Φ. So,

    <x>RDS(F,E)ˇLE. Hence, [<x>R(F,E)DS(F,E)]ˇLE. Therefore,

    <x>R¯Apr2S(F,E)ˇLE.

    Definition 4.1. Let (X,RE,LE) be a soft ideal approximation space and (F,E)ˇSS(X)E. The soft exterior of (F,E) is ExtiS(F,E)=Apr_iS(F,E)c, i{1,2,3}.

    Lemma 4.2. Let (X,RE,LE) be a soft ideal approximation space and (F,E),(G,E)ˇSS(X)E. For i{1,2,3}, we have

    (1) ExtiS(Φ)=˜X and ExtiS(˜X)=Φ,

    (2) ExtiS(F,E)(F,E)c,

    (3) (F,E)(G,E)ExtiS(F,E)ExtiS(G,E),

    (4) ExtiS[(F,E)(G,E)]=ExtiS(G,E)ExtiS(F,E),

    (5) \underline{Apr}_S^i(F, E) = Ext_S^i[Ext_S^i(F, E)],

    (6) Ext_S^i(F, E) = Ext_S^i([Ext_S^i(F, E)]^c).

    Proof. Straightforward from Theorems 3.2, 3.4, and 3.6.

    Definition 4.2. Let (X, R_E, {\mathcal{L}_E}) be a soft ideal approximation space and (F, E) \; \check{\in }\; SS(X)_E . Then, a soft point x_E\; \check{\in }\; SS(X)_E is called:

    (i) A soft accumulation point of (F, E) if (<x > R-x_E)\sqcap (F, E)\neq\Phi.

    The set of all soft ideal accumulation points of (F, E) is written as D_S(F, E) , that is,

    D_S(F, E) = \{x_E\; \check{\in }\; SS(X)_E \ : \ ( < x > R-x_E)\sqcap (F, E)\; \neq\; \Phi\}.

    (ii) A * -soft ideal accumulation point of (F, E) if (<x > R-x_E)\sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E}.

    The set of all * -soft ideal accumulation points of (F, E) is written as D_S^*(F, E) , that is,

    D_S^*(F, E) = \{x_E\; \check{\in }\; SS(X)_E \ : \ ( < x > R-x_E)\sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E}\}.

    (iii) A ** -soft ideal accumulation point of (F, E) if (R < x > R-x_E)\sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E}.

    The set of all ** -soft ideal accumulation points of (F, E) is written as D_S^{**}(F, E) , that is,

    D_S^{**}(F, E) = \{x_E\; \check{\in }\; SS(X)_E \ : \ (R < x > R-x_E)\sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E}\}.

    Lemma 4.3. Let (X, R_E, {\mathcal{L}_E}) be a soft ideal approximation space and (F, E) \; \check{\in }\; SS(X)_E . Then,

    (1) \overline{Apr}_S^1(F, E) = (F, E)\sqcup D_S(F, E),

    (2) \overline{Apr}_S^1(F, E) = (F, E) iff D_S(F, E)\sqsubseteq (F, E),

    (3) \overline{Apr}_S^2(F, E) = (F, E)\sqcup D_S^*(F, E),

    (4) \overline{Apr}_S^2(F, E) = (F, E) iff D_S^*(F, E)\sqsubseteq (F, E),

    (5) \overline{Apr}_S^3(F, E) = (F, E)\sqcup D_S^{**}(F, E),

    (6) \overline{Apr}_S^3(F, E) = (F, E) iff D_S^{**}(F, E)\sqsubseteq (F, E).

    Proof. We will prove that the third and forth statements and the others will be similar.

    (3) Let x\; \check{\in }\; \overline{Apr}_S^2(F, E). Then, x\; \check{\in }\; [(F, E)\sqcup \{y_E\; \check{\in }\; SS(X)_E \ : \ < y > R \sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E}\}] . Then, we have either x\; \check{\in }\; (F, E), that is,

    \begin{equation} x\; \check{\in }\; (F, E)\sqcup D_S^*(F, E) \end{equation} (4.1)

    or x\; \check{\notin }\; (F, E) . So, x\; \check{\in }\; \{y_E\; \check{\in }\; SS(X)_E: < y > R\sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E}\} . In the latter case, we have (<x > R-x_E)\sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E}. Hence, x\; \check{\in }\; D_S^*(F, E), that is,

    \begin{equation} x\; \check{\in }\; (F, E)\sqcup D_S^*(F, E). \end{equation} (4.2)

    From Eqs (4.1) and (4.2), \overline{Apr}_S^2(F, E)\sqsubseteq (F, E)\sqcup D_S^*(F, E). Conversely, let x\; \check{\in }\; (F, E)\sqcup D_S^*(F, E) . Then, we have either x\; \check{\in }\; (F, E), that is,

    \begin{equation} x\; \check{\in }\; \overline{Apr}_S^2(F, E) \end{equation} (4.3)

    or x\; \check{\notin }\; (F, E) . Thus, x\; \check{\in }\; D_S^*(F, E). So (<x > R-x_E)\sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E}. Hence, x\; \check{\in }\; \overline{Apr}_S^2(F, E), that is,

    \begin{equation} x\; \check{\in }\; \overline{Apr}_S^2(F, E). \end{equation} (4.4)

    From Eqs (4.3) and (4.4), (F, E) \sqcup D_S^*(F, E) \sqsubseteq \overline{Apr}_S^2(F, E).

    Therefore, \overline{Apr}_S^2(F, E) = (F, E)\sqcup D_S^*(F, E).

    (4) Let x\; \check{\notin }\; (F, E), that is, x\; \check{\notin }\; \overline{Apr}_S^2(F, E). Then, < x > R\sqcap (F, E)\; \check{\in }\; {\mathcal{L}_E}. Thus,

    (<x > R-x_E)\sqcap (F, E)\; \check{\in }\; {\mathcal{L}_E} and x\; \check{\notin }\; D_S^*(F, E). Conversely, let D_S^*(F, E)\sqsubseteq (F, E). Then, by part (1), D_S^*(F, E)\sqcup (F, E) = \overline{Apr}_S^2(F, E) = (F, E) .

    Lemma 4.4. Let (X, R_E, {\mathcal{L}_E}) be a soft ideal approximation space and (F, E), (G, E)\; \check{\in }\; SS(X)_E . Then,

    (1) if (F, E)\sqsubseteq (G, E), then D_S^{*}(F, E)\sqsubseteq D_S^{*}(G, E) and D_S^{**}(F, E)\sqsubseteq D_S^{**}(G, E),

    (2) D_S^*[(F, E)\sqcup (G, E)] = D_S^*(F, E)\sqcup D_S^*(F, E) and D_S^{**}[(F, E)\sqcup (G, E)] = D_S^{**}(F, E)\sqcup D_S^{**}(F, E),

    (3) D_S^*[(F, E)\sqcap (G, E)]\sqsubseteq D_S^*(F, E)\sqcap D_S^*(F, E) and D_S^{**}[(F, E)\sqcap (G, E)]\sqsubseteq D_S^{**}(F, E)\sqcap D_S^{**}(F, E),

    (4) D_S^{*}[(F, E)\sqcup D_S^{*}(F, E)]\sqsubseteq (F, E)\sqcup D_S^{*}(F, E) and D_S^{**}[(F, E)\sqcup D_S^{**}(F, E)]\sqsubseteq (F, E)\sqcup D_S^{**}(F, E) .

    Proof. (1) Suppose (F, E)\sqsubseteq (G, E) and let x\; \check{\in }\; D_S^*(F, E). Then, [< x > R-x_E]\sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E}. Thus, [< x > R-x_E]\sqcap (G, E)\; \check{\notin }\; {\mathcal{L}_E} . So, x\; \check{\in }\; D_S^*(G, E). The second part is easily proved.

    (2) Since (F, E)\sqsubseteq (F, E)\sqcup (G, E) and (G, E)\sqsubseteq (F, E)\sqcup (G, E), by part (1), we have D_S^*(F, E)\sqcup D_S^*(G, E)\sqsubseteq D_S^*(F, E)\sqcup (G, E)).

    Conversely, let x\; \check{\notin }\; (D_S^*(F, E)\sqcup D_S^*(G, E). Then, x\; \check{\notin }\; D_S^*(F, E) and x\; \check{\notin }\; D_S^*(G, E). Thus, (<x > R-x_E)\sqcap (F, E)\; \check{\in }\; {\mathcal{L}_E} and (<x > R-x_E)\sqcap (G, E)\; \check{\in }\; {\mathcal{L}_E} . So, (<x > R-x_E)\sqcap (F, E)\sqcup (G, E))\; \check{\in }\; {\mathcal{L}_E} . Hence, x\; \check{\in }\; D_S^*[(F, E)\sqcup (G, E)] . The proof of the second part is similar.

    (3) Similar to part (2).

    (4) Let x\; \check{\notin }\; (F, E)\sqcup D_S^*(F, E). It is obvious that x\; \check{\notin }\; (F, E) and (<x > R-x_E)\sqcap (F, E)\; \check{\in }\; {\mathcal{L}_E} . Then, < x > R\sqcap (F, E)\; \check{\in }\; {\mathcal{L}_E} . Thus, x\; \check{\notin }\; \overline{Apr}_S^2(F, E). So, x\; \check{\notin }\; \overline{Apr}_S^2(\overline{Apr}_S^2(F, E)). Hence, x\; \check{\notin }\; D_S^*(\overline{Apr}_S^2(F, E)) = D_S^*(F, E)\sqcup D_S^*(F, E)). Therefore, D_S^*(F, E)\sqcup D_S^*(F, E)\sqsubseteq (F, E)\sqcup D_S^*(F, E). The proof of the second part is similar.

    Corollary 4.1. Let (X, R_E, {\mathcal{L}_E}) be any soft ideal approximation space and (F, E)\; \check{\in }\; SS(X)_E . Then,

    \begin{equation*} D_S^{**}(F, E)\sqsubseteq D_S^*(F, E)\sqsubseteq D_S(F, E). \end{equation*}

    Proof. Let x\; \check{\notin }\; D_S(F, E). Then, (<x > R-x_E)\sqcap (F, E) = \Phi. Thus, (<x > R-x_E)\sqcap (F, E)\; \check{\in }\; {\mathcal{L}_E} . So, x\; \check{\notin }\; D_S^*(F, E) and (R < x > R-x_E)\sqcap (F, E)\; \check{\in }\; {\mathcal{L}_E}, where R < x > R\sqsubseteq < x > R . Hence, x\; \check{\notin }\; D_S^{**}(F, E). Therefore, D_S^{**}(F, E)\sqsubseteq D_S^*(F, E)\sqsubseteq D_S(F, E).

    Remark 4.1. The converse of the previous result is not true.

    Example 4.1. Let X = \{a, b, c\} associated with a set of parameters E = \{e_1, e_2\} . Let R_E be a soft relation of X and {\mathcal{L}_E} be a soft ideal on X , defined respectively by:

    R = \{(e_1, \{(a, a), (a, b), (a, c), (b, b), (b, c), (c, c)\}), (e_2, \{(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, b),
    (c, c)\})\}

    {\mathcal{L}_E} = \{ \Phi, (F_1, E), (F_2, E), (F_3, E)\} where,

    (F_1, E) = \{(e_1, \{c\}), (e_2, \phi)\}, (F_2, E) = \{(e_1, \phi), (e_2, \{c\})\}, (F_3, E) = \{(e_1, \{c\}), (e_2, \{c\})\}.

    Then, < a > R = \{(e_1, \{a, b, c\}), (e_2, \{a, b, c\})\}, \; < b > R = \{(e_1, \{b, c\}), (e_2, \{b, c\})\} ,

    \; < c > R = c_E. Also, R < a > = a_E, \; R < b > = \{(e_1, \{a, b\}), (e_2, \{a, b\})\}, \; R < c > = < a > R. Thus, R < a > R = a_E, \; R < b > R = b_E, \; R < c > R = c_E. Suppose (F, E) = \{(e_1, \{b, c\}), (e_2, \{b, c\})\}. Then, we have:

    ( < a > R-a_E)\sqcap (F, E) = (F, E)\neq\Phi,
    ( < b > R-b_E)\sqcap (F, E) = c_E\neq\Phi,
    ( < c > R-c_E)\sqcap (F, E) = \Phi.

    Thus, a\; \check{\in }\; D_S(F, E), \; b\; \check{\in }\; D_S(F, E), \; c\; \check{\notin }\; D_S(F, E). So, D_S(F, E) = \{(e_1, \{a, b\}), (e_2, \{a, b\})\} . On the other hand, we get:

    ( < a > R-a_E)\sqcap (F, E) = (F, E)\; \check{\notin }\; {\mathcal{L}_E},
    ( < b > R-b_E)\sqcap (F, E) = (F_3, E)\; \check{\in }\; {\mathcal{L}_E},
    ( < c > R-c_E)\sqcap (F, E) = \Phi\; \check{\in }\; {\mathcal{L}_E}.

    Thus, a\; \check{\in }\; D_S^*(F, E), \; b\; \check{\notin }\; D_S^*(F, E), \; c\; \check{\notin }\; D_S^*(F, E). Hence, D_S^*(F, E) = a_E . Also, we have:

    (R < a > R-a_E)\sqcap(F, E) = \Phi\; \check{\in }\; {\mathcal{L}_E},
    (R < b > R-b_E)\sqcap (F, E) = \Phi\; \check{\in }\; {\mathcal{L}_E},
    (R < c > R-c_E)\sqcap (F, E) = \Phi\; \check{\in }\; {\mathcal{L}_E}.

    Then, a\; \check{\notin }\; D_S^{**}(F, E), \; b\; \check{\notin }\; D_S^{**}(F, E), \; c\; \check{\notin }\; D_S^{**}(F, E). Thus, D_S^{**}(F, E) = \Phi. So, D_S(F, E)\not \sqsubseteq D_S^*(F, E)\not \sqsubseteq D_S^{**}(F, E).

    Definition 4.3. Let (X, R_E, {\mathcal{L}_E}) be any soft ideal approximation space and (F, E)\; \check{\in }\; SS(X)_E . Then, (F, E) is said to be:

    (i) soft dense if \overline{Apr}_S^1(F, E) = \tilde{X},

    (ii) * -soft ideal dense if \overline{Apr}_S^2(F, E) = \tilde{X},

    (iii) ** -soft ideal dense if \overline{Apr}_S^3(F, E) = \tilde{X},

    (iv) soft nowhere dense if \underline{Apr}_S^1(\overline{Apr}_S^1(F, E)) = \Phi,

    (v) * -soft ideal nowhere dense if \underline{Apr}_S^1(\overline{Apr}_S^2(F, E)) = \Phi,

    (vi) ** -soft ideal nowhere dense if \underline{Apr}_S^1(\overline{Apr}_S^3(F, E)) = \Phi.

    Corollary 4.2. Let (X, R_E, {\mathcal{L}_E}) be any soft ideal approximation space and (F, E)\; \check{\in }\; SS(X)_E . Then,

    (1) ** -soft ideal dense \Longrightarrow* -soft ideal dense \Longrightarrow soft dense,

    (2) soft nowhere dense \Longrightarrow* -soft ideal nowhere dense \Longrightarrow ** -soft ideal nowhere dense.

    Proof. Immediately from Definition 4.3 and part (3) of Theorem 3.5.

    Example 4.2. Let X = \{a, b, c\} , E = \{e_1, e_2\},

    R_E = \{(e_1, \{(a, a), (a, b), (b, b), (b, c), (c, c), (d, d), (d, b)), (e_2, \{(a, a), (a, b), (a, c), (b, b), (b, c), (c, c), \\ (d, d), (d, b))\}\} and {\mathcal L}_{E} = \{\Phi, (F_1, E), (F_2, E), (F_3, E)\} , where

    (F_1, E) = \{(e_1, \{a\}), (e_2, \phi)\}, (F_2, E) = \{(e_1, \phi), (e_2, \{a\})\}, (F_3, E) = \{(e_1, \{a\}), (e_2, \{a\})\}.

    Therefore, we have < a > R = \{(e_1, \{a, b\}), (e_2, \{a, b\})\}, \; < b > R = \{(e_1, \{b\}), (e_2, \{b\})\}, \; < c > R = \{(e_1, \{c\}), (e_2, \{c\})\}, \; < d > R = \{(e_1, \{b, d\}), (e_2, \{b, d\})\}. Also, R < a > = \{(e_1, \{a\}), (e_2, \{a\})\}, \; R < b > = \{(e_1, \{b\}), (e_2, \{b\})\}, \; R < c > = \{(e_1, \{b, c\}), (e_2, \{b, c\})\}\; R < d > = \{(e_1, \{d\}), (e_2, \{d\})\}. Thus, R < a > R = \{(e_1, \{a\}), (e_2, \{a\})\}, \; R < b > R = \{(e_1, \{b\}), (e_2, \{b\})\}, \; R < c > R = \{(e_1, \{c\}), (e_2, \{c\})\}\; R < d > R = \{(e_1, \{d\}), (e_2, \{d\})\}. Suppose (F, E) = \{(e_1, \{b, c\}), (e_2, \{b, c\})\}. Then, \overline{Apr}_S^2(F, E) = (F, E)\sqcup \{x \; \check{\in }\; \tilde{X}\; : < x > R \sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E} \} = \tilde{X}. Also, \overline{Apr}_S^3(F, E) = (F, E)\sqcup \{x \; \check{\in }\; \tilde{X}\; :R < x > R \sqcap (F, E)\; \check{\notin }\; {\mathcal{L}_E} \} = (F, E)\neq \tilde{X}. Hence, (F, E) is * -soft ideal dense but not ** -soft ideal dense.

    Corollary 4.3. Let (X, R_E, {\mathcal{L}_E}) be any soft ideal approximation space and (F, E)\; \check{\in }\; SS(X)_E . Then,

    (1) If (F, E) is soft dense, then [\overline{Apr}_S^1(F, E)]^c is soft nowhere dense.

    (2) If (F, E) is * -soft ideal dense, then [\overline{Apr}_S^2(F, E)]^c is * -soft ideal nowhere dense.

    (3) If (F, E) is ** -soft ideal dense, then [\overline{Apr}_S^3(F, E)]^c is ** -nowhere dense.

    Proof. (1) Suppose (F, E) is soft dense. Then, \overline{Apr}_S^1(F, E) = \tilde{X} . Thus, [\overline{Apr}_S^1(F, E)]^c = \Phi and

    \overline{Apr}_S^1[(\overline{Apr}_S^1(F, E))^c] = \Phi. So, \underline{Apr}_S^1[\overline{Apr}_S^1(\overline{Apr}_S^1(F, E))^c)] = \Phi. Hence,

    [\overline{Apr}_S^1(F, E)]^c is nowhere soft dense.

    (2) Suppose (F, E) is * -soft ideal dense. Then, \overline{Apr}_S^2(F, E) = \tilde{X}. Thus, [\overline{Apr}_S^2(F, E)]^c = \Phi. So,

    \overline{Apr}_S^2[(\overline{Apr}_S^2(F, E))^c] = \Phi and \underline{Apr}_S^1[\overline{Apr}_S^2((\overline{Apr}_S^2(F, E))^c)] = \Phi. Hence,

    [\overline{Apr}_S^2(F, E)]^c is * -soft ideal nowhere dense.

    (3) Similar to part (2).

    In this section, we introduce soft lower separation axioms via soft binary relations and soft ideal as a generalization of lower separation axioms given in [31]. We scrutinize its essential characterizations and infer some of its relationships associated with the soft ideal closure operators. Some illustrative examples are given. In an approximation space (X, R) where R is an equivalence relation on X , a general topology is generated by the lower approximations L(A) or the upper approximations U(A) of any subset as follows. \tau_R = \{A \subseteq X : \ A = L(A)\} or \tau_R = \{A \subseteq X : \ A^c = U(A^c)\} . In the soft case, it is an extension of the same definitions.

    Definition 5.1. (1) A soft approximation space (X, R_E) is said to be a soft- T_0 space if \forall x\neq y\; \check{\in }\; \tilde{X} , there exists (F, E)\; \check{\in }\; SS(X)_E such that

    x\; \check{\in }\; \underline{Apr}_S^1(F, E), \; y\; \check{\notin }\; (F, E)\; \text{or} \; y\; \check{\in }\; \underline{Apr}_S^1(F, E), \; x\; \check{\notin }\; (F, E).

    (2) A soft ideal approximation space (X, R_E, {\mathcal{L}_E}) is said to be a soft- T_0^* space if \forall x\neq y\; \check{\in }\; \tilde{X} , there exists (F, E)\; \check{\in }\; SS(X)_E such that

    x\; \check{\in }\; \underline{Apr}_S^2(F, E), \; y\; \check{\notin }\; (F, E)\; \text{or} \; y\; \check{\in }\; \underline{Apr}_S^2(F, E), \; x\; \check{\notin }\; (F, E).

    (3) A soft ideal approximation space (X, R_E, {\mathcal{L}_E}) is said to be a soft- T_0^{**} space if \forall x\neq y\; \check{\in }\; \tilde{X} , there exists (F, E)\; \check{\in }\; SS(X)_E such that

    x\; \check{\in }\; \underline{Apr}_S^3(F, E), \; y\; \check{\notin }\; (F, E)\; \text{or}\; y\; \check{\in }\; \underline{Apr}_S^3(F, E), \; x\; \check{\notin }\; (F, E).

    Proposition 5.1. For a soft ideal approximation space (X, R_E, {\mathcal{L}_E}) , these properties are equivalent:

    (1) \tilde{X} is a soft- T_0^* space.

    (2) \overline{Apr}_S^2(x_E)\neq\overline{Apr}_S^2(y_E) for all x\neq y \; \check{\in }\; \tilde{X}.

    Proof.

    (1) \Rightarrow (2) : For each x\neq y \; \check{\in }\; \tilde{X}, by part (1), there exists (F, E)\; \check{\in }\; SS(X)_E such that x\; \check{\in }\; \underline{Apr}_S^2(F, E) , y\; \check{\notin }\; (F, E). Thus, < x > R \sqcap(F, E)^c\; \check{\in }\; {\mathcal{L}_E}, y\; \check{\in }\; (F, E)^c. So, < x > R \sqcap y_E\; \check{\in }\; {\mathcal{L}_E} and by Proposition 4.1 part (1), x\; \check{\notin }\; \overline{Apr}_S^2(y_E). Similarly, we can prove that y\; \check{\notin }\; \overline{Apr}_S^2(x_E). Therefore, \overline{Apr}_S^2(x_E)\neq\overline{Apr}_S^2(y_E).

    (2) \Rightarrow (1) : Suppose part (2) holds and let x\neq y \; \check{\in }\; \tilde{X}. Then, x\; \check{\notin }\; \overline{Apr}_S^2(y_E) or y\; \check{\notin }\; \overline{Apr}_S^2(x_E). By Proposition 4.1 part (2), < x > R \sqcap y_E\; \check{\in }\; {\mathcal{L}_E} or < y > R \sqcap x_E\; \check{\in }\; {\mathcal{L}_E}. Thus, [x\; \check{\in }\; \underline{Apr}_S^2(y_E)^c, y\; \check{\notin }\; (y_E)^c] or [y\; \check{\in }\; \underline{Apr}_S^2(x_E)^c, x\; \check{\notin }\; (x_E)^c]. Therefore, \tilde{X} is soft- T_0^* space.

    Corollary 5.1. For a soft approximation space (X, R_E) , these properties are equivalent:

    (1) \tilde{X} is a soft- T_0 space.

    (2) \overline{Apr}_S^1(x_E)\neq\overline{Apr}_S^1(y_E) for each x\neq y \; \check{\in }\; \tilde{X}.

    Corollary 5.2. For a soft ideal approximation space (X, R_E, {\mathcal{L}_E}) , these properties are equivalent:

    (1) \tilde{X} is a soft- T_0^{**} space.

    (2) \overline{Apr}_S^3(x_E)\neq\overline{Apr}_S^3(y_E) for all x\neq y \; \check{\in }\; \tilde{X}.

    Definition 5.2. (1) A soft approximation space (X, R_E) is said to be a soft- T_1 space if \forall x\neq y\; \check{\in }\; \tilde{X} , there exist (F, E), (G, E)\; \check{\in }\; SS(X)_E such that

    x\; \check{\in }\; \underline{Apr}_S^1(F, E), \; y\; \check{\notin }\; (F, E)\; \text{and} \; y\; \check{\in }\; \underline{Apr}_S^1(G, E), \; x\; \check{\notin }\; (G, E).

    (2) A soft ideal approximation space (X, R_E, {\mathcal{L}_E}) is said to be a soft- T_1^* space if \forall x\neq y\; \check{\in }\; \tilde{X} , there exist (F, E), (G, E)\; \check{\in }\; SS(X)_E such that

    x\; \check{\in }\; \underline{Apr}_S^2(F, E), \; y\; \check{\notin }\; (F, E)\; \text{and} \; y\; \check{\in }\; \underline{Apr}_S^2(G, E), \; x\; \check{\notin }\; (G, E).

    (3) A soft ideal approximation space (X, R_E, {\mathcal{L}_E}) is said to be a soft- T_1^{**} space if \forall x\neq y\; \check{\in }\; \tilde{X} , there exist (F, E), (G, E)\; \check{\in }\; SS(X)_E such that

    x\; \check{\in }\; \underline{Apr}_S^3(F, E), \; y\; \check{\notin }\; (F, E)\; \text{and} \; y\; \check{\in }\; \underline{Apr}_S^3(G, E), \; x\; \check{\notin }\; (G, E).

    Proposition 5.2. For a soft ideal approximation space (X, R_E, {\mathcal{L}_E}) , these properties are equivalent:

    (1) \tilde{X} is a soft- T_1^* space.

    (2) \overline{Apr}_S^2(x_E) = x_E for all x \; \check{\in }\; \tilde{X}.

    (3) D_S^*(x_E) = \Phi for each x \; \check{\in }\; \tilde{X}.

    Proof. (1) \Rightarrow (2) : Suppose (X, R_E, {\mathcal{L}_E}) is a soft- T_1^* space and let x \; \check{\in }\; \tilde{X} . Thus, for y\; \check{\in }\; \tilde{X}-x_E, \; x\neq y and \exists(F, E)\; \check{\in }\; SS(X)_E such that y\; \check{\in }\; \underline{Apr}_S^2(F, E), \; x\; \check{\notin }\; (F, E). Thus, < y > R \sqcap(F, E)^c\; \check{\in }\; {\mathcal{L}_E}, \; x\; \check{\in }\; (F, E)^c. So, < y > R \sqcap x_E\; \check{\in }\; {\mathcal{L}_E}, that is, y\; \check{\notin }\; \overline{Apr}_S^2(x_E). Hence, \overline{Apr}_S^2(x_E) = x_E.

    (2) \Rightarrow (3) : Suppose part (2) holds and let x\; \check{\in }\; \tilde{X}. Then, \overline{Apr}_S^2(x_E) = x_E\sqcup D_S^*x_E but x\; \check{\notin }\; D_S^*x_E. Thus, D_S^*x_E = \Phi.

    (3) \Rightarrow (1) : Suppose part (3) holds and x\neq y\; \check{\in }\; \tilde{X}. By part (3), D_S^*x_E = D_S^*y_E = \Phi. Thus, \overline{Apr}_S^2(x_E) = x_E and \overline{Apr}_S^2(y_E) = y_E, that is, \underline{Apr}_S^2(x_E)^c = (x_E)^c and \underline{Apr}_S^2(y_E)^c = (y_E)^c. So, there exist (x_E)^c and (y_E)^c\; \check{\in }\; SS(X)_E such that y\; \check{\in }\; \underline{Apr}_S^2(x_E)^c, \; x\; \check{\notin }\; (x_E)^c\; \text{and}\; x\; \check{\in }\; \underline{Apr}_S^2(y_E)^c, \; y\; \check{\notin }\; (y_E)^c. Therefore, \tilde{X} is a soft- T_1^* space.

    Corollary 5.3. For a soft approximation space (X, R_E) , these properties are equivalent:

    (1) \tilde{X} is a soft- T_1 space.

    (2) \overline{Apr}_S^1(x_E) = x_E for all x \; \check{\in }\; \tilde{X}.

    (3) D_S(x_E) = \Phi for each x \; \check{\in }\; \tilde{X}.

    Corollary 5.4. For a soft ideal approximation space (X, R_E, {\mathcal{L}_E}) , these properties are equivalent:

    (1) \tilde{X} is a soft- T_1^{**} space.

    (2) \overline{Apr}_S^3(x_E) = x_E for all x \; \check{\in }\; \tilde{X}.

    (3) D_S^{**}(x_E) = \Phi for each x \; \check{\in }\; \tilde{X}.

    Definition 5.3. (1) A soft approximation space (X, R_E) is said to be a soft- R_0 space if,

    for all x\neq y\; \check{\in }\; \tilde{X} ,

    \overline{Apr}_S^1(x_E) = \overline{Apr}_S^1(y_E)\; \text{or}\; \overline{Apr}_S^1(x_E)\sqcap\overline{Apr}_S^1(y_E) = \Phi.

    (2) A soft ideal approximation space (X, R_E, {\mathcal{L}_E}) is said to be a soft- R_0^* space if, for all x\neq y\; \check{\in }\; \tilde{X},

    \overline{Apr}_S^2(x_E) = \overline{Apr}_S^2(y_E)\; \text{or}\; \overline{Apr}_S^2(x_E)\sqcap\overline{Apr}_S^2(y_E) = \Phi.

    (3) A soft ideal approximation space (X, R_E, {\mathcal{L}_E}) is said to be a soft- R_0^{**} space if, for all x\neq y\; \check{\in }\; \tilde{X},

    \overline{Apr}_S^3(x_E) = \overline{Apr}_S^3(y_E)\; \text{or}\; \overline{Apr}_S^3(x_E)\sqcap\overline{Apr}_S^3(y_E) = \Phi.

    Proposition 5.3. For a soft ideal approximation space (X, R_E, {\mathcal{L}_E}) , these properties are equivalent:

    (1) \tilde{X} is a soft- R_0^{*} space,

    (2) if x\; \check{\in }\; \overline{Apr}_S^2(y_E) , then y\; \check{\in }\; \overline{Apr}_S^2(x_E) for all x\neq y \; \check{\in }\; \tilde{X}.

    Proof.

    (1) \Rightarrow (2) : Suppose statement (1) holds, and let x \neq y be two soft points in (X, R_E, {\mathcal{L}_E}) . Then, \overline{Apr}_S^2(x_E) = \overline{Apr}_S^2(y_E) or \overline{Apr}_S^2(x_E)\sqcap\overline{Apr}_S^2(y_E) = \Phi.

    If \overline{Apr}_S^2(x_E) = \overline{Apr}_S^2(y_E) , then y\; \check{\in }\; \overline{Apr}_S^2(x_E) and x\; \check{\in }\; \overline{Apr}_S^2(y_E).

    If \overline{Apr}_S^2(x_E)\sqcap\overline{Apr}_S^2(y_E) = \Phi, then x_E\sqcap\overline{Apr}_S^2(y_E) = \Phi and y_E\sqcap\overline{Apr}_S^2(x_E) = \Phi. Thus, x\; \check{\notin }\; \overline{Apr}_S^2(y_E) and y\; \check{\notin }\; \overline{Apr}_S^2(x_E)). So, x\; \check{\notin }\; \overline{Apr}_S^2(y_E) and y\; \check{\notin }\; \overline{Apr}_S^2(x_E). Hence, in either case, statement (2) holds.

    (2) \Rightarrow (1) : Suppose that statement (2) holds and let x\neq y \; \check{\in }\; \tilde{X}. Then, we have

    \text{either}\; [x\; \check{\in }\; \overline{Apr}_S^2(y_E)\; \text{and} \; y\; \check{\in }\; \overline{Apr}_S^2(x_E)]\; \text{or} \; [x\; \check{\notin }\; \overline{Apr}_S^2(y_E)\; \text{and} \; y\; \check{\notin }\; \overline{Apr}_S^2(x_E)].

    If x\; \check{\in }\; \overline{Apr}_S^2(y_E)\; \text{and} \; y\; \check{\in }\; \overline{Apr}_S^2(x_E), then

    \begin{equation} \overline{Apr}_S^2(x_E) = \overline{Apr}_S^2(y_E). \end{equation} (5.1)

    If x\; \check{\notin }\; \overline{Apr}_S^2(y_E)\; \text{and} \; y\; \check{\notin }\; \overline{Apr}_S^2(x_E), then

    \begin{equation} \overline{Apr}_S^2(x_E)\sqcap\overline{Apr}_S^2(y_E) = \Phi. \end{equation} (5.2)

    From (5.1) and (5.2), the proof is complete.

    Corollary 5.5. For a soft approximation space (X, R_E) , these properties are equivalent:

    (1) \tilde{X} is a soft- R_0 space,

    (2) if x\; \check{\in }\; < y > R , then y\; \check{\in }\; < x > R for any x\neq y \; \check{\in }\; \tilde{X}.

    Corollary 5.6. For a soft ideal approximation space (X, R_E, {\mathcal{L}_E}) , these properties are equivalent:

    (1) \tilde{X} is a soft- R_0^{**} space,

    (2) if x\; \check{\in }\; \overline{Apr}_S^3(y_E) , then y\; \check{\in }\; \overline{Apr}_S^3(x_E) for all x\neq y \; \check{\in }\; \tilde{X}.

    Definition 5.4. (1) A soft approximation space (X, R_E) is said to be a soft- T_2 space if \forall x\neq y\; \check{\in }\; \tilde{X} , there exist (F, E), (G, E)\; \check{\in }\; SS(X)_E such that

    \begin{equation*} x\; \check{\in }\; \underline{Apr}_S^1(F, E), \; y\; \check{\in }\; \underline{Apr}_S^1(G, E)\; \text{and}\; (F, E)\sqcap (G, E) = \Phi. \end{equation*}

    (2) A soft ideal approximation space (X, R_E, {\mathcal{L}_E}) is said to be a soft- T_2^* space if \forall x\neq y\; \check{\in }\; \tilde{X} , there exist (F, E), (G, E)\; \check{\in }\; SS(X)_E such that

    \begin{equation*} x\; \check{\in }\; \underline{Apr}_S^2(F, E), \; y\; \check{\in }\; \underline{Apr}_S^2(G, E)\; \text{and}\; (F, E)\sqcap (G, E) = \Phi. \end{equation*}

    (3) A soft ideal approximation space (X, R_E, {\mathcal{L}_E}) is said to be a soft- T_2^{**} space if \forall x\neq y\; \check{\in }\; \tilde{X} , there exist (F, E), (G, E)\; \check{\in }\; SS(X)_E such that

    \begin{equation*} x\; \check{\in }\; \underline{Apr}_S^3(F, E), \; y\; \check{\in }\; \underline{Apr}_S^3(G, E), \; \text{and}\; (F, E)\sqcap (G, E) = \Phi. \end{equation*}

    Theorem 5.1. For a soft ideal approximation space (X, R_E, {\mathcal{L}_E}) , these properties are equivalent:

    (1) \tilde{X} is a soft- T_2^* space,

    (2) \exists (F, E)\; \check{\in }\; SS(X)_E: x\; \check{\in }\; \underline{Apr}_S^2(F, E), y\; \check{\in }\; [\overline{Apr}_S^2(F, E)]^c for any x\neq y \; \check{\in }\; \tilde{X}.

    Proof. (1) \Rightarrow (2) : Suppose \tilde{X} is a soft- T_2^* space and let x\neq y\; \check{\in }\; \tilde{X} . Then, there exist

    (F, E), \; (G, E)\; \check{\in }\; SS(X)_E such that x\; \check{\in }\; \underline{Apr}_S^2(F, E), \; y\; \check{\in }\; \underline{Apr}_S^2(G, E) and (F, E)\sqcap (G, E) = \Phi. Thus, < y > R \sqcap (G, E)^c\; \check{\in }\; {\mathcal{L}_E} and (F, E)\sqsubseteq (G, E)^c. So [< y > R-x_E]\sqcap (F, E)\; \check{\in }\; {\mathcal{L}_E}, that is, y\; \check{\notin }\; D_S^*(F, E). Hence, \underline{Apr}_S^2(G, E)\sqcap D_S^*(F, E) = \Phi and \underline{Apr}_S^2(G, E)\sqcap (F, E) = \Phi, that is, \underline{Apr}_S^2(G, E)\sqcap \overline{Apr}_S^2(F, E) = \Phi. Therefore, x\; \check{\in }\; \underline{Apr}_S^2(F, E), y\; \check{\in }\; \underline{Apr}_S^2(G, E)\sqsubseteq[\overline{Apr}_S^2(F, E)]^c.

    (2) \Rightarrow (1) : Suppose part (2) holds and let x\neq y\; \check{\in }\; \tilde{X}. Then, there exists (F, E)\; \check{\in }\; SS(X)_E such that x\; \check{\in }\; \underline{Apr}_S^2(F, E), \; y\; \check{\in }\; [\overline{Apr}_S^2(F, E)]^c . Let (G, E) = [\overline{Apr}_S^2(F, E)]^c . Then, (G, E) = \underline{Apr}_S^2(F, E)^c (from Theorem 3.3 part (7)) and so \underline{Apr}_S^2(G, E) = \underline{Apr}_S^2[\underline{Apr}_S^2(F, E)^c] = \underline{Apr}_S^2(F, E)^c = (G, E). Also, (F, E)\sqcap (G, E) = (F, E)\sqcap \underline{Apr}_S^2(F, E)^c\sqsubseteq (F, E)\sqcap (F, E)^c = \Phi. Hence, \tilde{X} is a soft- T_2^* space.

    Corollary 5.7. For a soft approximation space (X, R_E) , these properties are equivalent:

    (1) \tilde{X} is a soft- T_2 space,

    (2) \exists (F, E)\; \check{\in }\; SS(X)_E: x\; \check{\in }\; \underline{Apr}_S^1(F, E), y\; \check{\in }\; [\overline{Apr}_S^1(F, E)]^c for all x\neq y \; \check{\in }\; \tilde{X}.

    Corollary 5.8. For a soft ideal approximation space (X, R_E, {\mathcal{L}_E}) , these properties are equivalent:

    (1) \tilde{X} is a soft- T_2^{**} space,

    (2) \exists (F, E)\; \check{\in }\; SS(X)_E: x\; \check{\in }\; \underline{Apr}_S^3(F, E), y\; \check{\in }\; [\overline{Apr}_S^3(F, E)]^c for all x\neq y \; \check{\in }\; \tilde{X}.

    Corollary 5.9. For a soft ideal approximation space (X, R_E, {\mathcal{L}_E}) , these conditions hold:

    (1) soft- T_1 = soft- R_0 + soft- T_0 ,

    (2) soft- T_1^{*} = soft- R_0^{*} + soft- T_0^{*} ,

    (3) soft- T_1^{**} = soft- R_0^{**} + soft- T_0^{**} .

    Proof. Straightforward from Definition 5.3, Propositions 5.1 and 5.2, and Corollaries 5.1–5.4.

    Remark 5.1. From Definitions 5.1, 5.2, and 5.4 we have the following implication.

    Example 5.1 (1) Let X = \mathbb{Z} with E = \{e_1, e_2\} and R:E\longrightarrow P(\mathbb{Z}\times \mathbb{Z}) be a soft relation over \mathbb{Z}\times \mathbb{Z} defined by R(e_1) = \mathbb{Z}\times \mathbb{Z}, R(e_2) = \mathbb{N}\times \mathbb{N} and \mathcal{L}_E = \{(F, E)\; \check{\in }\; SS(X)_E: (F, E) ~~{ is~~ a ~~finite~~ soft ~~set}\}. Thus,

    \begin{equation*} \underline{Apr}_S^1(F, E) = \left\{ \begin{array}{rcl} (F, E) & \; \; \; {if} \; (F, E)^c\; \check{\in }\; \mathcal{L}_E, \\ \Phi& {otherwise.} \end{array}\right. \end{equation*}

    Thus, \forall x\neq y\; \check{\in }\; \tilde{\mathbb{Z}} , we have:

    \begin{equation*} x\; \check{\in }\; \underline{Apr}_S^1(y_E)^c = (y_E)^c, \; y\; \check{\notin }\; (y_E)^c\; \text{and} \; y\; \check{\in }\; \underline{Apr}_S^1(x_E)^c = (x_E)^c, \; x\; \check{\notin }\; (x_E)^c. \end{equation*}

    So, \tilde{\mathbb{Z}} is a soft- T_1 space. But \tilde{\mathbb{Z}} is not a soft- T_2 space, since if x\; \check{\in }\; \underline{Apr}_S^1(F, E), \; y\; \check{\in }\; \underline{Apr}_S^1(G, E) and (F, E)\sqcap (G, E) = \Phi, then \underline{Apr}_S^1(F, E)\sqcap \underline{Apr}_S^1(G, E) = \Phi and \underline{Apr}_S^1(F, E)\sqsubseteq [\underline{Apr}_S^1(G, E)]^c which is impossible because \underline{Apr}_S^1(F, E) is infinite soft set and [\underline{Apr}_S^1(G, E)]^c is finite soft set.

    (2) From part (1), we have

    \begin{equation*} \underline{Apr}_S^2(F, E) = \underline{Apr}_S^3(F, E) = \left\{ \begin{array}{rcl} (F, E) & {if} \; (F, E)^c\; \check{\in }\; \mathcal{L}_E, \\ \Phi& {otherwise.} \end{array}\right. \end{equation*}

    Then, \forall x\neq y\; \check{\in }\; \tilde{\mathbb{Z}} , we have:

    x\; \check{\in }\; \underline{Apr}_S^2(y_E)^c = (y_E)^c, \; y\; \check{\notin }\; (y_E)^c\; \text{and} \; y\; \check{\in }\; \underline{Apr}_S^2(x_E)^c = (x_E)^c, \; x\; \check{\notin }\; (x_E)^c.
    x\; \check{\in }\; \underline{Apr}_S^3(y_E)^c = (y_E)^c, \; y\; \check{\notin }\; (y_E)^c\; \text{and} \; y\; \check{\in }\; \underline{Apr}_S^3(x_E)^c = (x_E)^c, \; x\; \check{\notin }\; (x_E)^c.

    Hence, \tilde{\mathbb{Z}} is soft- T_1^{*} and soft- T_1^{**} . However, \tilde{\mathbb{Z}} is neither soft- T_2^{*} space nor soft- T_2^{**} . By the same way, any one can add examples to show that the above implication is not reversible.

    Definition 5.5. Let (X, R_E) and (Y, (R_2)_H) be two soft approximation spaces and let {\mathcal{L}_E} a soft ideal on X . Then,

    (1) a function f_{ \rho \varrho}: SS(X)_E\longrightarrow SS(Y)_H is said to be soft continuous if (\underline{Apr}_S^1)_{E}[f_{ \rho \varrho}^{-1}(G, H)]\sqsupseteq f_{ \rho \varrho}^{-1}[(\underline{Apr}_S^1)_{H}(G, H)] , that is, (\overline{Apr}_S^1)_{E}[f_{ \rho \varrho}^{-1}(G, H)]\sqsubseteq f_{ \rho \varrho}^{-1}[(\overline{Apr}_S^1)_{H}(G, H)] for all (G, H)\; \check{\in }\; SS(Y)_H.

    (2) A function f_{ \rho \varrho}: SS(X)_E\longrightarrow SS(Y)_H is said to be * -soft continuous if (\underline{Apr}_S^2)_{E}[f_{ \rho \varrho}^{-1}(G, H)]\sqsupseteq f_{ \rho \varrho}^{-1}[(\underline{Apr}_S^1)_{H}(G, H)] , that is, (\overline{Apr}_S^2)_{E}[f_{ \rho \varrho}^{-1}(G, H)]\sqsubseteq f_{ \rho \varrho}^{-1}[(\overline{Apr}_S^1)_{H}(G, H)] for all (G, H)\; \check{\in }\; SS(Y)_H.

    (1) A function f_{ \rho \varrho}: SS(X)_E\longrightarrow SS(Y)_H is said to be ** -soft continuous if (\underline{Apr}_S^3)_{E}[f_{ \rho \varrho}^{-1}(G, H)]\sqsupseteq f_{ \rho \varrho}^{-1}[(\underline{Apr}_S^1)_{H}(G, H)] , that is, (\overline{Apr}_S^3)_{E}[f_{ \rho \varrho}^{-1}(G, H)]\sqsubseteq f_{ \rho \varrho}^{-1}[(\overline{Apr}_S^1)_{H}(G, H)] for all (G, H)\; \check{\in }\; SS(Y)_H.

    Remark 5.2. From Corollary 3.4, we have the following implications:

    \begin{equation*} \text{Soft continuous}\Longrightarrow \text{ * -soft continuous}\Longrightarrow \text{ ** -soft continuous}. \end{equation*}

    Example 5.2. Let X = \{a, b, c\} associated with the parameters E = \{e_1, e_2\} . Let (R_1)_E be a soft relation of X , and {\mathcal{L}_E} be a soft ideal on X , defined respectively by:

    (R_1)_E = \{(e_1, \{(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, c)\}), (e_2, \{(a, a), (a, b), (a, c), (b, b), (b, c)\})\},

    {\mathcal{L}_E} = \{ \Phi, (F_1, E), (F_2, E), (F_3, E), (F_4, E), (F_5, E), (F_6, E), (F_7, E), (F_8, E), (F_9, E), (F_{10}, E), \\ (F_{11}, E), (F_{12}, E), (F_{13}, E), (F_{14}, E)(F_{15}, E)\}

    where

    (F_1, E) = \{(e_1, \{b\}), (e_2, \phi)\}, (F_2, E) = \{(e_1, \{c\}), (e_2, \phi)\}, (F_3, E) = \{(e_1, \{b, c\}), (e_2, \phi)\},
    (F_4, E) = \{(e_1, \phi), (e_2, \{b\})\}, (F_5, E) = \{(e_1, \phi), (e_2, \{c\})\}, (F_6, E) = \{(e_1, \phi), (e_2, \{b, c\})\},
    (F_7, E) = \{(e_1, \{b\}), (e_2, \{b\})\}, (F_8, E) = \{(e_1, \{b\}), (e_2, \{c\})\}, (F_9, E) = \{(e_1, \{b\}), (e_2, \{b, c\})\},
    (F_{10}, E) = \{(e_1, \{c\}), (e_2, \{b\})\}, (F_{11}, E) = \{(e_1, \{c\}), (e_2, \{c\})\}, (F_{12}, E) = \{(e_1, \{c\}), (e_2, \{b, c\})\},
    (F_{13}, E) = \{(e_1, \{b, c\}), (e_2, \{b\})\}, (F_{14}, E) = \{(e_1, \{b, c\}), (e_2, \{c\})\},
    (F_{15}, E) = \{(e_1, \{b, c\}), (e_2, \{b, c\})\}.

    Then, < a > R_1 = \{(e_1, \{a, b, c\}), (e_2, \{a, b, c\})\}, \; < b > R_1 = \{(e_1, \{b, c\}), (e_2, \{b, c\})\} = < c > R_1. Also, R_1 < a > = a_E, \; R_1 < b > = \{(e_1, \{a, b\}), (e_2, \{a, b\})\}, \; R_1 < c > = \Phi. Thus, R_1 < a > R_1 = a_E, \; R_1 < b > R_1 = b_E, \; R_1 < c > R_1 = \Phi. On the other hand, let Y = \{u, v, w\} associated with the parameters H = \{h_1, h_2\} . Let (R_2)_H be a soft relation over Y defined by:

    (R_2)_H = \{(h_1, \{(u, u), (u, v), (v, u), (v, v), (v, w), (w, u), (w, w)\}), (h_2, \{(u, u), (u, v), (v, u), (v, v),

    (w, w)\})\}. Then, < u > R_2 = \{(h_1, \{u, v\}), (h_2, \{u, v\})\} = < v > R_2, \; < w > R_2 = w_H. Also, R_2 < u > = \{(h_1, \{u, v\}), (h_2, \{u, v\})\} = R_2 < v >, \; R_2 < w > = w_H. Thus,

    R_2 < u > R_2 = \{(h_1, \{u, v\}), (h_2, \{u, v\})\} = R_2 < v > R_2, \; R_2 < w > R_2 = w_H. Now, define the function f_{ \rho \varrho}: SS(X)_E\longrightarrow SS(Y)_H, where \rho: E\longrightarrow H is a function defined by \rho(e_1) = h_1, \rho(e_2) = h_2 and \varrho: X\longrightarrow Y is a function defined by \varrho(a) = \varrho(b) = u, \; \varrho(c) = w. By calculating (\underline{Apr}_S^2)_{E}[f_{ \rho \varrho}^{-1}(G, H)] and f_{ \rho \varrho}^{-1}[(\underline{Apr}_S^1)_{H}(G, H)] of a soft set (G, H)\; \check{\in }\; SS(Y)_H , it is clear that f_{ \rho \varrho} is * -soft continuous. However, f_{ \rho \varrho} is not soft continuous, where

    (\underline{Apr}_S^1)_{E}[f_{ \rho \varrho}^{-1}(w_H)] = \Phi\not \sqsupseteq f_{ \rho \varrho}^{-1}[(\underline{Apr}_S^1)_{H}(w_H)] = c_E.

    Theorem 5.2. Let f_{ \rho \varrho}: SS(X)_E\longrightarrow SS(Y)_H be an injective soft continuous function. Then,

    (X, (R_1)_E, {\mathcal{L}_E}) is a soft T_i^* -space if (Y, (R_2)_H) is a soft- T_i space for i\in\{0, 1, 2\}.

    Proof. Suppose (Y, (R_2)_H) is a soft- T_i space for i\in\{0, 1, 2\} and let x_1 \neq x_2 in \tilde{X} . For i = 2, since f_{ \rho \varrho} is injective, f_{ \rho \varrho}(x_1, E) \neq f_{ \rho \varrho}(x_2, E) \; \check{\in }\; SS(Y)_H . Then, by the hypothesis, there exist (G_1, H), \; (G_2, H)\; \check{\in }\; SS(Y)_H such that f_{ \rho \varrho}(x_1, E)\sqsubseteq (\underline{Apr}_S^1)_{H}(G_1, H), \; f_{ \rho \varrho}(x_2, E)\sqsubseteq (\underline{Apr}_S^1)_{H}(G_2, H)) and (G_1, H)\sqcap (G_2, H) = \Phi_H, that is, x_1\; \check{\in }\; f_{ \rho \varrho}^{-1}[(\underline{Apr}_S^1)_{H}(G_1, H)], \; x_2\; \check{\in }\; f_{ \rho \varrho}^{-1}[(\underline{Apr}_S^1)_{H}(G_2, H)] and

    f_{ \rho \varrho}^{-1}(G_1, H)\sqcap f_{ \rho \varrho}^{-1}(G_2, H) = \Phi_H.

    Since f_{ \rho \varrho} is soft continuous, x_1\; \check{\in }\; (\underline{Apr}_S^1)_{E}[f_{ \rho \varrho}^{-1}(G_1, H)], \; x_2\; \check{\in }\; (\underline{Apr}_S^1)_{E}[f_{ \rho \varrho}^{-1}(G_2, H)]. Thus, x_1\; \check{\in }\; (\underline{Apr}_S^2)_{E}[f_{ \rho \varrho}^{-1}(G_1, H)], x_2\; \check{\in }\; (\underline{Apr}_S^2)_{E}[f_{ \rho \varrho}^{-1}(G_2, H)] that is there exist

    (F_1, E) = f_{ \rho \varrho}^{-1}(G_1, H), \; (F_2, E) = f_{\rho \varrho}^{-1}(G_2, H)\; \check{\in }\; SS(X)_E such that x_1\; \check{\in }\; (\underline{Apr}_S^2)_{E}(F_1, E), \; x_2\; \check{\in }\; (\underline{Apr}_S^2)_{E}(F_2, E) and (F_1, E)\sqcap(F_2, E) = \Phi_E. So, (X, R_E, {\mathcal{L}_E}) is a soft -T_2^* space. For i\in\{0, 1\} the proofs are similar.

    Corollary 5.10. Let f_{ \rho \varrho}: SS(X)_E\longrightarrow SS(Y)_H be an injective soft continuous function. Thus, (X, R_E) is a soft T_i -space if (Y, (R_2)_H) is a soft- T_i space for i\in\{0, 1, 2\}.

    Corollary 5.11. Let f_{ \rho \varrho}: SS(X)_E\longrightarrow SS(Y)_H be an injective soft continuous function. Then, (X, R_E, {\mathcal{L}_E}) is a soft T_i^{**} -space if (Y, (R_2)_H) is a soft- T_i space for i\in\{0, 1, 2\}.

    In this section, We reformulate and study soft connectedness in [31] with respect to these soft ideal approximation spaces. Some examples are submitted to explain the definitions.

    Definition 6.1. Let (X, R_E) be a soft approximation space. Then,

    (1) (F, E), \; (G, E)\; \check{\in }\; SS(X)_E are called soft separated sets if \overline{Apr}_S^1(F, E)\sqcap (G, E) = (F, E)\sqcap\overline{Apr}_S^1(G, E) = \Phi.

    (2) \tilde{A}\; \check{\in }\; SS(X)_E is said to be a soft disconnected set if there exist soft separated sets (F, E), \; (G, E)\; \check{\in }\; SS(X)_E such that \tilde{A}\sqsubseteq (F, E)\sqcup (G, E) . \tilde{A} is said to be soft connected if it is not soft disconnected.

    (3) (X, R_E) is said to be a soft disconnected space if there exist soft separated sets (F, E), (G, E) \check{\in } SS(X)_E such that (F, E) \sqcup (G, E) = \tilde{X} . (X, R_E) is said to be a soft connected space if it is not soft disconnected space.

    Definition 6.2. Let (X, R_E, {\mathcal{L}_E}) be a soft ideal approximation space. Then,

    (1) (F, E), \; (G, E)\; \check{\in }\; SS(X)_E are called * - soft separated (resp. ** - soft separated) sets if \overline{Apr}_S^2(F, E)\sqcap (G, E) = (F, E)\sqcap\overline{Apr}_S^2(G, E) = \Phi (resp. \overline{Apr}_S^3(F, E)\sqcap (G, E) = (F, E)\sqcap\overline{Apr}_S^3(G, E) = \Phi ).

    (2) \tilde{A}\; \check{\in }\; SS(X)_E is called a * -soft disconnected (resp. ** -soft disconnected) set if there exist * -soft separated (resp. ** -soft separated) sets (F, E), \; (G, E)\; \check{\in }\; SS(X)_E such that \tilde{A}\sqsubseteq (F, E)\sqcup (G, E) . \tilde{A} is said to be * -soft connected (resp. ** -soft connected) if it is not * -soft disconnected (resp. ** -soft disconnected).

    (3) (X, R_E, {\mathcal{L}_E}) is called a * -soft disconnected (resp. ** -soft disconnected) space if there exist * -soft separated (resp. ** -soft separated) sets (F, E), \; (G, E)\; \check{\in }\; SS(X)_E such that (F, E)\sqcup (G, E) = \tilde{X} . (X, R_E, {\mathcal{L}_E}) is called a * -soft connected (resp. ** -soft connected) space if it is not a * -soft disconnected (resp. ** -soft disconnected) space.

    Remark 6.1. The following implications are correct:

    \begin{equation*} \text{soft separated}\Longrightarrow \text{ * -soft separated}\Longrightarrow \text{ ** -soft separated}, \end{equation*}

    and so

    \begin{equation*} \text{ ** -soft connected}\Longrightarrow \text{ * -soft connected}\Longrightarrow \text{soft connected}. \end{equation*}

    Example 6.1. Let X = \{a, b, c\} associated with a set of parameters E = \{e_1, e_2\} . Let R_E be a soft relation over X defined by:

    \begin{equation*} R_E = \{(e_1, \{(a, a), (a, b), (a, c), , (b, a), (b, b), (b, c), (c, c)\}), (e_2, \{(a, a), (a, b), (a, c), (b, b), (b, c)\})\} \end{equation*}

    Then, < a > R = \{(e_1, \{a, b, c\}), (e_2, \{a, b, c\})\}, \; < b > R = \{(e_1, \{b, c\}), (e_2, \{b, c\})\}, \\ < c > R = \{(e_1, \{b, c\}), (e_2, \{b, c\})\}. Also, R < a > = a_E, \; R < b > = \{(e_1, \{a, b\}), (e_2, \{a, b\})\}, \; R < c > = \Phi. Thus, R < a > R = a_E, \; R < b > R = b_E, \; R < c > R = \Phi.

    (1) Let {\mathcal{L}_E} be a soft ideal on X defined by:

    \begin{equation*} {\mathcal{L}_E} = \{ \Phi, (F_1, E), (F_2, E), (F_3, E), (F_4, E), (F_5, E), (F_6, E)\} \end{equation*}

    where

    \begin{equation*} (F_1, E) = \{(e_1, \{b\}), (e_2, \phi)\}, (F_2, E) = \{(e_1, \phi), (e_2, \{b\})\}, (F_3, E) = \{(e_1, \{c\}), (e_2, \phi)\}, \end{equation*}
    \begin{equation*} (F_4, E) = \{(e_1, \phi), (e_2, \{c\})\}, (F_5, E) = \{(e_1, \{b\}), (e_2, \{c\})\}, (F_6, E) = \{(e_1, \{b, c\}), (e_2, \{b, c\})\}. \end{equation*}

    Then, we have

    \begin{equation*} \overline{Apr}_S^1b_E = \overline{Apr}_S^1c_E = \overline{Apr}_S^1(\widetilde{\{b, c\}}_E) = \overline{Apr}_S^1(\widetilde{\{a, b\}}_E) = \overline{Apr}_S^1(\widetilde{\{a, c\}}_E) = \tilde{X}, \; \overline{Apr}_S^1a_E = a_E. \end{equation*}

    Thus, (X, R_E) is a soft connected space. However, we get

    \begin{equation*} \tilde{X} = a_E\sqcup\widetilde{\{b, c\}}_E, \; \overline{Apr}_S^2a_E\sqcap \widetilde{\{b, c\}}_E = a_E\sqcap\overline{Apr}_S^2(\widetilde{\{b, c\}}_E) = \Phi. \end{equation*}

    So, (X, R_E, {\mathcal{L}_E}) is not a * -soft connected space.

    (2) Consider {\mathcal{L}_E} = \{ \Phi, (F_1, E), (F_2, E), (F_3, E)\} where

    (F_1, E) = \{(e_1, \{a\}), (e_2, \phi)\}, (F_2, E) = \{(e_1, \phi), (e_2, \{a\})\}, (F_3, E) = \{(e_1, \{a\}), (e_2, \{a\})\}.

    Then, we get

    \begin{equation*} \overline{Apr}_S^2b_E = \overline{Apr}_S^2c_E = \overline{Apr}_S^2(\widetilde{\{b, c\}}_E) = \overline{Apr}_S^2(\widetilde{\{a, b\}}_E) = \overline{Apr}_S^2(\widetilde{\{a, c\}}_E) = \tilde{X}, \; \overline{Apr}_S^2a_E = a_E. \end{equation*}

    Thus, (X, R_E, {\mathcal{L}_E}) is a * -soft connected space. However, we have

    \begin{equation*} \tilde{X} = a_E\sqcup\widetilde{\{b, c\}}_E, \; \overline{Apr}_S^3a_E\sqcap \widetilde{\{b, c\}}_E = a_E\sqcap\overline{Apr}_S^3(\widetilde{\{b, c\}}_E) = \Phi. \end{equation*}

    So, (X, R_E, {\mathcal{L}_E}) is not a ** -soft connected space.

    Proposition 6.1. Let (X, R_E, {\mathcal{L}_E}) be a soft ideal approximation space. Then, these properties are equivalent:

    (1) (X, R_E, {\mathcal{L}_E}) is * -soft connected,

    (2) for each (F, E), \; (G, E)\; \check{\in }\; SS(X)_E with (F, E)\sqcap (G, E) = \Phi, \; \underline{Apr}_S^2(F, E) = (F, E), \; \underline{Apr}_S^2(G, E) = (G, E) and (F, E)\sqcup (G, E) = \tilde{X}, (F, E) = \Phi or (G, E) = \Phi,

    (3) for each (F, E), \; (G, E)\; \check{\in }\; SS(X)_E with (F, E)\sqcap (G, E) = \Phi, \; \overline{Apr}_S^2(F, E) = (F, E), \; \overline{Apr}_S^2(G, E) = (G, E) and (F, E)\sqcup (G, E) = \tilde{X}, (F, E) = \Phi or (G, E) = \Phi.

    Proof. (1) \Rightarrow (2) : Suppose part (1) holds and let (F, E), \; (G, E)\; \check{\in }\; SS(X)_E with \underline{Apr}_S^2(F, E) = (F, E) ,

    \; \underline{Apr}_S^2(G, E) = (G, E) such that (F, E)\sqcap (G, E) = \Phi and (F, E)\sqcup (G, E) = \tilde{X}. Then,

    \begin{equation*} \overline{Apr}_S^2(F, E)\sqsubseteq \overline{Apr}_S^2(G, E)^c = [\underline{Apr}_S^2(G, E)]^c = (G, E)^c, \end{equation*}
    \begin{equation*} \overline{Apr}_S^2(G, E)\sqsubseteq \overline{Apr}_S^2(F, E)^c = [\underline{Apr}_S^2(F, E)]^c = (F, E)^c. \end{equation*}

    Thus, \overline{Apr}_S^2(F, E)\sqcap (G, E) = (F, E)\sqcap\overline{Apr}_S^2(G, E) = \Phi. So, (F, E), \; (G, E) are * -soft separated sets. Since (F, E)\sqcup (G, E) = \tilde{X} , (F, E) = \Phi or (G, E) = \Phi by part (1).

    (2) \Rightarrow (3) and (3)\Rightarrow (1) Clear.

    Corollary 6.1. Let (X, R_E) be a soft approximation space. Then, these properties are equivalent:

    (1) (X, R_E) is soft connected,

    (2) for each (F, E), \; (G, E)\; \check{\in }\; SS(X)_E with (F, E)\sqcap (G, E) = \Phi, \; \underline{Apr}_S^1(F, E) = (F, E), \; \underline{Apr}_S^1(G, E) = (G, E) and (F, E)\sqcup (G, E) = \tilde{X}, (F, E) = \Phi or (G, E) = \Phi,

    (3) for each (F, E), \; (G, E)\; \check{\in }\; SS(X)_E with (F, E)\sqcap (G, E) = \Phi, \; \overline{Apr}_S^1(F, E) = (F, E), \; \overline{Apr}_S^1(G, E) = (G, E) and (F, E)\sqcup (G, E) = \tilde{X}, (F, E) = \Phi or (G, E) = \Phi.

    Corollary 6.2. Let (X, R_E, {\mathcal{L}_E}) be a soft ideal approximation space. Then, these properties are equivalent:

    (1) (X, R_E, {\mathcal{L}_E}) is ** -soft connected.

    (2) For each (F, E), \; (G, E)\; \check{\in }\; SS(X)_E with (F, E)\sqcap (G, E) = \Phi, \; \underline{Apr}_S^3(F, E) = (F, E), \; \underline{Apr}_S^3(G, E) = (G, E) and (F, E)\sqcup (G, E) = \tilde{X}, (F, E) = \Phi or (G, E) = \Phi.

    (3) For each (F, E), \; (G, E)\; \check{\in }\; SS(X)_E with (F, E)\sqcap (G, E) = \Phi, \; \overline{Apr}_S^3(F, E) = (F, E), \; \overline{Apr}_S^3(G, E) = (G, E) and (F, E)\sqcup (G, E) = \tilde{X}, (F, E) = \Phi or (G, E) = \Phi.

    Theorem 6.1. Let (X, R_E, {\mathcal{L}_E}) be a soft ideal approximation space and (F, E)\; \check{\in }\; SS(X)_E be * -soft connected. If (F_1, E), \; (F_2, E)\; \check{\in }\; SS(X)_E are * -soft separated sets with (F, E)\sqsubseteq (F_1, E)\sqcup (F_2, E), then either (F, E)\sqsubseteq (F_1, E) or (F, E)\sqsubseteq (F_2, E).

    Proof. Suppose (F_1, E), \; (F_2, E) are * -soft separated sets with (F, E)\sqsubseteq (F_1, E)\sqcup (F_2, E). Then, we have

    \overline{Apr}_S^2(F_1, E)\sqcap (F_2, E) = (F_1, E)\sqcap\overline{Apr}_S^2(F_2, E) = \Phi, \; (F, E) = [(F, E)\sqcap (F_1, E)]\sqcup[(F, E)\sqcap (F_2, E)].

    On the other hand, we get

    \overline{Apr}_S^2[(F, E)\sqcap (F_1, E)]\sqcap [(F, E)\sqcap (F_2, E)]\sqsubseteq\overline{Apr}_S^2(F, E)\sqcap\overline{Apr}_S^2(F_1, E)\sqcap [(F, E)\sqcap (F_2, E)] = \overline{Apr}_S^2(F, E)\sqcap (F, E)\sqcap\overline{Apr}_S^2(F_1, E)\sqcap (F_2, E) = (F, E)\sqcap\Phi = \Phi. Also,

    \overline{Apr}_S^2[(F, E)\sqcap (F_2, E)]\sqcap [(F, E)\sqcap (F_1, E)]\sqsubseteq\overline{Apr}_S^2(F, E)\sqcap\overline{Apr}_S^2(F_2, E)\sqcap [(F, E)\sqcap (F_1, E)] = \overline{Apr}_S^2(F, E)\sqcap (F, E)\sqcap\overline{Apr}_S^2(F_2, E)\sqcap (F_1, E) = (F, E)\sqcap\Phi = \Phi. Thus, [(F, E)\sqcap (F_1, E)] and [(F, E)\sqcap (F_2, E)] are * -soft separated sets with (F, E) = [(F, E)\sqcap (F_1, E)]\sqcup[(F, E)\sqcap (F_2, E)]. However, (F, E) is * -soft connected, which implies that (F, E)\sqsubseteq (F_1, E) or (F, E)\sqsubseteq (F_2, E).

    Corollary 6.3. Let (X, R_E) be a soft approximation space and (F, E)\; \check{\in }\; SS(X)_E be soft connected. If (F_1, E), \; (F_2, E)\; \check{\in }\; SS(X)_E are soft separated sets with (F, E)\sqsubseteq (F_1, E)\sqcup (F_2, E), then either (F, E)\sqsubseteq (F_1, E) or (F, E)\sqsubseteq (F_2, E).

    Corollary 6.4. Let (X, R_E, {\mathcal{L}_E}) be a soft ideal approximation space and (F, E)\; \check{\in }\; SS(X)_E be ** -soft connected. If (F_1, E), \; (F_2, E)\; \check{\in }\; SS(X)_E are ** -soft separated sets with (F, E)\sqsubseteq (F_1, E)\sqcup (F_2, E), then either (F, E)\sqsubseteq (F_1, E) or (F, E)\sqsubseteq (F_2, E).

    Theorem 6.2. Let f_{ \rho \varrho}: (X, R_E, {\mathcal{L}_E})\longrightarrow (Y, (R_2)_H) be a * -soft continuous function. Then,

    f_{ \rho \varrho}(F, E)\; \check{\in }\; SS(Y)_H is a soft connected set if (F, E)\; \check{\in }\; SS(X)_E is * -soft connected.

    Proof. Assume that (F, E) is * -soft connected in (X, R_E, {\mathcal{L}_E}). Suppose that f_{ \rho \varrho}(F, E) is soft disconnected. Thus, there exist two soft separated sets (G_1, H), \; (G_2, H)\; \check{\in }\; SS(Y)_H with f_{ \rho \varrho}(F, E)\sqsubseteq (G_1, H)\sqcup (G_2, H), that is, (\overline{Apr}_S^1)_{H}(G_1, H)\sqcap (G_2, H) = (G_1, H)\sqcap(\overline{Apr}_S^1)_{H}(G_2, H) = \Phi. Since f_{ \rho \varrho} is * -soft continuous, (F, E)\sqsubseteq f_{ \rho \varrho}^{-1}(G_1, H)\sqcup f_{ \rho \varrho}^{-1}(G_2, H). Thus, we have

    (\overline{Apr}_S^2)_{E}[f_{ \rho \varrho}^{-1}(G_1, H)]\sqcap f_{ \rho \varrho}^{-1}(G_2, H)\sqsubseteq f_{ \rho \varrho}^{-1}[(\overline{Apr}_S^1)_{H}(G_1, H)]\sqcap f_{ \rho \varrho}^{-1}(G_2, H)

    = f_{ \rho \varrho}^{-1}[(\overline{Apr}_S^1)_{H}(G_1, H)\sqcap (G_2, H)] = f_{ \rho \varrho}^{-1}(\Phi) = \Phi. Also, we have

    (\overline{Apr}_S^2)_{E}[f_{ \rho \varrho}^{-1}(G_2, H)]\sqcap f_{ \rho \varrho}^{-1}(G_1, H)\sqsubseteq f_{ \rho \varrho}^{-1}[(\overline{Apr}_S^1)_{H}(G_2, H)]\sqcap f_{ \rho \varrho}^{-1}(G_1, H)

    = f_{ \rho \varrho}^{-1}[(\overline{Apr}_S^1)_{H}(G_2, H)\sqcap (G_1, H)] = f_{ \rho \varrho}^{-1}(\Phi) = \Phi.

    So, f_{ \rho \varrho}^{-1}(G_1, H) and f_{ \rho \varrho}^{-1}(G_2, H) are * -soft separated sets in (X, R_E, {\mathcal{L}_E}) , that is,

    (F, E)\sqsubseteq f_{ \rho \varrho}^{-1}(G_1, H)\sqcup f_{ \rho \varrho}^{-1}(G_2, H). Hence, (F, E) is * -soft disconnected, which contradicts that (F, E) is * -soft connected. Therefore, f_{ \rho \varrho}(F, E) is a soft connected set in (Y, (R_2)_H) .

    Corollary 6.5. Let f_{ \rho \varrho} : (X, R_E) \rightarrow (Y, (R_2)_H) be a soft continuous function. Then, f_{ \rho \varrho}(F, E)\; \check{\in }\; SS(Y)_H is soft connected set, if (F, E)\; \check{\in }\; SS(X)_E is soft connected.

    Corollary 6.6. Let f_{ \rho \varrho}: (X, R_E, {\mathcal{L}_E})\longrightarrow (Y, (R_2)_H) be a ** -soft continuous function. Then,

    f_{ \rho \varrho}(F, E)\; \check{\in }\; SS(Y)_H is soft connected set if (F, E)\; \check{\in }\; SS(X)_E is ** -soft connected.

    Herein, we first compare the current purposed methods in Definitions 3.4–3.6 and demonstrate that the method given in Definition 3.6 is the best in terms of developing the soft approximation operators and the values of soft accuracy. Then, we clarify that the third approach in Definition 3.6 produces soft accuracy measures of soft subsets higher than their counterparts displayed in the previous method 2.4 in [17]. Moreover, we applied these approaches to handle real-life problems.

    Definition 7.1. Let (X, R_E, {\mathcal{L}_E}) be a soft ideal approximation space. Then, the soft boundary region Bnd_S^i(F, E) of a soft set (F, E)\; \check{\in }\; SS(X)_E and the soft accuracy measure Acc_S^i(F, E) of an absolute soft set (F, E)\; \check{\in }\; SS(X)_E, i\in\{1, 2, 3\} with respect to the soft binary relation R_E are defined respectively by:

    \begin{equation*} Bnd_S^i(F, E) = \overline{Apr}_S^i(F, E)-\underline{Apr}_S^i(F, E), \; Acc_S^i(F, E) = \frac{|\underline{Apr}_S^i(F, E)|}{|\overline{Apr}_S^i(F, E)|}, i\in\{1, 2, 3\} \end{equation*}

    where (F, E)\neq\Phi. Note that |\tilde{A}_E| = |A| denotes the cardinality of set A\subseteq X.

    Proposition 7.1. Let (X, R_E, {\mathcal{L}_E}) be a soft ideal approximation space and (F, E)\; \check{\in }\; SS(X)_E. Then,

    (1) Bnd_S^3(F, E)\sqsubseteq Bnd_S^2(F, E)\sqsubseteq Bnd_S^1(F, E).

    (2) Acc_S^1(F, E)\leq Acc_S^2(F, E)\leq Acc_S^3(F, E).

    Proof. (1) Let x\; \check{\in }\; Bnd_S^3(F, E) = \overline{Apr}_S^3(F, E)-\underline{Apr}_S^3(F, E). Then, from Corollary 3.4, we have

    x\; \check{\in }\; \overline{Apr}_S^2(F, E)-\underline{Apr}_S^2(F, E) = Bnd_S^2(F, E) . Again, by Corollary 3.4,

    if x\; \check{\in }\; Bnd_S^2(F, E) = \overline{Apr}_S^2(F, E)-\underline{Apr}_S^2(F, E), then x\; \check{\in }\; \overline{Apr}_S^1(F, E)-\underline{Apr}_S^1(F, E) = Bnd_S^1(F, E). Hence, Bnd_S^3(F, E)\sqsubseteq Bnd_S^2(F, E)\sqsubseteq Bnd_S^1(F, E).

    (2) From Corollary 3.4, we have

    \begin{eqnarray*} Acc_S^1(F, E)& = &\frac{|\underline{Apr}_S^1(F, E)|}{|\overline{Apr}_S^1(F, E)|}\leq\frac{|\underline{Apr}_S^2(F, E)|}{|\overline{Apr}_S^2(F, E)|} = Acc_S^2(F, E)\\&\leq&\frac{|\underline{Apr}_S^1(F, E)|}{|\overline{Apr}_S^1(F, E)|} = Acc_S^1(F, E). \end{eqnarray*}

    Proposition 7.2. Let (X, R_E, {(\mathcal{L}_1)_E}) and (X, R_E, {(\mathcal{L}_2)_E}) be soft ideal approximation spaces such that {(\mathcal{L}_1)_E}\sqsubseteq{(\mathcal{L}_2)_E}. Thus, for each (F, E)\; \check{\in }\; SS(X)_E we have

    (1) (\underline{Apr}_S^2)_{{(\mathcal{L}_1)_E}}(F, E)\sqsubseteq(\underline{Apr}_S^2)_{{(\mathcal{L}_2)_E}}(F, E).

    (2) (\overline{Apr}_S^2)_{{(\mathcal{L}_2)_E}}(F, E)\subseteq (\overline{Apr}_S^2)_{{(\mathcal{L}_1)_E}}(F, E).

    (3) (Bnd_S^2)_{{(\mathcal{L}_2)_E}}(F, E)\sqsubseteq (Bnd_S^2)_{{(\mathcal{L}_1)_E}}(F, E).

    (4) (Acc_S^2)_{{(\mathcal{L}_1)_E}}(F, E)\leq (Acc_S^2)_{{(\mathcal{L}_2)_E}}(F, E).

    Proof.

    (1) Let x\; \check{\in }\; (\underline{Apr}_S^2)_{{(\mathcal{L}_1)_E}}(F, E). Then, < x > R \sqcap(F, E)^c \; \check{\in }\; {(\mathcal{L}_1)_E}. Since {(\mathcal{L}_1)_E}\sqsubseteq{(\mathcal{L}_2)_E}. Thus, < x > R \sqcap(F, E)^c \; \check{\in }\; {(\mathcal{L}_2)_E}. Therefore, x\; \check{\in }\; (\underline{Apr}_S^2)_{{(\mathcal{L}_2)_E}}(F, E). Hence, (\underline{Apr}_S^2)_{{(\mathcal{L}_1)_E}}(F, E)\sqsubseteq(\underline{Apr}_S^2)_{{(\mathcal{L}_2)_E}}(F, E).

    (2) Let x\; \check{\in }\; (\overline{Apr}_S^2)_{{(\mathcal{L}_2)_E}}(F, E). Then, < x > R \sqcap(F, E)^c \; \check{\notin }\; {(\mathcal{L}_2)_E}. Since {(\mathcal{L}_1)_E}\sqsubseteq{(\mathcal{L}_2)_E}. Thus, < x > R \sqcap(F, E)^c \; \check{\notin }\; {(\mathcal{L}_1)_E}. Therefore, x\; \check{\in }\; (\overline{Apr}_S^2)_{{(\mathcal{L}_1)_E}}(F, E). Hence, (\overline{Apr}_S^2)_{{(\mathcal{L}_2)_E}}(F, E)\subseteq (\overline{Apr}_S^2)_{{(\mathcal{L}_1)_E}}(F, E).

    (3), (4): It is immediately obtained by parts (1) and (2).

    Corollary 7.1. Let (X, R_E, {(\mathcal{L}_1)_E}), and (X, R_E, {(\mathcal{L}_2)_E}) be soft ideal approximation spaces such that {(\mathcal{L}_1)_E}\sqsubseteq{(\mathcal{L}_2)_E}. Thus, for each (F, E)\; \check{\in }\; SS(X)_E we have

    (1) (\underline{Apr}_S^3)_{{(\mathcal{L}_1)_E}}(F, E)\sqsubseteq(\underline{Apr}_S^3)_{{(\mathcal{L}_2)_E}}(F, E).

    (2) (\overline{Apr}_S^3)_{{(\mathcal{L}_2)_E}}(F, E)\subseteq (\overline{Apr}_S^3)_{{(\mathcal{L}_1)_E}}(F, E).

    (3) (Bnd_S^3)_{{(\mathcal{L}_2)_E}}(F, E)\sqsubseteq (Bnd_S^3)_{{(\mathcal{L}_1)_E}}(F, E).

    (4) (Acc_S^3)_{{(\mathcal{L}_1)_E}}(F, E)\leq (Acc_S^3)_{{(\mathcal{L}_2)_E}}(F, E).

    Remark 7.1. Proposition 7.2 shows that the soft boundary region of a soft set (F, E)\; \check{\in }\; SS(X)_E decreases as the soft ideal increases as illustrated in the next example.

    Example 7.1. Let X = \{a, b, c\} associated with a set of parameters E = \{e_1, e_2\} . Let R_E be a soft relation over X . Let {(\mathcal{L}_1)_E}, {(\mathcal{L}_2)_E} be soft ideals on X , defined respectively by:

    R_E = \{(e_1, \{(a, a), (a, b), (a, c), (b, b), (b, c)\}), (e_2, \{(a, a), (a, c), (b, a), (b, b), (b, c)\})\}
    {(\mathcal{L}_1)_E} = \{ \Phi, \{(e_1, \{a\}), (e_2, \phi)\}\}
    ({\mathcal{L}_2)_E} = SS(\{a, c\})_E = \{(F, E): (F, E) \text{ is a soft set over } \{a, c\}\}.

    Therefore, < a > R = \{(e_1, \{a\}), (e_2, \{a\})\}, \; < b > R = \{(e_1, \{b, c\}), (e_2, \{b, c\})\} = < c > R.

    Let (F, E) = \{(e_1, \{c\}), (e_2, \phi)\}. Then,

    (Bnd_S^2)_{{(\mathcal{L}_1)_E}}(F, E) = (\overline{Apr}_S^2)_{{(\mathcal{L}_1)_E}}(F, E)-(\underline{Apr}_S^2)_{{(\mathcal{L}_1)_E}}(F, E) = (\widetilde{\{b, c\}})_E-\Phi = (\widetilde{\{b, c\}})_E.

    Also

    (Bnd_S^2)_{{(\mathcal{L}_2)_E}}(F, E) = (\overline{Apr}_S^2)_{{(\mathcal{L}_2)_E}}(F, E)-(\underline{Apr}_S^2)_{{(\mathcal{L}_2)_E}}(F, E) = \{(e_1, \{c\}), (e_2, \phi)\}-\Phi
    = \{(e_1, \{c\}), (e_2, \Phi)\}.

    It is clear that (Bnd_S^2)_{{(\mathcal{L}_2)_E}}(F, E)\sqsubseteq (Bnd_S^2)_{{(\mathcal{L}_1)_E}}(F, E).

    Remark 7.2. From Proposition 5.2, one can deduce that Definition 3.6 improves the soft boundary region which means decreasing for a soft set (F, E)\; \check{\in }\; SS(X)_E , and improves the soft accuracy measure which means increasing for that soft set (F, E)\; \check{\in }\; SS(X)_E by increasing the soft lower approximation and decreasing the soft upper approximation in comparison to the methods in Definitions 3.4, 3.5, and Definition 2.4 in [17]. So, the suggested method in Definition 3.6 is more accurate in decision-making. As a special case:

    (1) If R_E is soft symmetric relation, then the soft approximations in Definition 3.6 coincide with the soft approximations in Definition 3.5.

    (2) If {\mathcal{L}_E} = \Phi and R_E is soft symmetric relation, then the soft approximations in Definition 3.5 coincide with the soft approximations in Definition 3.5.

    (3) If {\mathcal{L}_E} = \Phi, E = \{e\} and R_E is soft reflexive and soft transitive relation, then the soft approximations in Definition 3.6 coincide with the previous soft approximations in [17].

    Example 7.2. Selection of a house:

    Considering X = \{x_1, x_2, x_3, x_4, x_5, x_6\} is a collection of six houses where H = {expensive, beautiful, cheap, in green surroundings, wooden modern, in good repair, in bad repair} be a set of parameters.

    Suppose Mr.Z wants to purchase a house on the following parametric set E = {beautiful, cheap, in green surroundings, wooden, in good repair}. Consider E = \{e_1, e_2, e_3, e_4, e_5\}.

    Define a soft equivalence relation R : E \longrightarrow P(X \times X) . The soft equivalence classes for each e \in E are obtained as follows:

    {For } R(e_1) : { are } \{x_1, x_3\}, \{x_2, x_4, x_5, x_6\}.{ For } R(e_2): { are } \{x_1, x_2, x_4, x_5\}, \{x_3\}, \{x_6\}.
    { For } R(e_3): { are } \{x_1, x_2, x_4, x_5, x_6\}, \{x_3\}.{ For } R(e_4): { are } \{x_1, x_3, x_6\}, \{x_2, x_4, x_5\}.
    {For } R(e_5): { is } \{x_1, x_2, x_3, x_4, x_5, x_6\}.

    Therefore, R < x_1 > R = (\widetilde{\{x_1\}})_E, \; R < x_2 > R = R < x_4 > R = R < x_5 > R = (\widetilde{\{x_2, x_4, x_5\}})_E, \; R < x_3 > R = (\widetilde{\{x_3\}})_E, \; R < x_6 > R = (\widetilde{\{x_6\}})_E. Consider {\mathcal{L}_E} = SS(\{x_1, x_3, x_5\})_E = \{(F, E): (F, E) is a soft set over \{x_1, x_3, x_5\}\} be a soft ideal over X . The soft representation of the equivalence relation R_E is explained in Table 1. In Table 2, the soft approximations, soft boundary region, and soft accuracy measure of a soft set (F, E)\; \check{\in }\; SS(X)_E by using our suggested method in Definition 3.6. This method is the best tool to help Mr.Z in his decision-making about selecting the house that is most suitable to his choice of parameters. For example, take (\widetilde{\{x_2, x_3, x_4\}})_E, then from Table 2, the soft lower and soft upper approximations, soft boundary region, and soft accuracy measure are (\widetilde{\{x_3\}})_E, (\widetilde{\{x_2, x_3, x_4, x_5\}})_E, (\widetilde{\{x_2, x_4, x_5\}})_E , and 1/4 , respectively. One can see that Mr.Z will decide to buy the house x_3 according to his choice parameters in E .

    Table 1.  Soft equivalence relation representation of houses under consideration.
    e_1 e_2 e_3 e_4 e_5
    (x_1, x_1) 1 1 1 1 1
    (x_1, x_2) 0 1 1 0 1
    (x_1, x_3) 1 0 0 1 1
    (x_1, x_4) 0 1 1 0 1
    (x_1, x_5) 0 1 1 0 1
    (x_1, x_6) 0 0 1 1 1
    (x_2, x_1) 0 1 1 0 1
    (x_2, x_2) 1 1 1 1 1
    (x_2, x_3) 0 0 0 0 1
    (x_2, x_4) 1 1 1 1 1
    (x_2, x_5) 1 1 1 1 1
    (x_2, x_6) 1 0 1 0 1
    (x_3, x_1) 1 0 0 1 1
    (x_3, x_2) 0 0 0 0 1
    (x_3, x_3) 1 1 1 1 1
    (x_3, x_4) 0 0 0 0 1
    (x_3, x_5) 0 0 0 0 1
    (x_3, x_6) 0 0 0 1 1
    (x_4, x_1) 0 1 1 0 1
    (x_4, x_2) 1 1 1 1 1
    (x_4, x_3) 0 0 0 0 1
    (x_4, x_4) 1 1 1 1 1
    (x_4, x_5) 1 1 1 1 1
    (x_4, x_6) 1 0 1 0 1
    (x_5, x_1) 0 1 1 0 1
    (x_5, x_2) 1 1 1 1 1
    (x_5, x_3) 0 0 0 0 1
    (x_5, x_4) 1 1 1 1 1
    (x_5, x_5) 1 1 1 1 1
    (x_5, x_6) 1 0 1 0 1
    (x_6, x_1) 0 0 1 1 1
    (x_6, x_2) 1 0 1 0 1
    (x_6, x_3) 0 0 0 1 1
    (x_6, x_4) 1 0 1 0 1
    (x_6, x_5) 1 0 1 0 1
    (x_6, x_6) 1 1 1 1 1

     | Show Table
    DownLoad: CSV
    Table 2.  Soft approximations, soft boundary region and soft accuracy measure of a soft set (F, E) \check{\in } SS(X)_E of Definition 3.6.
    (F, E) \check{\in } SS(X)_E \underline{Apr}_S^3(F, E) \overline{Apr}_S^3(F, E) Bnd_S^3(F, E) Acc_S^3(F, E)
    (\widetilde{\{x_1, x_2, x_3\}})_E (\widetilde{\{x_1, x_3\}})_E (\widetilde{\{x_1, x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 2/5
    (\widetilde{\{x_1, x_3, x_4\}})_E (\widetilde{\{x_1, x_3\}})_E (\widetilde{\{x_1, x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 2/5
    (\widetilde{\{x_1, x_2, x_5\}})_E (\widetilde{\{x_1\}})_E (\widetilde{\{x_1, x_2, x_4, x_5\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 1/4
    (\widetilde{\{x_1, x_3, x_6\}})_E (\widetilde{\{x_1, x_3, x_6\}})_E (\widetilde{\{x_1, x_3, x_6\}})_E \Phi 1
    (\widetilde{\{x_1, x_4, x_6\}})_E (\widetilde{\{x_1, x_6\}})_E (\widetilde{\{x_1, x_2, x_4, x_5, x_6\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 2/5
    (\widetilde{\{x_2, x_3, x_4\}})_E (\widetilde{\{x_3\}})_E (\widetilde{\{x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 1/4
    (\widetilde{\{x_2, x_3, x_5\}})_E (\widetilde{\{x_3\}})_E (\widetilde{\{x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 1/4
    (\widetilde{\{x_2, x_4, x_5\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E \Phi 1
    (\widetilde{\{x_2, x_4, x_6\}})_E (\widetilde{\{x_2, x_4, x_6\}})_E (\widetilde{\{x_2, x_4, x_6\}})_E \Phi 1
    (\widetilde{\{x_2, x_5, x_6\}})_E (\widetilde{\{x_6\}})_E (\widetilde{\{x_2, x_4, x_5, x_6\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 1/4
    (\widetilde{\{x_3, x_4, x_5\}})_E (\widetilde{\{x_3\}})_E (\widetilde{\{x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 1/4
    (\widetilde{\{x_4, x_5, x_6\}})_E (\widetilde{\{x_6\}})_E (\widetilde{\{x_2, x_4, x_5, x_6\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 1/4
    (\widetilde{\{x_1, x_2, x_3, x_4\}})_E (\widetilde{\{x_1, x_2, x_3, x_4\}})_E (\widetilde{\{x_1, x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_5\}})_E 4/5
    (\widetilde{\{x_1, x_2, x_5, x_6\}})_E (\widetilde{\{x_1, x_6\}})_E (\widetilde{\{x_1, x_2, x_4, x_5, x_6\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 2/5
    (\widetilde{\{x_1, x_3, x_4, x_5\}})_E (\widetilde{\{x_1, x_3\}})_E (\widetilde{\{x_1, x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 2/5
    (\widetilde{\{x_1, x_4, x_5, x_6\}})_E (\widetilde{\{x_1, x_6\}})_E (\widetilde{\{x_1, x_2, x_4, x_5, x_6\}})_E (\widetilde{\{x_2, x_4, x_5\}})_E 2/5
    (\widetilde{\{x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_2, x_3, x_4, x_5\}})_E \Phi 1
    (\widetilde{\{x_2, x_4, x_5, x_6\}})_E (\widetilde{\{x_2, x_4, x_5, x_6\}})_E (\widetilde{\{x_2, x_4, x_5, x_6\}})_E \Phi 1
    (\widetilde{\{x_1, x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_1, x_2, x_3, x_4, x_5\}})_E (\widetilde{\{x_1, x_2, x_3, x_4, x_5\}})_E \Phi 1
    (\widetilde{\{x_1, x_2, x_3, x_4, x_6\}})_E (\widetilde{\{x_1, x_2, x_3, x_4, x_6\}})_E (\widetilde{\{x_1, x_2, x_3, x_4, x_6\}})_E \Phi 1
    (\widetilde{\{x_1, x_3, x_4, x_5, x_6\}})_E (\widetilde{\{x_1, x_3, x_6\}})_E \widetilde{X} (\widetilde{\{x_4, x_5\}})_E 1/2
    (\widetilde{\{x_2, x_3, x_4, x_5, x_6\}})_E (\widetilde{\{x_2, x_3, x_4, x_5, x_6\}})_E (\widetilde{\{x_2, x_3, x_4, x_5, x_6\}})_E \Phi 1

     | Show Table
    DownLoad: CSV

    Example 7.3. Selection of a car:

    Suppose a person Mr.Z wants to buy a car from the alternatives x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10}. Let X = \{x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10}\} be the universe of ten different cars and let E = \{e_1, e_2, e_3\} be the set of attributes, where e_1 refers to price, e_2 refers to color, and e_3 refers to car brands.

    The parameters are characterized as follows:

    The price of a car includes under 30 lacs, between 31 and 35 lacs, and between 36 and 40 lacs.

    The car brand includes Honda Accord, Audi, Mercedes Benz, and BMW.

    The color of a car includes black, white, and silver.

    Define a soft equivalence relation R : E \longrightarrow P(X \times X) for each e \in E which describes the advantages of the car for which the person Mr.Z will buy. The soft equivalence classes for each e \in E are obtained as follows:

    {For } R(e_1): { are } \{x_{1}, x_{10}\}, \{x_2, x_4, x_6, x_7\}, \{x_3, x_5, x_8, x_9\} ,

    which means that the price of cars x_1 and x_{10} is under 30 lacs; the price of cars x_2, x_4, x_6 , and x_7 is between 31 and 35 lacs; and the price of cars x_3, x_5, x_8 , and x_9 is between 36 and 40 lacs.

    {For } R(e_2): { are } \{x_1\}, \{x_2\}, \{x_3, x_4, x_5, x_7, x_8, x_9, x_{10}\}, \{x_6\},

    which represents that the brand of car x_1 is Honda Accord; the brand of car x_2 is Audi; the brand of cars x_3, x_4, x_5, x_7, x_8, x_9, and x_{10} is Mercedes Benz; and the brand of car x_6 is BMW. For R(e_3) : are \{x_{10}\}, \{x_6\}, \{x_1, x_2, x_3, x_4, x_5, x_7, x_8, x_9\}, which represents that the color of cars x_1, x_2, x_3, x_4, x_5, x_7, x_8, and x_9 is black; the color of car x_{10} is white; and the color of car x_6 is silver.

    Therefore, R < x_1 > R = (\widetilde{\{x_1\}})_E, \; R < x_2 > R = (\widetilde{\{x_2\}})_E, \; R < x_6 > R = (\widetilde{\{x_6\}})_E, \; R < x_{10} > R = (\widetilde{\{x_{10}\}})_E, \; R < x_4 > R = R < x_7 > R = (\widetilde{\{x_4, x_7\}})_E, \; R < x_3 > R = R < x_5 > R = R < x_8 > R = R < x_9 > R = (\widetilde{\{x_3, x_5, x_8, x_9\}})_E.

    Consequently, anyone can offer a soft ideal to extend an example similar to the one in Table 2 to help Mr.Z in his decision-making about selecting the car that is most suitable according to the given parameters.

    For example, let {\mathcal{L}_E} = SS(\{x_2, x_6, x_{10}\})_E = \{(F, E): (F, E) \text{ is a soft set over } \{x_2, x_6, x_{10}\}\} be a soft ideal over X and (F, E) = (\widetilde{\{x_1, x_4, x_8\}})_E\; \check{\in }\; SS(X)_E consisting of these cars which are most acceptable for Mr.Z. Thus, \underline{Apr}_S^3(F, E) = (\widetilde{\{x_1\}})_E, \overline{Apr}_S^3(F, E) = (\widetilde{\{x_1, x_3, x_4, x_5, x_7, x_8, x_9\}})_E,

    Bnd_S^3(F, E) = (\widetilde{\{x_3, x_4, x_5, x_7, x_8, x_9\}})_E and Acc_S^3(F, E) = 1/7. Mr.Z will buy the car x_1 which is under 30 lacs, a Honda Accord, and is white.

    This paper introduced new soft closure operators based on soft ideals, defining soft topological spaces. To that end, soft accumulation points, soft subspaces, and soft lower separation axioms of such spaces are defined and studied. Moreover, soft connectedness in these spaces is defined, which enables us to make more generalizations and studies. The obtained results are newly presented and could enrich soft topology theory. Finally, applications in multi criteria group decision making by using our methods to present the importance of our soft ideals approximations have been presented.

    As it is well-known that the soft interior and soft closure topological operators behave similarly to the lower and upper soft approximations. So, in forthcoming works, we plan to study the counterparts of these models via topological structures. In addition, we will benefit from the hybridization of rough set theory with some approaches, such as fuzzy sets and soft fuzzy sets, to introduce these approximation spaces via these hybridized frames and show their role in efficiently dealing with uncertain knowledge.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors extend their appreciation to the Deputy-ship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number ISP-2024.

    The authors declare that they have no conflicts of interest.



    [1] Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci., 11 (1982), 341–356.
    [2] T. M. Al-shami, An improvement of rough sets' accuracy measure using containment neighborhoods with a medical application, Inform. Sciences, 569 (2021), 110–124. https://doi.org/10.1016/j.ins.2021.04.016 doi: 10.1016/j.ins.2021.04.016
    [3] T. M. Al-shami, Topological approach to generate new rough set models, Complex Intell. Syst., 8 (2022), 4101–4113. https://doi.org/10.1007/s40747-022-00704-x doi: 10.1007/s40747-022-00704-x
    [4] M. Hosny, T. M. Al-shami, A. Mhemdi, Novel approaches of generalized rough approximation spaces inspired by maximal neighbourhoods and ideals, Alex. Eng. J., 69 (2023), 497–520. https://doi.org/10.1016/j.aej.2023.02.008 doi: 10.1016/j.aej.2023.02.008
    [5] H. Mustafa, T. M. Al-shami, R. Wassef, Rough set paradigms via containment neighborhoods and ideals, Filomat, 37 (2023), 4683–4702. https://doi.org/10.2298/FIL2314683M doi: 10.2298/FIL2314683M
    [6] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for basic rough set concepts, Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, RSFDGrC 2005, Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 3641 (2005), 64–73. https://doi.org/10.1007/11548669_7
    [7] A. A. Allam, M. Y. Bakeir, E. A. Abo-Tabl, New approach for closure spaces by relations, Acta Math. Acad. Paedag. Nyregyhziensis, 22 (2006), 285–304. https://doi.org/10.1016/j.midw.2006.03.006 doi: 10.1016/j.midw.2006.03.006
    [8] A. Kandil, S. A. El-Sheikh, M. Hosny, M. Raafat, Bi-ideal approximation spaces and their applications, Soft Comput., 24 (2020), 12989–13001. https://doi.org/10.1007/s00500-020-04720-2 doi: 10.1007/s00500-020-04720-2
    [9] E. A. Abo-Tabl, A comparison of two kinds of definitions of rough approximations based on a similarity relation, Inform. Sciences, 181 (2011), 2587–2596. https://doi.org/10.1016/j.ins.2011.01.007 doi: 10.1016/j.ins.2011.01.007
    [10] J. H. Dai, S. C. Gao, G. J. Zheng, Generalized rough set models determined by multiple neighborhoods generated from a similarity relation, Soft Comput., 22 (2018), 2081–2094. https://doi.org/10.1007/s00500-017-2672-x doi: 10.1007/s00500-017-2672-x
    [11] T. M. Al-Shami, Maximal rough neighborhoods with a medical application, J. Amb. Intel. Hum. Comput., 2022, 1–12. https://doi.org/10.1007/s12652-022-03858-1
    [12] D. Molodtsov, Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
    [13] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6
    [14] M. I. Ali, F. Feng, X. Y. Liu, W. K. Min, M. Shabir, On some new operations in soft set theory, Comput. Math. Appl., 57 (2009), 1547–1553. https://doi.org/10.1016/j.camwa.2008.11.009 doi: 10.1016/j.camwa.2008.11.009
    [15] M. Shabir, M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786–1799. https://doi.org/10.1016/j.camwa.2011.02.006
    [16] M. I. Ali, M. Shabir, Logic connectives for so13 set431d f442 soft sets, IEEE T. Fuzzy Syst., 22 (2013), 1431–1442. https://doi.org/10.1007/s40278-013-1808-8 doi: 10.1007/s40278-013-1808-8
    [17] M. Shabir, R. S. Kanwal, M. I. Ali, Reduction of nformation system, Soft Comput., 24 (2020), 10801–10813. https://doi.org/10.1007/s00500-019-04582-3 doi: 10.1007/s00500-019-04582-3
    [18] N. Rehman, A. Ali, M. I. Ali, C. Park, SDMGRS: Soft dominance based multi granulation rough sets and their applications in conflict analysis problems, IEEE Access, 6 (2018), 31399–31416. https://doi.org/10.1109/ACCESS.2018.2841876 doi: 10.1109/ACCESS.2018.2841876
    [19] N. Malik, M. Shabir, T. M. Al-shami, R. Gul, A. Mhemdi, Medical decision-making techniques based on bipolar soft information, AIMS Math., 8 (2023), 18185–18205. https://doi.org/10.3934/math.2023924 doi: 10.3934/math.2023924
    [20] Z. A. Ameen, T. M. Al-shami, R. Abu-Gdairi, A. Mhemdi, The relationship between ordinary and soft algebras with an application, MDPI Math., 11 (2023), 2035. https://doi.org/10.3390/math11092035 doi: 10.3390/math11092035
    [21] A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. A. El-latif, Soft ideal theory soft local function and generated soft topological spaces, Appl. Math. Inform. Sci., 8 (2014), 1595–1603. https://doi.org/10.12785/amis/080413 doi: 10.12785/amis/080413
    [22] A. C. Guler, E. D. Yildirim, O. Ozbakir, Rough approximations based on different topologies via ideals, Turk. J. Math., 46 (2022), 1177–1192. https://doi.org/10.55730/1300-0098.3150 doi: 10.55730/1300-0098.3150
    [23] M. Hosny, Topological approach for rough sets by using J-nearly concepts via ideals, Filomat, 34 (2020), 273–286. https://doi.org/10.2298/FIL2002273H doi: 10.2298/FIL2002273H
    [24] M. Hosny, Idealization of j-approximation spaces, Filomat, 34 (2020), 287–301. https://doi.org/10.2298/FIL2002287H doi: 10.2298/FIL2002287H
    [25] M. Hosny, T. M. Al-shami, Rough set models in a more general manner with applications, AIMS Math., 7 (2022), 18971–19017. https://doi.org/10.3934/math.20221044 doi: 10.3934/math.20221044
    [26] T. M. Al-shami, D. Ciucci, Subset neighborhood rough sets, Knowl.-Based Syst., 137 (2022), 07868. https://doi.org/10.1016/j.knosys.2021.107868 doi: 10.1016/j.knosys.2021.107868
    [27] T. M. Al-Shami, M. Hosny, Improvement of approximation spaces using maximal left neighborhoods and ideals, IEEE Access, 10 (2022), 79379–79393. https://doi.org/10.1109/ACCESS.2022.3194562 doi: 10.1109/ACCESS.2022.3194562
    [28] A. A. Azzam, Z. A. Ameen, T. M. Al-shami, M. E. El-Shafei, Generating soft topologies via soft set operators, MDPI Symmetry, 14 (2022), 914. https://doi.org/10.3390/sym14050914 doi: 10.3390/sym14050914
    [29] I. Ibedou, S. E. Abbas, Generalization of rough fuzzy sets based on a fuzzy ideal, Iran. J. Fuzzy Syst., 20 (2023), 27–38. https://doi.org/10.22111/ijfs.2023.7344 doi: 10.22111/ijfs.2023.7344
    [30] S. E. Abbas, S. El-Sanowsy, H. M. Khiamy, Certain approximation spaces using local functions via idealization, Sohag J. Sci., 8 (2023), 311–321. https://doi.org/10.21608/sjsci.2023.201184.1072 doi: 10.21608/sjsci.2023.201184.1072
    [31] S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, New approach for closure spaces by relations via ideals, Ann. Fuzzy Math. Inform., 26 (2023), 59–81.
    [32] F. Feng, M. I. Ali, M. Shabir, Soft relations applied to semigroups, Filomat, 27 (2013), 1183–1196. https://doi.org/10.2298/FIL1307183F doi: 10.2298/FIL1307183F
    [33] S. Al Ghour, On soft generalized \omega-closed sets and soft T 1/2 spaces in soft topological spaces, MDPI Axioms, 11 (2022), 194. https://doi.org/10.3390/axioms11050194 doi: 10.3390/axioms11050194
    [34] S. Al Ghour, Soft complete continuity and soft strong continuity in soft topological spaces, MDPI Axioms, 12 (2023), 78. https://doi.org/10.3390/axioms12010078 doi: 10.3390/axioms12010078
    [35] T. M. Al-shami, Z. A. Ameen, A. A. Azzam, M. E. El-Shafei, Soft separation axioms via soft topological operators, AIMS Math., 7 (2022), 15107–15119. https://doi.org/10.3934/math.2022828 doi: 10.3934/math.2022828
    [36] J. B. Liu, Y. Bao, W. T. Zheng, Analyses of some structural properties on a class of hierarchical scale-free networks, Fractals, 30 (2022), 2250136. https://doi.org/10.1142/S0218348X22501365 doi: 10.1142/S0218348X22501365
    [37] J. B. Liu, N. Salamat, M. Kamran, S. Ashraf, R. H. Khan, Single-valued neutrosophic set with quaternion information: A promising approach to assess image quality, Fractals, 31 (2023), 1–10. https://doi.org/10.1142/S0218348X23400741 doi: 10.1142/S0218348X23400741
    [38] G. Nordo, A soft embedding theorem for soft topological spaces, In: Developments and Novel Approaches in Nonlinear Solid Body Mechanics, Springer, Cham, 2020, 37–57. https://doi.org/10.1007/978-3-030-50460-1_5
    [39] A. Allam, T. H. Ismail, R. Muhammed, A new approach to soft belonging, J. Ann. Fuzzy Math. Inform., 13 (2017), 145–152. https://doi.org/10.30948/afmi.2017.13.1.145 doi: 10.30948/afmi.2017.13.1.145
    [40] I. Zorlutuna, M. Akdag, W. K. Min, S. Atmaca, Remarks on soft topological spaces, Ann. Fuzzy Math. Inform., 3 (2012), 171–185. https://doi.org/10.1136/vr.e5655 doi: 10.1136/vr.e5655
  • This article has been cited by:

    1. 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
    2. Ahmed Ramadan, Anwar Fawakhreh, Enas Elkordy, Novel categorical relations between $ \mathcal{L} $-fuzzy co-topologies and $ \mathcal{L} $-fuzzy ideals, 2024, 9, 2473-6988, 20572, 10.3934/math.2024999
  • Reader Comments
  • © 2024 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(1702) PDF downloads(94) Cited by(2)

Figures and Tables

Tables(2)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog