International Journal of Computational Intelligence Systems

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Volume 10, Issue 1, 2017, Pages 90 - 103

Soft points, s-relations and soft rough approximate operations

Authors
*Corresponding author: Guangji Yu
Corresponding Author
Received 8 October 2014, Accepted 2 September 2016, Available Online 1 January 2017.
DOI
10.2991/ijcis.2017.10.1.7How to use a DOI?
Keywords
Soft sets; Soft points; Soft point sets; s-relations; Soft rough approximate operations; Soft topologies
Abstract

Soft set theory is a new mathematical tool to deal with uncertain problems. Since soft sets are defined by mappings and they lack “points”, managing them is not convenient. In this paper, the concept of soft points is introduced and the relationship between soft points and soft sets is investigated. We prove that soft sets can be translated into soft point sets and may be expediently handled like ordinary sets. Moreover, we propose s-relations on soft sets. By means of soft points and these results, a pair of soft rough approximate operations is defined. Serial, reflexive, symmetric, transitive and Euclidean s-relations are characterized by using soft rough approximate operations. In addition, we research soft topologies induced by a reflexive s-relation on a special soft set and gives their structure.

Copyright
© 2017, the Authors. Published by Atlantis Press.
Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Most of traditional methods for formal modeling, reasoning and computing are crisp, deterministic and precise in character. However, many practical problems within fields such as economics, engineering, environmental science, medical science and social sciences involve data that contain uncertainties. We cannot use traditional methods because of various types of uncertainties present in these problems.

There are several theories: probability theory, fuzzy set theory27, interval mathematics, and rough set theory22, which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties (see 21). For example, probability theory can deal only with stochastically stable phenomena. To overcome these kinds of difficulties, Molodtsov 21 proposed a completely new approach, which is called soft set theory, for modeling uncertainty.

Presently, works on soft set theory are progressing rapidly. Maji et al.18,20,19 further studied soft set theory, used this theory to solve decision making problems and devoted fuzzy soft sets combining soft sets with fuzzy sets. Roy et al.24 presented a fuzzy soft set theoretic approach towards decision making problems. Li et al.12 investigated decision making based on intuitionistic fuzzy soft sets. Jiang et al.10 extended soft sets with description logics. Aktas et al.2 defined soft groups. Li et al.16 proposed L-fuzzy soft sets based on complete Boolean lattices. Feng et al.6,7 investigated the relationship among soft sets, rough sets and fuzzy sets. Ge et al.8 discussed relationships between soft sets and topological spaces. Shabir et al.25 proposed soft topological spaces which are defined on the universe with a fixed set of parameters. Babitha et al.5 introduced relations on soft sets. Li et al.13,14 considered roughness of fuzzy soft sets and obtained the relationship among soft sets, soft rough sets and topologies. Li et al.15 studied parameter reductions of soft coverings.

Rough set theory was proposed by Pawlak 22. It is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. The foundation of its object classification is an equivalence relation. The upper and lower approximation operations are two core notions in rough set theory. They can also be seen as a closure operator and an interior operator of the topology induced by an equivalence relation on a universe. We may relax equivalence relations so that rough set theory is able to solve more complicated problems in practice. Pawlak rough set theory has been extended to tolerance relations, similarity relations, binary relations 17,26,30.

Since soft sets are defined by mappings and then lack “points”, managing them is not convenient. Thus, we try to attempt introducing the concept of “soft points” and deal with them as same as ordinary sets.

Feng et al.7 proposed soft rough approximate operations. But the introduction of these operations seemed suddenly and disposing them is not convenient as soft sets lacks “points” and “soft points” are not proposed. In this paper, we introduce the concept of soft points, prove that soft sets can be translate into soft point sets and then it is convenient to deal with soft sets as same as ordinary sets. We propose s-relations on soft sets. By means of soft points and these results, soft rough approximate operations are defined. And because we do the above work, it is very convenient to deal with the operations introduced by us.

The organization of this paper is as follows: In Section 2, we briefly recall basic concepts about rough sets, soft sets and soft topological spaces. In Section 3, we introduce the concept of soft points and investigate the relationship between soft points and soft sets. In Section 4, we introduce the concepts of serial, reflexive, symmetric, transitive and Euclidean s-relations on soft sets, and investigate the relationships between these s-relations and soft point sets. In Section 5, we propose two soft rough approximate operations. In Section 6, we investigate soft topologies induced by a reflexive s-relation on a special soft set and give their structure. Section 7 concludes this paper and highlights the prospects for potential future development.

2. Overview of rough sets, soft sets and soft topological spaces

In this section, we briefly recall basic concepts about rough sets, soft sets and soft topological spaces.

Throughout this paper, U refers to an initial universe, E refers to the set of parameters and 2U denotes the power set of U. We only consider the case where both U and E are nonempty finite sets.

2.1. Rough sets

Let R be an equivalence relation on U. The pair (U, R) is called a Pawlak approximation space. Using the equivalence relation R, one can define the following rough approximations:

R*(X)={x∈U:[x]R⊆X},R*(X)={x∈U:[x]R∩X≠∅}.

Then R*(X) and R*(X) called the Pawlak lower approximation and the Pawlak upper approximation of X, respectively.

The Pawlak boundary region of X, defined by the difference between these Pawlak rough approximations, that is BndR(X) = R*(X) − R*(X). It can easily be seen that R*(X) ⊆ X ⊆ R*(X).

A set is Pawlak rough if its boundary region is not empty. Otherwise, the set is crisp. Thus X is Pawlak rough if R*(X) ≠ R*(X).

We may relax equivalence relations so that rough set theory is able to solve more complicated problems in practice. Pawlak rough set theory has been extended to binary relations 17,26,30.

Definition 2.1 (30)

Let R be a binary relation on U. The pair (U, R) is called a approximation space. Based on the approximation space (U, R), we define a pair of operations R¯ , R¯:2U→2U as follows:

R¯(X)={x∈U:R(x)⊆X},R¯(X)={x∈U:R(x)∩X≠∅},
where X ∈ 2U and R(x) = {y ∈ U : xRy} is the successor neighborhood of x. Then R¯(X) and R¯(X) are called the lower approximation and the upper approximation of X, respectively.

X is called a definable set if R¯(X)=R¯(X) ; X is called a rough set if R¯(X)≠R¯(X) .

2.2. Soft sets

Definition 2.2 (21)

Let A ⊆ E. A pair (f, A) is called a soft set over U, if f is a mapping given by f : A → 2U. We denote (f,A) by fA.

In other words, a soft set over U is the parameterized family of subsets of the universe U. For ε ∈ A, f(ε) may be considered as the set of ε-approximate elements of the soft set fA. Obviously, a soft set is not a ordinary set.

Denote S(U, E) = {fE : fE is a soft set over U}.

Definition 2.3 (18)

Let A,B ⊆ E, fA ∈ S(U,A) and gB ∈ S(U,B).

  1. (1)

    fA is called a soft subset of gB, if A ⊆ B and ∀ ε ∈ A, f(ε) ⊆ g(ε). We write fA⊂˜gB .

  2. (2)

    fA is called a soft super set of gB, if gB⊂˜fA . We write fA⊃˜gB .

  3. (3)

    fA and gB are called soft equal, if A = B and ∀ ε ∈ A, f(ε) = g(ε). We write fA = gB.

Obviously, fA = gB if and only if fA⊂˜gB and fA⊃˜gB .

Definition 2.4 (3, 18)

Let A,B ⊆ E, fA ∈ S(U,A) and gB ∈ S(U,B).

  1. (1)

    hA∪B is called the union of fA and gB, if

    h(ε)={f(ε), if ε∈A−B,g(ε), if ε∈B−A,f(ε)∪g(ε), if ε∈A∩B.

    We write fA∪˜gB=hA∪B.

  2. (2)

    hA∩B is called the soft intersection of fA and gB, if ∀ ε ∈ A ∩ B, h(ε) = f(ε) ∩ g(ε). We write fA∩˜gB=hA∩B .

Remark 2.5

Let A,B,C ⊆ E, fA ∈ S(U,A), gB ∈ S(U,B) and hC ∈ S(U,C). Then

  1. (1)

    fA∩˜gB⊂˜fA (or gB) ⊂˜fA∪˜gB .

  2. (2)

    If hC⊂˜fA and hC⊂˜gB , then hC⊂˜fA∩˜gB .

  3. (3)

    If hC⊃˜fA and hC⊃˜gB , then hC⊃˜fA∪˜gB .

Definition 2.6 (25)

Let A ⊆ E, fA,gA,hA ∈ S(U,A). hA is called the difference of fA and gA, if ∀ ε ∈ A, h(ε) = f(ε) − g(ε). We write hA = fA − gA.

Definition 2.7 (3)

Let A ⊆ E, fA, gA ∈ S(U,A). gA is called the relative complement of fA, if ∀ ε ∈ A, g(ε) = U − f(ε). We write gA=fA′ or (fA)′.

Proposition 2.8 (3)

Let A ⊆ E, fA,gA ∈ S(U,A). Then

  1. (1)

    (fA∪˜gA)′=fA′∩˜gA′ .

  2. (2)

    (fA∩˜gA)′=fA′∪˜gA′ .

Remark 2.9

Let A ⊆ E, fA,gA ∈ S(U,A). Then

  1. (1)

    (fA′)′=fA .

  2. (2)

    fA⊂˜gA⇔(fA)′⊃˜(gA)′ .

Definition 2.10 (25)

Let X ∈ 2U. The soft set XE over U is defined by ∀ ε ∈ E, X(ε) = X.

In this paper, UE and ∅E are also denoted by Ũ and ∅˜ , respectively.

Remark 2.11

Let fA,gA ∈ S(U,A). Then

  1. (1)

    UA−fA=fA′ ,

  2. (2)

    fA∩˜gA=∅A⇔fA⊂˜gA′ ,

  3. (3)

    fA−gA=fA∩˜gA′ .

2.3. Soft topological spaces

In what follows we consider problems on the universe U and the fixed set E of parameters.

Definition 2.12 (25)

τ ⊆ S(U,E) is called a soft topology over U, if (i) ∅˜ , Ũ ∈ τ; (ii) the union of any number of soft sets in τ belongs to τ; (iii) the intersection of any two soft sets in τ belongs to τ.

The triplet (U, τ, E) is called a soft topological space over U. Every element of τ is called a soft open set in U and its relative complement is called a soft closed set in U.

In this paper, the family of all soft closed sets is denoted by τ′.

Definition 2.13 (25)

Let (U, τ, E) be a soft topological space over U. ∀ fE ∈ S(U,E), the soft closure of fE is defined by

cl(fE)=∩˜  {gE:fE⊂˜gE and gE∈τ′}.

Definition 2.14 (9)

Let (U, τ, E) be a soft topological space over U. ∀ fE ∈ S(U,E), the soft interior of fE is defined by

int(fE)=∪˜  {gE:gE⊂˜fE and gE∈τ}.

Proposition 2.15 (9)

Let (U, τ, E) be a soft topological space over U. Then ∀ fE ∈ S(U,E), int(fE) = Ũ − cl(Ũ − fE).

3. Soft points

In this section, we will introduce the concept of soft points and investigate the relationship between soft points and soft sets.

3.1. The concept of soft points

In this subsection we define soft points, which originate from the concept of fuzzy points (see 11,23).

Definition 3.1

Let fE*∈S(U,E) . fE* is called a soft point over U, if there exist e ∈ E and x ∈ U such that

f*(ε)={{x},if ε=e,∅,if ε∈E−{e}.

We denote fE* by (xe)E.

In this case, x is called the support point of (xe)E, {x} is called the support point set of (xe)E and e is called the expressive parameter of (xe)E.

Example 3.2

Let U = {x1,x2,x3,x4,x5} and E = {e1,e2,e3,e4}. We define f*(e1) = ∅, f*(e2) = ∅, f*(e3) = {x5}, f*(e4) = ∅.

Then fE* is a soft point over U. We denote fE* by ((x5)e3)E, where x5 is the support point of ((x5)e3)E, {x5} is the support point set of ((x5)e3)E and e3 is the expressive parameter of ((x5)e3)E.

For fE ∈ S(U,E), denote

ℱ(E)={(xe)E:x∈f(e)  and  e∈E},                              P(U,E)={(xe)E :(xe)E is a soft points over U}.

Remark 3.3

  1. (1)

    (xe)E ∈ ℱ(E) ⇔ x ∈ f(e) and e ∈ E.

  2. (2)

    |ℱ(E)|=∑e∈E|f(e)| .

  3. (3)

    If fE = (xe)E, then ℱ(E) = {(xe)E}.

Example 3.4

Let U = {x1,x2,x3,x4,x5} and E = {e1,e2,e3,e4}. We define f(e1) = {x1,x4}, f(e2) = U, f(e3) = {x5}, f(e4) = ∅. Then

       ℱ(E)={((x1)e1)E,((x4)e1)E,((x1)e2)E,((x2)e2)E,((x3)e2)E,((x4)e2)E,((x5)e2)E,((x5)e3)E}  and       P(U,E)={((xi)ej)E:1≤i≤5,1≤j≤4}.

To illustrate the fact that the soft contain relation, the soft intersection operation, the soft union operation and the soft difference operation on two soft sets can be be translated into the contain relation, the intersection operation, the union operation and the difference operation on two soft point sets (i.e., two ordinary sets), respectively, we give the following Proposition 3.5.

Proposition 3.5

Let fE, gE, hE ∈ S(U,E).

  1. (1)

    If gE⊂˜fE , then 𝒢(E) ⊆ ℱ(E).

  2. (2)

    If fE=gE∩˜hE , then ℱ(E) = 𝒢(E) ∩ ℋ(E).

  3. (3)

    If fE=gE∪˜hE , then ℱ(E) = 𝒢(E) ∪ ℋ(E).

  4. (4)

    If fE = gE − hE, then ℱ(E) = 𝒢(E) − ℋ(E).

Proof.

  1. (1)

    This is obvious.

  2. (2)

    Let (xe)E ∈ ℱ(E). Then x ∈ f(e). Since fE=gE∩˜hE , we have x ∈ g(e) and x ∈ h(e). Thus (xe)E ∈ 𝒢(E) and (xe)E ∈ ℱ(E). Hence (xe)E ∈ 𝒢(E) ∩ ℋ(E). Conversely, the proof is similar.

  3. (3)

    The proof is similar to (2).

  4. (4)

    The proof is similar to (2).

Proposition 3.6

  1. (1)

    If fE = UE, then P(U, E) = ℱ(E).

  2. (2)

    P(U,E) = ∪ {ℱ(E) : fE ∈ S(U, E)}.

Proof.

  1. (1)

    This is obvious.

  2. (2)

    Let fE ∈ S(U,E). Since fE⊂˜UE , by Proposition 3.5 and (1), ℱ(E) ⊆ P(U,E). Thus P(U,E) ⊇ ∪{ℱ(E) : fE ∈ S(U,E)}.

Conversely, since UE ∈ S(U,E), by (1), we have P(U,E) ⊆ ∪{ℱ(E) : fE ∈ S(U,E)}.

Hence ℱ(E) = ∪{ℱ(E) : fE ∈ S(U,E)}.

3.2. Soft points and soft sets

In this subsection, we will investigate the relationship between soft points and soft sets.

Definition 3.7

Let fE ∈ S(U,E) and (xe)E ∈ P(U,E). We define (xe)E∈˜fE by (xe)E⊂˜fE .

Note that (xe)E∉˜fE , if (xe)E⊄˜fE .

Remark 3.8

  1. (1)

    (xe)E = (x′e′)E ⇔ x = x′ and e = e′.

  2. (2)

    (xe)E∈˜fE⇔x∈f(e) and e ∈ E ⇔ (xe)E ∈ ℱ(E).

  3. (3)

    (xe)E∈˜fE and fE⊂˜gE⇒(xe)E∈˜gE .

  4. (4)

    (xe)E∈˜(xe)E .

  5. (5)

    (xe)E∈˜fE⇔(xe)E∉˜fE′ .

Theorem 3.9

Let fE ∈ S(U,E). Then fE=∪˜ ℱ(E) .

Proof.

Denote hE=∪˜  ℱ(E) . Then hE=∪˜{(xe)E:x∈f(e) and e∈E} . Thus

hE=∪e∈E˜∪x∈f(e)˜(xe)E.
∀ ε ∈ E,
        h(ε)=∪e∈E∪x∈f(e)xe(ε)=(∪x∈f(ε)xε(ε))∪(∪e∈E−{ε}∪x∈f(e)xe(ε))=(∪x∈f(ε){x})  ∪   ∅=f(ε).

This shows hE = fE. Hence fE=∪˜ ℱ(E) .

Remark 3.10

Theorem 3.9 reveals the fact that a soft set can be translated into a soft point set and vice versa.

Theorem 3.11

Let fE,gE ∈ S(U,E). Then

  1. (1)

    fE⊂˜gE⇔ℱ(E)⊆𝒢(E) .

  2. (2)

    fE = gE ⇔ ℱ(E) = 𝒢(E).

Proof.

These hold by Proposition 3.5 and Theorem 3.9.

Remark 3.12

Theorem 3.11 illustrates that the soft contain relation and the soft equal relation can be respectively translated into the contain relation and the equal relation on two soft point sets (i.e., two ordinary sets) and vice versa.

When we study some problems of soft sets by using soft points in this paper, we will abide by the following logic thinking: firstly, the soft contain relation, the soft intersection operation, the soft union operation and the soft difference operation on soft sets are translated into the contain relation, the intersection operation, the union operation and the difference operation on soft point sets by Proposition 3.5, respectively; secondly, the relations and operations on ordinary sets (i.e., soft point sets) are realized; thirdly, the results of the relations and operations on ordinary sets are translated into the results on soft sets by Theorem 3.9.

4. s-relations on soft sets

In this section, we introduce the concepts of serial, reflexive, symmetric, transitive and Euclidean s-relations on soft sets, and investigate the relationships between these s-relations and soft point sets.

Definition 4.1 (5)

Let A,B ⊆ E, fA ∈ S(U,A) and gB ∈ S(U,B). hA×B is called the cartesian product of fA and gB, if ∀ (a,b) ∈ A × B, h(a,b) = f(a) × g(b). We write hA×B = fA × gB.

Definition 4.2 (5)

Let A,B ⊆ E, fA ∈ S(U,A) and gB ∈ S(U,B).

  1. (1)

    R is called a relation from fA to gB, if R⊂˜fA×gB .

  2. (2)

    R is called a relation on fA, if R⊂˜fA×fA .

In other words, a relation R from fA to gB is of the form lP, where P ⊆ A × B and ∀ (a,b) ∈ P, l(a,b) ⊆ f(a) × g(b).

Definition 4.3

Let fE ∈ S(U,E). R is called a surjective relation (brief. s-relation) on fE, if there exists a soft set lE×E over U × U such that R=lE×E⊂˜fE×fE .

Remark 4.4

R is a s-relation on fE ⇒ R is a relation on fE.

Example 4.5

Let U = {x1,x2,x3,x4,x5} and E = {e1,e2}. We define f(e1) = {x1,x3,x5}, f(e2) = {x2,x4}. Then fE ∈ S(U,E) and E × E = {(e1,e1), (e1,e2), (e2,e1), (e2,e2)}.

Let hE×E = fE × fE. Then

h(e1,e1)=f(e1)× f(e1), h(e1,e2)=f(e1)× f(e2),h(e2,e1)=f(e2)× f(e1)  and  h(e2,e2)=f(e2)× f(e2).
  1. (1)

    Define l : E × E → 2U×U by

    l(e1,e1)={(x1,x1),(x1,x3),(x1,x5),(x3,x3),(x3,x5),(x5,x5)},l(e1,e2)=f(e1)×f(e2),l(e2,e1)={(x2,x1),(x2,x3),(x2,x5),(x4,x3),(x4,x5)}
    and
    l(e2,e2)=f(e2)×f(e2).

    Then

    l(e1,e1)⊆h(e1,e1),  l(e1,e2)⊆h(e1,e2),l(e2,e1)⊆h(e2,e1)  and  l(e2,e2)⊆h(e2,e2).

    So lE×E⊂˜fE×fE

    Put R1 = lE×E. Then R1 is a s-relation on fE.

  2. (2)

    Put P = {(e1,e1), (e1,e2)}. Then P ⊊ E × E. Define k : P → 2U×U by

    k(e1,e1)=f(e1)× f(e1) and k(e1,e2)=f(e1)× f(e2).

    Put R2 = kP. Since R2⊂˜fE×fE , R2 is a relation on fE. But R2 is not a s-relation on fE.

    Since soft sets can be translated into soft point sets, every relation on a soft set can be translated into a relation on a soft point set. We introduce the following Definition 4.6 for this reason.

Definition 4.6

Let R be a s-relation on fE ∈ S(U,E). Define a relation R* on ℱ(E) as follows: for any (xe)E, (xe′′)E∈ℱ(E) ,

(xe)ER*(xe′′)E⇔(xe)E×(xe′′)E⊂˜R.

Then R* is called the relation induced by R.

Remark 4.7

  1. (1)

    (xe)E×(xe′′)E⊂˜fE×fE⇔x∈f(e) and x′ ∈ f(e′).

  2. (2)

    Let R=lE×E⊂˜fE×fE . Then

    (xe)ER*(xe′′)E⇔(x,x′)∈l(e,e′)       ⇒x∈f(e),x′∈f(e′).

Definition 4.8

Let R be a s-relation on fE. R is called serial (resp. reflexive, symmetric, transitive, Euclidean), if R* is serial (resp. reflexive, symmetric, transitive, Euclidean).

Let fE ∈ S(U,E). Denote Sf(U,E)={gE∈S(U,E):gE⊂˜fE} .

Let R be a s-relation on fE and R* the relation induced by R. ∀ (xe)E ∈ ℱ(E), gE ∈ Sf(X,E), put

R*((xe)E)={(xe′′)E∈ℱ(E):(xe)ER*(xe′′)E},Pg(fE,R)={(xe)E∈ℱ(E):R*((xe)E)⊆𝒢(E)},Pg(fE,R)={(xe)E∈ℱ(E):R*((xe)E)∩𝒢(E)≠∅}.

Remark 4.9

Let R be a s-relation on fE ∈ S(U,E) and R* the relation induced by R. Then

  1. (1)

    R is serial ⇔ ∀ (xe)E ∈ ℱ(E), R*((xe)E) ≠ ∅.

  2. (2)

    R is reflexive ⇔ ∀ (xe)E ∈ ℱ(E), (xe)E ∈ R*((xe)E).

  3. (3)

    R is symmetric ⇔ ∀ (xe)E, (xe′′)E∈ℱ(E) , (xe′′)E∈R*((xe)E) implies (xe)E∈R*((xe′′)E) .

  4. (4)

    R is transitive ⇔ ∀ (xe)E, (xe′′)E , (xe″″)E∈ℱ(E) , (xe)E∈R*((xe′′)E) and (xe′′)E∈R*((xe″″)E) implies (xe)E∈R*((xe″″)E)

    ⇔∀  (xe)E,(xe′′)E∈ℱ(E),(xe′′)E∈R*((xe)E)
    implies R*((xe′′)E)⊆R*((xe)E) .

  5. (5)

    R is Euclidean ⇔ ∀ (xe)E, (xe′′)E , (xe″″)E∈ℱ(E) , (xe′′)E∈R*((xe)E) and (xe″″)E∈R*((xe)E) implies R*((xe″″)⊆R*((xe′′)E)

    ⇔∀  (xe)E,(xe′′)E∈ℱ(E),(xe′′)E∈R*((xe)E)
    implies R*((xe)E)⊆R*((xe′′)E).

Lemma 4.10

Let R be a s-relation on fE ∈ S(U,E). Then ∀ gE,hE ∈ Sf(U,E),

  1. (1)

    Pf(fE,R) = ℱ(E).

  2. (2)

    1. a)

      R is serial ⇒ Pg(fE,R) ⊆ Pg(fE,R).

    2. b)

      R is reflexive ⇒ Pg(fE,R) ⊆ 𝒢 (E) ⊆ Pg(fE,R).

  3. (3)

    1. a)

      gE⊂˜hE⇒Pg(fE,R)⊆Ph(fE,R) ;

    2. b)

      gE⊂˜hE⇒Pg(fE,R)⊆Ph(fE,R) .

  4. (4)

    1. a)

      Pl(fE,R) = Pg(fE,R) ∪ Ph(fE,R) where lE=gE∪˜hE ;

    2. b)

      Pl(fE,R) = Pg(fE,R) ∩ Ph(fE,R) where lE=gE∩˜hE .

Proof.

  1. (1)

    This is obvious.

  2. (2)

    1. a)

      Let (xe)E ∈ Pg(fE,R). Thus R*((xe)E) ⊆ 𝒢(E). Since R is serial, by Remark 4.9, R*((xe)E) ≠ ∅. This implies R*((xe)E) ∩ 𝒢(E) ≠ ∅. So (xe)E ∈ Pg(fE,R). Thus Pg(fE,R) ⊆ Pg(fE,R).

    2. b)

      Let (xe)E ∈ Pg(fE,R). Then R*((xe)E) ⊆ 𝒢(E). Since R is reflexive, by Remark 4.9, we have (xe)E ∈ R*((xe)E) ⊆ 𝒢(E). Thus Pg(fE,R) ⊆ 𝒢(E). Since (xe)E ∈ R*((xe)E) and (xe)E ∈ 𝒢(E), R*((xe)E) ∩ 𝒢(E) ≠ ∅. Thus 𝒢(E) ⊆ Pg(fE,R).

  3. (3)

    1. a)

      Let (xe)E ∈ Pg(fE,R). Then R*((xe)E) ⊆ 𝒢(E). Since gE⊂˜hE , 𝒢(E) ⊆ ℋ(E) and R*((xe)E) ⊆ ℋ(E). Thus (xe)E ∈ Ph(fE,R). Hence Pg(fE,R) ⊆ Ph(fE,R).

    2. b)

      The proof is similar to a).

  4. (4)

    1. a)

      Let (xe)E ∈ Pl(fE,R). Then R*((xe)E) ∩ ℒ(E) ≠ ∅. Since lE=gE∪˜hE , by Proposition 3.5, R*((xe)E) ∩ 𝒢(E) ≠ ∅ and R*((xe)E) ∩ ℋ(E) ≠ ∅. Thus (xe)E ∈ Pg(fE,R) and (xe)E ∈ Ph(fE,R). Hence Pl(fE,R) ⊆ Pg(fE,R)∪Ph(fE,R).

      Conversely, this is obvious.

    2. b)

      The proof is similar to a).

5. Soft rough approximate operations

In this section, we propose two soft rough approximate operations. Serial, reflexive, symmetric, transitive and Euclidean s-relations are characterized by using them.

Definition 5.1

Let R be a s-relation on fE ∈ S(U,E). Then the pair P = (fE,R) is called a soft approximation space. Based on P, we define the following operations apr¯P , apr¯P:Sf(U,E)→Sf(U,E) by

apr¯P(gE)=∪˜ Pg(fE,R),  apr¯P(gE)=∪˜ Pg(fE,R),
where gE ∈ Sf(U,E). Then, apr¯P and apr¯P are called the soft P-lower approximation operator and the soft P-upper approximation operator on fE, respectively; apr¯P(gE) and apr¯P(gE) are called the soft P-lower approximation of gE and the soft P-upper approximation of gE, respectively.

gE is called a soft P-definable set if apr¯P(gE)=apr¯P(gE) ; gE is called a soft P-rough set if apr¯P(gE)≠apr¯P(gE) .

Remark 5.2

In 7, Feng et al. proposed two operations apr¯P , apr¯P:2U→2U by

apr¯P(X)={u∈U:∃ e∈E, s.t. u∈f(e)⊆X},apr¯P(X)={u∈U:∃e∈E, s.t. u∈f(e)and f(e)∩X≠∅}.
where X ∈ 2U, P = (U, fE) and fE ∈ S(U,E).

Lemma 5.3

Let R be a s-relation on fE ∈ S(U,E). Then ∀ gE,hE ∈ Sf(U,E), we have

  1. (1)

    hE=apr¯P(gE)⇔ℋE=Pg(fE,R) .

  2. (2)

    hE=apr¯P(gE)⇔ℋE=Pg(fE,R) .

Proof.

  1. (1)

    Sufficiency. This holds by Theorem 3.11.

    Necessity. Denote Pg(fE,R) = {(ya)E : y ∈ X and a ∈ A} where X ⊆ U and A ⊆ E.

    Let (xe)E ∈ ℋE. Then x ∈ h(e) = ∪ {ya(e) : y ∈ X and a ∈ A}.

    We claim that e ∈ A. Otherwise, ya(e) = ∅ ∀ y ∈ X and a ∈ A. Then h(e) = ∪ {ya(e) : y ∈ X and a ∈ A} = ∅, a contradiction.

    Thus h(e) = ∪ {ye(e) : y ∈ X} = ∪ {{y} : y ∈ X} = X. This implies x ∈ X. So (xe)E ∈ Pg(fE,R).

    Conversely, (xe)E ∈ Pg(fE,R). Then x ∈ X and e ∈ A. Note that h(e) = ∪ {ya(e) : y ∈ X and a ∈ A} = ∪ {ye(e) : y ∈ X} = ∪ {{y} : y ∈ X} = X. So x ∈ h(e). This implies (xe)E ∈ ℋE.

    Hence ℋE = Pg(fE,R).

  2. (2)

    The proof is similar to (1).

Lemma 5.4

Let (fα)E ∈ S(U,E) for α ∈ A ∪ B. Then

      ∪˜  {(fα)E:α∈A∪B}=(∪˜  {(fα)E:α∈A})  ∪˜  (∪˜  {(fα)E:α∈B}).

Proof.

Denote C = A∪B, fEC=∪˜  {(fα)E:α∈C} , fEA=∪˜  {(fα)E:α∈A} , fEB=∪˜  {(fα)E:α∈B} and gE=fEA∪˜fEB .

Then fC(e) = ∪ {fα(e) : α ∈ C} ∀ e ∈ E and g(e) = fA(e) ∪ fB(e) ∀ e ∈ E. Thus g(e) = (∪ {fα(e) : α ∈ A}) ∪ (∪ {fα(e) : α ∈ B}) = ∪ {fα(e) : α ∈ A∪B} = ∪ {fα(e) : α ∈ C} = fC(e).

Lemma 5.5

Let R be a s-relation on fE, (xe)E ∈ Sf(U,E). Denote hE=apr¯P((xe)E) . Then ℋ(E) = {(yε)E ∈ ℱ(E) : (xe)E ∈ R*((yε)E)}.

Proof.

Denote gE = (xe)E. Then hE=apr¯P(gE) .

Let (yε)E ∈ ℋ(E). By Lemma 5.3, (yε)E ∈ Pg(fE,R). This implies R*((yε)E) ∩ 𝒢(E) ≠ ∅. By Remark 3.3, 𝒢(E) = {(xe)E}. So (xe)E ∈ R*((yε)E). Thus (yε)E ∈ {(yε)E ∈ ℱ(E) : (xe)E ∈ R*((yε)E)}.

Conversely, let (yε)E ∈ {(yε)E ∈ ℱ(E) : (xe)E ∈ R*((yε)E)}. Then (xe)E ∈ R*((yε)E). So {(xe)E} = R*((yε)E) ∩ 𝒢(E) ≠ ∅. This implies (yε)E ∈ Pg(fE,R). By Lemma 5.3, (yε)E ∈ ℋ(E).

Therefore, ℋ(E) = {(yε)E ∈ ℱ(E) : (xe)E ∈ R*((yε)E)}.

Proposition 5.6

Let R be a s-relation on fE ∈ S(U,E). Then ∀ gE,hE ∈ Sf(U,E),

  1. (1)

    If gE⊂˜hE , then

    1. a)

      apr¯P(gE)⊂˜apr¯P(hE) ;

    2. b)

      apr¯P(gE)⊂˜apr¯P(hE) .

  2. (2)

    1. a)

      apr¯P(gE∩˜hE)=apr¯P(gE)∩˜apr¯P(hE) ;

    2. b)

      apr¯P(gE∪˜hE)=apr¯P(gE)∪˜apr¯P(hE) .

Proof.

  1. (1)

    These hold by Lemma 4.10 and Theorem 3.11.

  2. (2)

    1. a)

      Denote qE=apr¯P(gE) , pE=apr¯P(hE) , kE=qE∩˜pE , lE=gE∩˜hE and wE=apr¯P(lE) . By Proposition 3.5 and Lemma 4.10, 𝒦(E) = 𝒬(E) ∩ 𝒫(E) and Pl(fE,R) = Pg(fE,R) ∩ Ph(fE,R).

      Let (xe)E ∈ 𝒦(E). Then (xe)E ∈ 𝒬(E) and (xe)E ∈ 𝒫(E). By Lemma 5.3, (xe)E ∈ Pg(fE,R) and (xe)E ∈ Ph(fE,R). Thus (xe)E ∈ Pl(fE,R). By Lemma 5.3, (xe)E ∈ 𝒲(E). By Theorem 3.11, apr¯P(gE)∩˜apr¯P(hE)⊂˜apr¯P(gE∩˜hE) .

      Conversely, apr¯P(gE∩˜hE)⊂˜apr¯P(gE)∩˜apr¯P(hE) is obvious.

    2. b)

      This holds by Lemma 4.10 and Lemma 5.4.

Proposition 5.7

Let R be a s-relation on fE. Then the following are equivalent.

  1. (1)

    R is serial;

  2. (2)

    ∀ gE ∈ Sf(U,E), apr¯P(gE)⊂˜apr¯P(gE) .

Proof.

(1) ⇒ (2) holds by Lemma 4.10 and Theorem 3.11.

(2) ⇒ (1). Let gE ∈ Sf(U,E).

Denote hE=apr¯P(gE) and lE=apr¯P(gE) .

Suppose ∀ (xe)E ∈ ℱ(E), R*((xe)E) = ∅. Then R*((xe)E) ⊆ 𝒢(E) ∀ gE ∈ Sf(U,E). This implies (xe)E ∈ Pg(fE,R). By Lemma 5.3, (xe)E ∈ ℋ(E). Since hE⊂˜lE , ℋ(E) ⊆ ℒ(E) and (xe)E ∈ ℒ(E). But R*((xe)E) ∩ 𝒢(E) = ∅. Thus (xe)E ∉ Pg(fE,R). By Lemma 5.3, (xe)E ∉ ℒ(E), a contradiction. Hence R*((xe)E) ≠ ∅.

Proposition 5.8

Let R be a s-relation on fE. Then the following are equivalent.

  1. (1)

    R is reflexive;

  2. (2)

    ∀ gE ∈ Sf(U,E),

    apr¯P(gE)⊂˜gE⊂˜apr¯P(gE).

Proof.

(1) ⇒ (2) holds by Lemma 4.10 and Theorem 3.11.

(2) ⇒ (1). Let gE ∈ Sf(U,E). Denote gE = (xe)E and hE=apr¯P(gE) .

By (2), gE⊂˜hE . Then 𝒢(E) ⊆ ℋ(E). This implies (xe)E ∈ ℋ(E). By Lemma 5.3, (xe)E ∈ Pg(fE,R). Thus R*((xe)E) ∩ 𝒢(E) ≠ ∅. So R*((xe)E) ∩ 𝒢(E) = (xe)E. Hence (xe)E ∈ R*((xe)E).

Proposition 5.9

Let R be reflexive on fE ∈ S(U,E). Then

  1. (1)

    apr¯P(fE)=apr¯P(fE)=fE .

  2. (2)

    apr¯P(∅˜)=apr¯P(∅˜)=∅˜ .

Proof.

  1. (1)

    By Lemma 4.10, apr¯P(fE)⊂˜apr¯P(fE) .

    Conversely, since Pf(fE,R) ⊆ ℱ(E), then apr¯P(fE)⊂˜fE . By Lemma 4.10, fE=apr¯P(fE) . Thus apr¯P(fE)⊂˜apr¯P(fE) . Therefore, apr¯P(fE)=apr¯P(fE)=fE.

  2. (2)

    This is obvious.

Proposition 5.10

Let R be a s-relation on fE. Then the following are equivalent.

  1. (1)

    R is symmetric;

  2. (2)

    ∀ gE ∈ Sf(U,E),

    apr¯P(apr¯P(gE))⊂˜gE⊂˜apr¯P(apr¯P(gE)).

Proof.

(1) ⇒ (2). Let gE ∈ Sf(U,E). Denote

kE=apr¯P(gE), wE=apr¯P(kE), hE=apr¯P(gE)and lE=apr¯P(hE).

Suppose 𝒲(E) − 𝒢(E) ≠ ∅. Pick (xe)E ∈ 𝒲(E) − 𝒢(E). Then (xe)E ∉ 𝒢(E) and (xe)E ∈ 𝒲(E). By Lemma 5.3, (xe)E ∈ Pk(fE,R). This implies R*((xe)E) ∩ 𝒦(E) ≠ ∅. Pick (xe′′)E∈R*((xe)E)∩𝒦(E) . Then (xe′′)E∈𝒦(E) . By Lemma 5.3, (xe)E ∈ Pg(fE,R). This implies R*((xe′′)E⊆𝒢(E) . Since R is symmetric, (xe)E∈R*((xe′′)E) . Thus (xe)E ∈ 𝒢(E), a contradiction. Hence 𝒲(E) ⊆ 𝒢(E). By Theorem 3.11, wE⊂˜gE .

Therefore, apr¯P(apr¯P(gE))⊂˜gE .

Suppose 𝒢(E) − ℒ(E) ≠ ∅. Pick (xe)E ∈ 𝒢(E) − ℒ(E). Then (xe)E ∈ 𝒢(E) and (xe)E ∉ ℒ(E). By Lemma 5.3, (xe)E ∉ Ph(fE,R). This implies R*((xe)E) ⊈ ℋ(E). Thus R*((xe)E) − ℋ(E) ≠ ∅. Pick (xe′′)E∈R*((xe)E)−ℋ(E) . Then (xe′′)E∉ℋ(E) . By Lemma 5.3, (xe′′)E∉Pg(fE,R) . This implies R*((xe′′)E)∩𝒢(E)=∅ . Since (xe′′)E∈R*((xe)E) , by R is symmetric, we have (xe)E∈R*((xe′′)E) . Thus 𝒢(E) = ∅, a contradiction. Hence 𝒢(E) ⊆ ℒ(E). By Theorem 3.11, gE⊂˜lE .

Therefore, gE⊂˜apr¯P(apr¯P(gE)) .

(2) ⇒ (1). Let (xe)E, (xe′′)E∈ℱ(E) with (xe′′)E∈R*((xe)E) . Denote

gE=(xe)E,hE=apr¯P(gE)  and  lE=apr¯P(hE).

By Lemma 5.3 and Lemma 5.5, ℒE = Ph(fE,R) and

ℋ(E)={(yε)E∈ℱ(E):(xe)E∈R*((yε)E)}.

Then gE⊂˜lE⊂˜hE . This implies (xe)E ∈ ℒ(E) ⊆ ℋ(E). So (xe)E ∈ Pg(fE,R). Thus R*((xe)E) ⊆ 𝒢(E). Since (xe′′)E∈R*((xe)E) , (xe′′)E∈𝒢(E) . Then (xe′′)E=(xe)E . Hence (xe′′)E∈ℋ(E) . By Lemma 5.5, (xe)E∈R*((xe′′)E) .

Therefore, R is symmetric.

Lemma

Let R be reflexive on fE ∈ S(U,E) and let gE,hE ∈ Sf(U,E).

  1. (1)

    If hE=apr¯P(gE) and R is transitive, then Pg(fE,R) = Ph(fE,R) ⊆ Ph(fE,R) ⊆ Pg(fE,R);

  2. (2)

    If hE=apr¯P(gE) and R is Euclidean, then Pg(fE,R) ⊆ Pg(fE,R) ⊆ Ph(fE,R) ⊆ Ph(fE,R).

Proof.

  1. (1)

    Let hE=apr¯P(gE) . Since apr¯P(gE)⊂˜gE , hE⊂˜gE . By Lemma 4.10, Pg(fE,R) ⊇ Ph(fE,R) ⊆ Ph(fE,R) ⊆ Pg(fE,R). It suffices to show that Pg(fE,R) ⊆ Ph(fE,R).

    Suppose Pg(fE,R) − Ph(fE,R) ≠ ∅. Pick (xe)E ∈ Pg(fE,R) − Ph(fE,R). Then R*((xe)E) ⊆ 𝒢(E) and R*((xe)E) ⊈ ℋ(E), and so R*((xe)E) − ℋ(E) ≠ ∅. Pick (xe′′)E∈R*((xe)E)−ℋ(E) . Since R is transitive, R*((xe′′)E)⊆R*((xe)E)⊆𝒢(E) . Thus (xe′′)E∈Pg(fE,R) . By Lemma 5.3, (xe′′)E∈ℋ(E) . But (xe′′)E∉ℋ(E) , a contradiction. Therefore, Pg(fE,R) ⊆ Ph(fE,R).

  2. (2)

    Let hE=apr¯P(gE) . Since gE⊂˜apr¯P(gE) , gE⊂˜hE . By Lemma 4.10, Pg(fE,R) ⊆ Pg(fE,R) and Ph(fE,R) ⊆ Ph(fE,R). It suffices to show that Pg(fE,R) ⊆ Ph(fE,R).

    Suppose Pg(fE,R) − Ph(fE,R) ≠ ∅. Pick (xe)E ∈ Pg(fE,R) − Ph(fE,R). Then R*((xe)E) ∩ 𝒢(E) ≠ ∅ and R*((xe)E) ⊈ ℋ(E). Pick (xe″″)E∈R*((xe)E)∩𝒢(E) and (xe′′)E∈R*((xe)E)−ℋ(E) . Then (xe′′)E∉ℋ(E) . Since R is Euclidean, (xe″″)E∈R*((xe′′)E) . That implies R*((xe′′)E)∩𝒢(E)≠∅ . Thus (xe′′)E∈Pg(fE,R) . By Lemma 5.3, (xe′′)E∈ℋ(E) , a contradiction. Therefore, Pg(fE, R) ⊆ Ph(fE,R).

Proposition 5.12

Let R be reflexive on fE ∈ S(U,E). Then the following are equivalent.

  1. (1)

    R is transitive;

  2. (2)

    ∀ gE ∈ Sf(U,E),

           apr¯P(gE)⊂˜apr¯P(apr¯P(gE))⊂˜apr¯P(apr¯P(gE))⊂˜apr¯P(gE).

Proof.

(1) ⇒ (2). Let gE ∈ Sf(U,E). Denote

hE=apr¯P(gE),kE=apr¯P(gE)  and  lE=apr¯P(kE).

We will prove hE⊂˜apr¯P(hE) . We can suppose Pg(fE,R) ≠ ∅. ∀ (xe)E ∈ Pg(fE,R), by Lemma 5.11, Pg(fE,R) = Ph(fE,R), then (xe)E ∈ Ph(fE,R). Hence apr¯P(gE)⊂˜apr¯P(apr¯P(gE)) . By Proposition 5.7,

apr¯P(apr¯P(gE))⊂˜apr¯P(apr¯P(gE)).

Suppose that ℒ(E) − 𝒦(E) ≠ ∅. Pick (xe)E ∈ ℒ(E) − 𝒦(E). Then (xe)E ∈ ℒ(E). By Lemma 5.3, (xe)E ∈ Pk(fE,R). This implies R*((xe)E) ∩ 𝒦(E) ≠ ∅. Pick (xe′′)E∈R*((xe)E)∩𝒦(E) . Then (xe′′)E∈𝒦(E) . By Lemma 5.3, (xe)E ∈ Pg(fE,R). This implies R*((xe′′)E∩𝒢(E)≠∅ . Thus 𝒢(E) ≠ ∅. Since (xe)E ∉ 𝒦(E), by Lemma 5.3, we have (xe)E ∉ Pg(fE,R). So R*((xe)E) ∩ 𝒢(E) = ∅. since R is reflexive, (xe)E ∈ R*((xe)E). Thus 𝒢(E) = ∅, a contradiction. Hence ℒ(E) ⊆ 𝒦(E). By Theorem 3.11, lE⊂˜kE .

Therefore, apr¯P(apr¯P(gE))⊂˜apr¯P(gE) .

(2) ⇒ (1). Let (xe)E, (xe′′)E , (xe″″)E∈ℱ(E) with (xe)E∈R*((xe′′)E) and (xe′′)E∈R*((xe″″)E) . Denote

gE=(xe)E,hE=apr¯P(gE)  and  lE=apr¯P(hE).

By Lemma 5.3 and Lemma 5.5, ℒE = Ph(fE,R) and

ℋ(E)={(yε)E∈ℱ(E):(xe)E∈R*((yε)E)}.

(xe)E∈R*((xe′′)E) implies (xe)E ∈ ℋ(E). Note that (xe′′)E∈R*((xe″″)E) . Then R*((xe″″)E)∩ℋ(E)≠∅ . So (xe″″)E∈Ph(fE,R)=ℒ(E) .

Since apr¯P(apr¯P((xe)E))⊂˜apr¯P((xe)E) , ℒ(E) ⊆ ℋ(E). Thus (xe″″)E∈ℋ(E) . By Lemma 5.5, (xe)E∈R*((xe″″)E) .

Therefore, R is transitive.

Proposition 5.13

Let R be reflexive on fE ∈ S(U,E). Then the following are equivalent.

  1. (1)

    R is Euclidean;

  2. (2)

    ∀ gE ∈ Sf(U,E),

           apr¯P(apr¯P(gE))⊂˜apr¯P(gE)⊂˜apr¯P(gE)⊂˜apr¯P(apr¯P(gE)).

Proof.

(1) ⇒ (2). Let gE ∈ Sf(U,E). Denote

kE=apr¯P(gE),  lE=apr¯P(kE)  and  hE=apr¯P(gE).

Suppose ℒ(E) − 𝒦(E) ≠ ∅. Pick (xe)E ∈ ℒ(E) − 𝒦(E). Then (xe)E ∉ 𝒦(E) and (xe)E ∈ ℒ(E). By Lemma 5.3, (xe)E ∈ Pk(fE,R). This implies R*((xe)E) ∩ 𝒦(E) ≠ ∅. Pick (xe′′)E∈R*((xe)E)∩𝒦(E) . Then (xe′′)E∈𝒦(E) . By Lemma 5.3, (xe)E ∈ Pg(fE,R). This implies R*((xe′′)E⊆𝒢(E) . Since R is Euclidean, R*((xe)E)⊆R*((xe′′)E) . Thus R*((xe)E) ⊆ 𝒢(E). Since (xe)E ∉ 𝒦(E), by Lemma 5.3, (xe)E ∉ Pg(fE,R). So R*((xe)E) ⊈ 𝒢(E), a contradiction. Hence ℒ(E) ⊆ 𝒦(E). By Theorem 3.11, lE⊂˜kE .

Therefore, apr¯P(apr¯P(gE))⊂˜apr¯P(gE) .

By Proposition 5.7, apr¯P(gE)⊂˜apr¯P(gE) .

We will prove hE⊂˜apr¯P(hE) . Suppose Pg(fE,R) ≠ ∅. ∀ (xe)E ∈ Pg(fE,R), by Lemma 5.11, Pg(fE,R) ⊆ Ph(fE,R). Then (xe)E ∈ Ph(fE,R). Hence apr¯P(gE)⊂˜apr¯P(apr¯P(gE)) .

(2) ⇒ (1) Let (xe)E, (xe′′)E , (xe″″)E∈ℱ(E) with (xe)E∈R*((xe″″)E) and (xe′′)E∈R*((xe″″)E) . Denote

gE=(xe)E,  hE=apr¯P(gE)  and  lE=apr¯P(hE).

By Lemma 5.3 and Lemma 5.5, ℒE = Ph(fE,R) and

ℋ(E)={(yε)E∈ℱ(E):(xe)E∈R*((yε)E)}.

(xe)E∈R*((xe″″)E) implies (xe″″)E∈ℋ(E) . Since apr¯P(gE)⊂˜apr¯P(apr¯P(gE))) , ℋ(E) ⊆ ℒ(E). So (xe″″)E∈ℒ(E)=Ph(fE,R) . Thus R*((xe″″)E)⊆ℋ(E) . Note that (xe′′)E∈R*((xe″″)E) . Then (xe′′)E∈ℋ(E) . By Lemma 5.5, (xe)E∈R*((xe′′)E) .

Therefore, R is Euclidean.

6. Soft topologies induced by s-relations on special soft sets

In this section, we investigate soft topologies induced by a reflexive s-relation on a special soft set and give their structure.

6.1. s-relations on Ũ

Proposition 6.1

Let R be a s-relation on Ũ. Then ∀ gE ∈ S(U,E), we have

  1. (1)

    apr¯P(gE)=U˜−apr¯P(U˜−gE) ;

  2. (2)

    apr¯P(gE)=U˜−apr¯P(U˜−gE) .

Proof.

  1. (1)

    Denote hE=U˜−gE , qE=apr¯P(hE) , lE=U˜−qE and kE=apr¯P(gE) .

    To prove lE⊂˜kE , by Theorem 3.11, it suffices to show ℒ(E) ⊆ 𝒦(E).

    Suppose ℒ(E) − 𝒦(E) ≠ ∅. Pick (xe)E ∈ ℒ(E) − 𝒦(E). Then (xe)E ∉ 𝒦(E). By Lemma 5.3, (xe)E ∉ Pg(fE,R). Thus R*((xe)E) ⊈ 𝒢(E). Pick (xe′′)E∈R*((xe)E)−𝒢(E) . Then (xe′′)E∉𝒢(E) . This implies x′ ∉ g(e′). So x′ ∈ U − g(e′) = h(e′).

    Since (xe)E ∈ ℒ(E), (xe)E ∈ 𝒰(E) − 𝒬(E). Then (xe)E ∉ 𝒬(E). By Lemma 5.3, (xe)E ∉ Ph(fE,R). So R*((xe)E) ∩ ℋ(E) = ∅. Since (xe′′)E∈R*((xe)E) , (xe′′)E∉ℋ(E) . This implies x′ ∉ h(e′), a contradiction. Hence ℒ(E) ⊆ 𝒦(E).

    Therefore, U˜−apr¯P(U˜−gE)⊂˜apr¯P(gE) .

    Conversely, to prove kE⊂˜lE=U˜−qE , by Remark 2.11, it suffices to show kE∩˜qE=∅ .

    Suppose wE=kE∩˜qE≠∅˜ . By Proposition 3.5, 𝒲(E) = 𝒦(E) ∩ 𝒬(E). Pick (xe)E ∈ 𝒲(E). Then (xe)E ∈ 𝒦(E) and (xe)E ∈ 𝒬(E). By Lemma 5.3, (xe)E ∈ Pg(fE,R) and (xe)E ∈ Ph(fE,R). Thus R*((xe)E) ⊆ 𝒢(E) and R*((xe)E) ∩ ℋ(E) ≠ ∅. Pick (xe′′)E∈R*((xe)E)∩ℋ(E) . Then (xe′′)E∈ℋ(E) . Thus x′ ∈ h(e′) = U − g(e′) and so x′ ∉ g(e′). But (xe′′)E∈R*((xe)E) . This implies (xe′′)E∈𝒢(E) . Thus x′ ∈ g(e′), a contradiction. Hence apr¯P(gE)⊂˜U˜−apr¯P(U˜−gE) .

    Therefore, apr¯P(gE)=U˜−apr¯P(U˜−gE) .

  2. (2)

    Denote hE = Ũ − gE, qE=apr¯P(hE) , lE = Ũ − qE and kE=apr¯P(gE) .

    To prove lE⊂˜kE , by Theorem 3.11, it suffices to show ℒ(E) ⊆ 𝒦(E).

    Suppose ℒ(E) − 𝒦(E) ≠ ∅. Pick (xe)E ∈ ℒ(E) − 𝒦(E). Then (xe)E ∉ 𝒦(E). By Lemma 5.3, (xe)E ∉ Pg(fE,R). Then R*((xe)E) ∩ 𝒢(E) = ∅.

    Since (xe)E ∈ ℒ(E), (xe)E ∈ 𝒰(E) − 𝒬(E). Then (xe)E ∉ 𝒬(E). By Lemma 5.3, (xe)E ∉ Ph(fE,R). Then R*((xe)E) ⊈ ℋ(E) and so R*((xe)E) − ℋ(E) ≠ ∅. Pick (xe′′)E∈R*((xe)E)−ℋ(E) . Then (xe′′)E∉ℋ(E) . Thus x′ ∉ h(e′) = g′(e′) = U − g(e′) and so x′ ∈ g(e′).

    Since (xe′′)E∈R*((xe)E) , (xe′′)E∉𝒢(E) . Thus x′ ∉ g(e′), a contradiction. Hence ℒ(E) ⊆ 𝒦(E).

    Therefore, U˜−apr¯P(U˜−gE)⊂˜apr¯P(gE) .

    Conversely, to prove kE⊂˜lE=U˜−qE , by Remark 2.11, it suffices to show kE∩˜qE=∅ .

    Suppose wE=kE∩˜qE≠∅˜ . By Proposition 3.5, 𝒲(E) = 𝒦(E) ∩ 𝒬(E). Pick (xe)E ∈ 𝒲(E). Then (xe)E ∈ 𝒦(E) and (xe)E ∈ 𝒬(E). By Lemma 5.3, (xe)E ∈ Pg(fE,R) and (xe)E ∈ Ph(fE,R). Thus R*((xe)E) ∩ 𝒢(E) ≠ ∅ and R*((xe)E) ⊆ ℋ(E). Pick (xe′′)E∈R*((xe)E)∩𝒢(E) . Then (xe′′)E∈𝒢(E) . Thus x′ ∈ g(e′). But (xe′′)E∈R*((xe)E) . This implies (xe′′)E∈ℋ(E) . Thus x′ ∈ h(e′) =U − g(e′) and so x′ ∉ g(e′), a contradiction. Hence apr¯P(gE)⊂˜U˜−apr¯P(U˜−gE) .

    Therefore, apr¯P(gE)=U˜−apr¯P(U˜−gE) .

Corollary 6.2

Let R be a s-relation on Ũ. Then the following are equivalent.

  1. (1)

    R is serial;

  2. (2)

    ∀ gE ∈ S(U,E), apr¯P(gE)⊂˜apr¯P(gE) ;

  3. (3)

    apr¯P(∅˜)=∅˜ ;

  4. (4)

    apr¯P(U˜)=U˜ .

Proof.

This follows from Proposition 5.7 and Proposition 6.1.

Corollary 6.3

Let R be a s-relation on Ũ. Then the following are equivalent.

  1. (1)

    R is reflexive;

  2. (2)

    ∀ gE ∈ S(U,E), apr¯P(gE)⊂˜gE ;

  3. (3)

    ∀ gE ∈ S(U,E), gE⊂˜apr¯P(gE) .

Proof.

This follows from Proposition 5.8 and Proposition 6.1.

Corollary 6.4

Let R be a s-relation on Ũ. Then the following are equivalent.

  1. (1)

    R is symmetric;

  2. (2)

    ∀ gE ∈ Sf(U,E), gE⊂˜apr¯P(apr¯P(gE)) ;

  3. (3)

    ∀ gE ∈ Sf(U,E), apr¯P(apr¯P(gE))⊂˜gE .

Proof.

This follows from Proposition 5.10 and Proposition 6.1.

Corollary 6.5

Let R be a s-relation on Ũ. Then the following are equivalent.

  1. (1)

    R is transitive;

  2. (2)

    apr¯P(gE)⊂˜apr¯P(apr¯P(gE)) ∀ gE∈Sf(U,E) ;

  3. (3)

    apr¯P(apr¯P(gE))⊂˜apr¯P(gE)  ∀  gE∈Sf(U,E) .

Proof.

This follows from Proposition 5.12 and Proposition 6.1.

Corollary 6.6

Let R be a s-relation on Ũ. Then the following are equivalent.

  1. (1)

    R is Euclidean;

  2. (2)

    ∀ gE ∈ Sf(U,E), apr¯P(gE)⊂˜apr¯P(apr¯P(gE)) ;

  3. (3)

    ∀ gE ∈ Sf(U,E), apr¯P(apr¯P(gE))⊂˜apr¯P(gE) .

Proof.

This follows from Proposition 5.13 and Proposition 6.1.

6.2. Soft topologies induced by relations on Ũ

Theorem 6.7

Let R be reflexive on Ũ. Then τR={gE∈S(U,E):apr¯P(gE)=gE} is a soft topology over U.

Proof.

  1. (1)

    By Proposition 5.9, ∅˜ , Ũ ∈ τR.

  2. (2)

    Let gE, hE ∈ τR. Since gE=apr¯P(gE) and hE=apr¯P(hE) , by Proposition 5.6, gE∩˜hE=apr¯P(gE)∩˜apr¯P(hE)=apr¯P(gE∩˜hE) .

  3. (3)

    Let (gα)E ∈ τR ∀ α ∈ ∧, we will show that ∪˜  {(gα)E:α∈∧}=apr¯P(∪˜  {(gα)E:α∈∧}) . Since R is reflexive, by Proposition 5.8, apr¯P(∪˜  {(gα)E:α∈∧})⊂˜∪˜  {(gα)E:α∈∧} .

    Conversely, since (gα)E=apr¯P((gα)E) , by Proposition 5.6, we have ∪˜  {(gα)E:α∈∧}=∪˜  {apr¯P((gα)E):α∈∧}⊂˜apr¯P(∪˜  {(gα)E:α∈∧}) .

    Therefore, τR={gE∈Sf(U,E):apr¯P(gE)=gE} is a soft topology on fE.

Definition 6.8

Let R be reflexive on Ũ. Then τR is called the soft topology induced by R on Ũ.

The following Theorem 6.9 gives the structure of the soft topology induced by a reflexive s-relation on Ũ.

Theorem 6.9

Let R be reflexive on Ũ and τR the soft topology induced by R on U. Then

  1. (1)

    1. a)

      τR={apr¯P(gE):gE∈S(U,E)} whenever R is transitive.

    2. b)

      {apr¯P(gE):gE∈S(U,E)}⊆τR whenever R is Euclidean.

  2. (2)

    apr¯P is a soft interior operator of τR.

  3. (3)

    apr¯P is a soft closure operator of τR.

Proof.

  1. (1)

    1. a)

      Let gE ∈ S(U,E). By Corollary 6.5, apr¯P(apr¯P(gE))=apr¯P(gE) . This implies apr¯P(gE)∈τR . Thus τR⊇{apr¯P(gE):gE∈S(U,E)} . Hence τR={apr¯P(gE):gE∈S(U,E)} .

    2. b)

      By Corollary 6.6, {apr¯P(gE):gE∈S(U,E)}⊆τR .

  2. (2)

    It suffices to show apr¯P(gE)=int(gE) for any gE ∈ S(U,E).

    By (1), apr¯P(gE)∈τR . By Corollary 6.3, apr¯P(gE)⊂˜gE . Thus apr¯P(gE)⊂˜int(gE) .

    Conversely, suppose hE ∈ τR and hE⊂˜gE , by Proposition 5.6, hE=apr¯P(hE)⊂˜apr¯P(gE) . By Remark 2.5,

    int(gE)=∪˜ {hE:hE∈τR and hE⊂˜gE}⊂˜apr¯P(gE).

    Thus apr¯P(gE)=int (gE) .

  3. (3)

    By Proposition 2.17 and Proposition 6.1,

    apr¯P(gE)=U˜−apr¯P(U˜−gE)=U˜−int (U˜−gE)=cl(gE).

Theorem 6.10

Let R be reflexive and transitive on Ũ and τR the soft topology induced by R on Ũ. Then ∀ gE ∈ S(U,E), gE∈τR⇔gE∈τR′ .

Proof.

Necessity. Let gE ∈ τR. Then apr¯P(gE)=gE . By Proposition 6.1 and Remark 2.9, apr¯P(gE′)=U˜−apr¯P((gE′)′)=U˜−apr¯P(gE)=U˜−gE=gE′ . By Theorem 6.9, gE′=apr¯P(gE′)∈τR . Thus gE∈τR′ .

Sufficiency. Let gE∈τR′ . Then gE′∈τR and apr¯P(gE′)=gE′ . By Proposition 6.1 and Remark 2.9, apr¯P(gE)=U˜−apr¯P(gE′)=gE . By Theorem 6.5, gE=apr¯P(gE)∈τR .

Definition 6.11

Let τ be a topology on U. τ is called a pseudo-discrete topology on U, if A ⊆ U is open in U if and only if A is closed in U.

Theorem 6.12

Let R be reflexive and transitive on Ũ. Then τR is a pseudo-discrete topology over U.

Proof.

This holds by Theorem 6.7 and Theorem 6.10.

7. Conclusions

In this paper, the fact that soft sets can be translated into soft point sets has been proved. Thus, we may expediently handled soft set like ordinary sets. We have proposed s-relations on soft sets. By means of soft points and s-relations, a pair of soft rough approximate operations has been defined. Serial, reflexive, symmetric, transitive and Euclidean s-relations have been characterized by using soft rough approximate operations. In addition, we have investigated soft topologies induced by a reflexive s-relation on a special soft set and given their structure. In the future, we will investigate the axiomatization of the proposed soft rough approximate operations and consider some concrete applications of our proposed notions.

Acknowledgements

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of the paper. This work is supported by Guangxi University Science and Technology Research Project (KY2015YB266), Given Point on Master of Applied Statistics in Guangxi University of Finance and Economics (2016TJYB03), Quantitative Economics Key Laboratory Program of Guangxi University of Finance and Economics (2014SYS11), Guangxi Province Universities and Colleges Excellence Scholar and Innovation Team Funded Scheme, and National Natural Science Foundation of China (11461005).

References

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Journal
International Journal of Computational Intelligence Systems
Volume-Issue
10 - 1
Pages
90 - 103
Publication Date
2017/01/01
ISSN (Online)
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risenwbib
TY  - JOUR
AU  - Guangji Yu
PY  - 2017
DA  - 2017/01/01
TI  - Soft points, s-relations and soft rough approximate operations
JO  - International Journal of Computational Intelligence Systems
SP  - 90
EP  - 103
VL  - 10
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.2017.10.1.7
DO  - 10.2991/ijcis.2017.10.1.7
ID  - Yu2017
ER  -