Further research of single valued neutrosophic rough set model

Further research of single valued neutrosophic rough set model Yan-Ling Liu, Hai-Long Yang∗ College of Mathematics and Information Science, Shaanxi Normal University, 710119, Xi’an, P.R. China

Abstract—Neutrosophic sets (NSs), as a new mathematical tool for dealing with problems involving incomplete, indeterminant and inconsistent knowledge, were proposed

by Smarandache. By simplifying NSs, Wang et al. proposed the concept of single valued neutrosophic sets (SVNSs) and studied some properties of SVNSs. In this paper, we mainly investigate the topological structures of single valued neutrosophic rough sets which is constructed by combining SVNSs and rough sets. Firstly, we introduce the concept of single valued neutrosophic topological spaces. Then, we discuss the relationships between single valued neutrosophic approximation spaces and single valued neutrosophic topological spaces. Concretely, a reflexive and transitive single valued neutrosophic relation can induce a single valued neutrosophic topological space such that its single valued neutrosophic interior and closure operators are the lower and upper approximation operators induced by this single valued neutrosophic relation, respectively. Conversely, a single valued neutrosophic interior (closure, respectively) operator derived from a single valued neutrosophic topological space is just the single valued neutrosophic lower (upper, respectively) approximation operator derived from a single valued neutrosophic approximation space under some conditions. Finally, we show there exists a one-to-one correspondence between the set of all reflexive and transitive single valued neutrosophic relations and the set of all single valued neutrosophic rough topologies.

Keywords—Neutrosophic sets; Single valued neutrosophic sets; Single valued neutrosophic rough sets; Single valued neutrosophic topological spaces

1

Introduction Rough set theory, initiated by Pawlak [15, 16], is an effective mathematical tool for

the study of intelligent systems characterized by insufficient and incomplete information. ∗

Corresponding author. E-mail address: yanghailong@snnu.edu.cn (H.-L. Yang).

1

Pawlak rough set model is established based on an equivalence relation. In the real application, the equivalence relation is a very stringent condition which limits the applications of rough sets in real world. For this reason, by replacing the equivalence relation with covering, similarity relation, tolerance relation, fuzzy relation, etc, different kinds of generalizations of Pawlak rough set model were proposed [3, 5, 9, 11, 19, 20, 23, 30, 31, 32, 33].

Rough set theory has been successfully applied to many fields, such as machine learning, knowledge acquisition, and decision analysis, etc.

Smarandache [21, 22] introduced the concept of NSs which consists of three membership functions (truth membership function, indeterminacy membership function and falsity membership function), where every function value is a real standard or non-standard subset of the nonstandard unit interval ]0− , 1+ [. The NS generalizes the concepts of the classical set, fuzzy set [37], interval-valued fuzzy set [24], intuitionistic fuzzy set [1] and interval-valued intuitionistic fuzzy set [2]. Wang et al. [25] proposed SVNSs by simplifying NSs. SVNSs can also be looked as an extension of intuitionistic fuzzy sets [1], in which three membership functions are unrelated and their function values belong to the unit closed interval. Many researchers have studied the theory and applications of SVNSs. Ye [34, 35] proposed decision making based on correlation coefficients and weighted correlation coefficient of SVNSs, and gave the application of proposed methods. Majumdar and Samant [13] studied distance, similarity and entropy of SVNSs from a theoretical aspect. Yang et al. [29] proposed SVNRs and studied some kinds of kernels and closures of SVNRs. Broumi and Smarandance [6] proposed single valued neutrosophic information systems based on rough set theory to exploit simultaneously the advantages of SVNSs and rough sets. They studied rough approximation of a SVNS in the single valued neutrosophic information systems and investigated the knowledge reduction and extension of the single valued neutrosophic information systems. Yang et al. [28] proposed single valued neutrosophic rough sets by combining SVNSs and rough sets, and explored a general framework of the study of single valued neutrosophic rough sets.

Topological structures and properties [10] of rough sets are important research issues for the study of rough sets. Many researchers have addressed the issues [4, 7, 8, 12, 14, 17, 18, 26, 27]. Wiweger [26] and Chuchro [7, 8] established the relationships between crisp rough sets and crisp topological spaces. Boixader et al. [4] investigated the connection between fuzzy rough sets and fuzzy topological spaces. Qin et al. [17, 18]

2

discussed topological structures of fuzzy rough sets. Wu and Zhou [27] generalized the results to IF rough sets and established relationships between IF rough approximations and IF topologies. Ma and Hu [14] studied topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets. Li and Cui [12] studied similarity of fuzzy relations which is based on fuzzy topologies induced by fuzzy rough approximation operators. Zhang et al. [38] discussed the topological structures of interval-valued hesitant fuzzy rough set and its application. Along this line, in the present paper, we shall study topological structures of single valued neutrosophic rough sets and establish the relationships between the single valued neutrosophic approximation spaces and single valued neutrosophic topological spaces.

The rest of this paper is organized as follows. In the next section, we recall some basic notions on Pawlak rough sets, NSs, SVNSs and single valued neutrosophic rough sets. In Section 3, we give notions of single valued neutrosophic topology and its interior operation and closure operation. Some related properties are also studied. Section 4 investigates the relationships between single valued neutrosophic approximation spaces and single valued neutrosophic topology spaces. The last section summarizes the conclusion.

2

Preliminaries In this section, we recall some basic notions and results which will be used in the

paper.

2.1. P awlak rough sets Definition 2.1([15, 16]). Let U be a nonempty finite universe and R be an equivalence relation in U . (U, R) is called a Pawlak approximation space. ∀X ⊆ U , the lower and upper approximations of X, denoted by R(X) and R(X), are defined as follows, respectively: R(X) = {x ∈ U | [x]R ⊆ X}, R(X) = {x ∈ U | [x]R ∩ X ̸= ∅}, where [x]R = {y ∈ U | (x, y) ∈ R}. R and R are called as lower and upper approximation operators, respectively. The pair (R(X), R(X)) is called a Pawlak rough set.

2.2. NSs and SVNSs

3

Definition 2.2([21]). Let U be a space of points (objects), with a generic element e in U is characterized by three membership functions, a in U denoted by x. A NS A truth-membership function TAe, an indeterminacy membership function IAe and a falsitymembership function FAe, where ∀x ∈ U , TAe(x), IAe(x) and FAe(x) are real standard or non-standard subsets of ]0− , 1+ [.

There is no restriction on the sum of TAe(x), IAe(x) and FAe(x), thus 0− ≤ sup TAe(x)+ sup IAe(x)+ sup FAe(x) ≤ 3+ .

Definition 2.3([25]). Let U be a space of points (objects), with a generic element in e in U is characterized by three membership functions, a U denoted by x. A SVNS A truth-membership function TAe, an indeterminacy membership function IAe, and a falsitymembership function FAe, where ∀x ∈ U , TAe(x), IAe(x), FAe(x) ∈ [0, 1]. e e can be denoted by A= e {⟨x, T e(x), I e(x), F e(x)⟩ | x ∈ U } or A= The SVNS A A A A e (TAe, IAe, FAe). ∀x ∈ U , A(x) = (TAe(x), IAe(x), FAe(x)), and (TAe(x), IAe(x), FAe(x)) is called a single valued neutrosophic number. e be a SVNS In this paper, SVNS(U ) will denote the family of all SVNSs in U . Let A e is called an empty SVNS, in U . If ∀x ∈ U , TAe(x) = 0 and IAe(x) = FAe(x) = 1, then A e is called a full denoted by e ∅. If ∀x ∈ U , TAe(x) = 1, and IAe(x) = FAe(x) = 0, then A e . ∀α1 , α2 , α3 ∈ [0, 1], α1\ SVNS, denoted by U , α2 , α3 denotes a constant SVNS satisfying, Tα1\ ,α2 ,α3 (x) = α1 , Iα1\ ,α2 ,α3 (x) = α2 , Fα1\ ,α2 ,α3 (x) = α3 .

For any y ∈ U , a single valued neutrosophic singleton set 1y and its complement 1U −{y} are, defined as: { ∀x ∈ U , { 1, x = y 0, x = y T1y (x) = , I1y (x) = F1y (x) = ; 0,{ x ̸= y 1, x{̸= y 0, x = y 1, x = y T1U −{y} (x) = , I1U −{y} (x) = F1U −{y} (x) = .

1, x ̸= y 0, x ̸= y e and B e be two SVNSs in U . If for any x ∈ U , T e(x) ≤ T e (x), Definition 2.4([36]). Let A A B e is contained in B, e i.e. A e b B. e IAe(x) ≥ IBe (x) and FAe(x) ≥ FBe (x), then we called A ebB e and B e b A, e then we called A e is equal to B, e denoted by A e = B. e If A e be a SVNS in U . The complement of A e is denoted by A ec , Definition 2.5([25]). Let A where ∀x ∈ U , TAec (x) = FAe(x), IAec (x) = 1 − IAe(x), and FAec (x) = TAe(x). e and B e be two SVNSs in U .

Definition 2.6([29]). Let A e and B e is a SVNS C, e denoted by C e=A e d B, e where ∀x ∈ U , (1) The union of A

4

TCe (x) = TAe(x) ∨ TBe (x), ICe (x) = IAe(x) ∧ IBe (x) and FCe (x) = FAe(x) ∧ FBe (x); e and B e is a SVNS D, e denoted by D e =A e e B, e where ∀x ∈ U , (2) The intersection of A TDe (x) = TAe(x) ∧ TBe (x), IDe (x) = IAe(x) ∨ IBe (x) and FDe (x) = FAe(x) ∨ FBe (x), where “ ∨ ” and “ ∧ ” denote maximum and minimum, respectively.

It is easy to verify that the union and intersection of SVNSs satisfy commutative law, associative law, and distributive law. e and B e be two SVNSs in U . The following results hold: Proposition 2.7([28]). Let A ebA edB e and B ebA e d B, e (1) A eeB ebA e and A eeB e b B, e (2) A ec )c = A, e (3) (A e d B) e c=A ec e B ec, (4) (A e e B) e c=A ec d B ec.

(5) (A 2.3. Single valued neutrosophic rough sets e in U × U is called a single valued neutrosophic relation Definition 2.8([28]). A SVNS R e {⟨(x, y), T e (x, y), I e (x, y), F e (x, y)⟩ | (x, y) ∈ U × U }, (SVNR) in U , denoted by R= R R R where TRe : U × U −→ [0, 1], IRe : U × U −→ [0, 1], and FRe : U × U −→ [0, 1] denote the truth-membership function, indeterminacy membership function, and falsity-membership e respectively. function of R, e be a SVNR in U , the complement R ec of R e is defined as follows Let R ec = {⟨(x, y), T ec (x, y), I ec (x, y), F ec (x, y)⟩ | (x, y) ∈ U × U }, R R R R where ∀(x, y) ∈ U × U , TRec (x, y) = FRe (x, y), IRec (x, y) = 1 − IRe (x, y) and FRec (x, y) = TRe (x, y). e be a SVNR in U . If ∀x ∈ U , T e (x, x) = 1 and I e (x, x) = Definition 2.9([28]). Let R R R e is called a reflexive SVNR. If ∀x, y ∈ U , T e (x, y) = T e (y, x), FRe (x, x) = 0, then R R R e is called a symmetric SVNR. If IRe (x, y) = IRe (y, x) and FRe (x, y) = FRe (y, x), then R ∨ ∧ ∧ e is called a seri∀x ∈ U , y∈U TRe (x, y) = 1 and y∈U IRe (x, y) = y∈U FRe (x, y) = 0, then R ∨ ∧ al SVNR. If ∀x, y, z ∈ U , y∈U (TRe (x, y)∧TRe (y, z)) ≤ TRe (x, z), y∈U (IRe (x, y)∨IRe (y, z)) ≥ ∧ e is called a transitive SVNR, IRe (x, z) and y∈U (FRe (x, y) ∨ FRe (y, z)) ≥ FRe (x, z), then R where “ ∨ ” and “ ∧ ” denote maximum and minimum, respectively.

5

e be a SVNR in U , the tuple (U, R) e is called a single valued Definition 2.10([28]). Let R e ∈ SVNS(U ), the lower and upper approximationneutrosophic approximation space. ∀A e w.r.t. (U, R), e denoted by R( e A) e and R( e A), e are two SVNSs whose membership s of A functions are defined as: ∀x ∈ U , ∧ TR( e(y)), e A) e (x, y) ∨ TA e (x) = y∈U (FR ∨ IR( e A) e (x) = e (x, y)) ∧ IA e(y)), y∈U ((1 − IR ∨ FR( e A) e (x) = e (x, y) ∧ FA e(y)); y∈U (TR ∨ TR( e (x, y) ∧ TA e(y)), e A) e (x) = y∈U (TR ∧ IR( e (x, y) ∨ IA e(y)), e A) e (x) = y∈U (IR ∧ FR( e (x, y) ∨ FA e(y)). e A) e (x) = y∈U (FR e A), e R( e A)) e is called the single valued neutrosophic rough set of A e w.r.t. (U, R). e The pair (R( e and R e are referred to as the single valued neutrosophic lower and upper approximation R operators, respectively.

Yang et al. [28] studied the properties of single valued neutrosophic lower and upper approximation operators as follows. e be a single valued neutrosophic approximation sProposition 2.11([28]). Let (U, R) pace. The single valued neutrosophic lower and upper approximation operators have the eB e ∈ SVNS(U ), ∀α1 , α2 , α3 ∈ [0, 1], following properties: ∀A, eU e) = U e , R( ee (1) R( ∅) = e ∅; e b B, e then R( e A) e b R( e B) e and R( e A) e b R( e B); e (2) If A eA e e B) e = R( e A) e e R( e B), e R( eA e d B) e = R( e A) e d R( e B); e (3) R( eA e d B) e c R( e A) e d R( e B), e R( eA e e B) e b R( e A) e e R( e B); e (4) R( eA ec ) = (R( e A)) e c , R( eA ec ) = (R( e A)) e c; (5) R( e d α1\ e A) e d α1\ e e α1\ e A) e e α1\ (6) R(A , α2 , α3 ) = R( , α2 , α3 , R(A , α2 , α3 ) = R( , α2 , α3 ; e α1\ ee (7) R( , α2 , α3 ) = α1\ , α2 , α3 ⇐⇒ R( ∅) = e ∅, e α1\ eU e) = U e.

R( , α2 , α3 ) = α1\ , α2 , α3 ⇐⇒ R( e be a single valued neutrosophic approximation space.

Proposition 2.12([28]). Let (U, R) e are the lower and upper approximation operators, then we have e and R R e is serial ⇐⇒ R( e α1\ (1) R , α2 , α3 ) = α1\ , α2 , α3 , ∀α1 , α2 , α3 ∈ [0, 1], ee ⇐⇒ R( ∅) = e ∅, e α1\ ⇐⇒ R( , α2 , α3 ) = α1\ , α2 , α3 , ∀α1 , α2 , α3 ∈ [0, 1], 6

eU e) = U e; ⇐⇒ R( e is reflexive ⇐⇒ R( e A) e b A, e ∀A e ∈ SVNS(U ), (2) R e b R( e A), e ∀A e ∈ SVNS(U ); ⇐⇒ A e is symmetric ⇐⇒ R(1 e U −{x} )(y) = R(1 e U −{y} )(x), ∀x, y ∈ U , (3) R e x )(y) = R(1 e y )(x), ∀x, y ∈ U ; ⇐⇒ R(1 e is transitive ⇐⇒ R( e A) e b R( e R( e A)), e ∀A e ∈ SVNS(U ), (4) R e R( e A)) e b R( e A), e ∀A e ∈ SVNS(U ). ⇐⇒ R(

3

Single valued neutrosophic topological spaces In this section, we will introduce the concept of single valued neutrosophic topological

spaces and basis concepts related to single valued neutrosophic topological spaces.

We first introduce the concept of single valued neutrosophic topology as follows.

Definition 3.1. A single valued neutrosophic topology on a nonempty set U is a family τ of SVNSs in U that satisfies the following conditions: e ∈ τ, (T1 ) e ∅, U (T2 )

eeB e ∈ τ for any A, eB e ∈ τ, A

(T3 )

ei ∈ τ for any A ei ∈ τ , i ∈ I, I is an index set. di∈I A

e The pair (U, τ ) is called a single valued neutrosophic topological space and each SVNS A in τ is referred to as a single valued neutrosophic open set in (U, τ ). The complement of a single valued neutrosophic open set in (U, τ ) is called a single valued neutrosophic closed set in (U, τ ). e B, e C, e D e are four SVNSs in U defined as follows: Example 3.2. Let U = {x1 , x2 }. A, e = {⟨x1 , 0.2, 0.8, 0.1⟩, ⟨x2 , 1, 0.3, 0.1⟩}, A e = {⟨x1 , 0.2, 0.8, 0.6⟩, ⟨x2 , 0.5, 0.4, 1⟩}, B e = {⟨x1 , 0.3, 0.7, 0.1⟩, ⟨x2 , 1, 0.2, 0.1⟩}, C e = {⟨x1 , 0.1, 0.9, 0.8⟩, ⟨x2 , 0.4, 0.5, 1⟩}.

D e , A, e B, e C, e D} e is a single valued neutrosophic topology

By Definitions 2.6 and 3.1, τ = {e ∅, U on U and (U, τ ) is a single valued neutrosophic topological space.

7

Now, we define the single valued neutrosophic interior and closure operators in a single valued neutrosophic topological space.

Definition 3.3. Let (U, τ ) be a single valued neutrosophic topological space. For any e ∈ SVNS(U ), the single valued neutrosophic interior and closure of A e are defined as A follows: e = d{M f|M f ∈ τ and M f b A}, e int(A) e = e{N e |N e c ∈ τ and A ebN e }, cl(A) where int and cl: SVNS(U ) −→ SVNS(U ) are called the single valued neutrosophic interior and closure operators of τ , respectively.

Next, we discuss the properties of the single valued neutrosophic interior and closure operators. e∈ Theorem 3.4. Let (U, τ ) be a single valued neutrosophic topological space. For any A SVNS(U ), we have e is a single valued neutrosophic open set in (U, τ ) iff int(A) e = A; e (1) A e is a single valued neutrosophic closed set in (U, τ ) iff cl(A) e = A. e (2) A Proof. It is straightforward from Definition 3.3.

Theorem 3.5. Let (U, τ ) be a single valued neutrosophic topological space. For any eB e ∈ SVNS(U ), the following results hold: A, e c = cl(A ec ); (Int0) (int(A)) e c = int(A ec ); (Cl0) (cl(A)) e) = U e , int(e (Int1) int(U ∅) = e ∅; e) = U e , cl(e (Cl1) cl(U ∅) = e ∅; e b A; e (Int2) int(A) e b cl(A); e (Cl2) A e = int(A); e (Int3) int(int(A)) e = cl(A); e (Cl3) cl(cl(A)) e e B) e = int(A) e e int(B); e (Int4) int(A e d B) e = cl(A) e d cl(B); e (Cl4) cl(A e b B, e then int(A) e b int(B); e (Int5) If A e b B, e then cl(A) e b cl(B). e (Cl5) If A

8

eB e ∈ SVNS(U ), by Definition 3.3 and Proof. We only prove (Int0) and (Int4). For any A, Proposition 2.7, we have e c = (d{M f|M f ∈ τ and M f b A}) e c (Int0) (int(A)) fc | M f ∈ τ and M f b A} e = e{M e |N e c ∈ τ and N e c b A} e = e{N e |N e c ∈ τ and A ec b N e} = e{N ec ); = cl(A e e B) e = d{M f|M f ∈ τ and M f b (A e e B)} e (Int4) int(A f|M f ∈ τ and M fbA e and M f b B} e = d{M f|M f ∈ τ and M f b A}) e e (d{M f|M f ∈ τ and M f b B}) e = (d{M e e int(B). e = int(A) The following Theorem 3.6 shows that under some conditions, a single valued neutrosophic operator is the single valued neutrosophic interior operator (the single valued neutrosophic closure operator) of a certain topology.

Theorem 3.6. (1) If a single valued neutrosophic operator int : SVNS(U ) −→ SVNS(U ) satisfies the properties (Int1)–(Int4), then there exists a single valued neutrosophic topology τ int on U such that intτint = int.

(2) If a single valued neutrosophic operator cl: SVNS(U ) −→ SVNS(U ) satisfies the properties (Cl1)–(Cl4), then there exists a single valued neutrosophic topology τ cl on U such that clτcl = cl. e ∈ SVNS(U ) | int(A) e = A}. e Next, we show τ int satisfies Proof. (1) Define τ int = {A (T1 )–(T3 ). e ∈ τ int .

(T1 ) By (Int1), e ∅, U eB e ∈ τ int , int(A) e =A e and int(B) e = B. e By (Int4), we have int(Ae e B) e = (T2 ) For any A, e e int(B) e =A e e B. e So A eeB e ∈ τ int . int(A) ei ∈ τ int , then int(A ei ) = A ei for any i ∈ I. By (Int2), we have (T3 ) Suppose that A ei . ei ) b di∈I A int(di∈I A ei ) b di∈I int(A ei ). By (Int3) and (Int5), we have int(di∈I int(A ei )) c Conversely, int(A ei )) = int(A ei ) for any i ∈ I. Thus int(di∈I int(A ei )) c di∈I int(A ei ). Since int(int(A ei . ei ) c di∈I A ei ) = A fi , we have int(di∈I A int(A ei . So di∈I (A ei ) ∈ τ int . ei )) = di∈I A Thus int(di∈I (A 9

Hence τ int is a single valued neutrosophic topology on U . It is obvious that intτint = int. e ∈ SVNS(U ) | cl(A ec ) = A ec }. Next, we show τ cl satisfies (T1 )–(T3 ).

(2) Define τ cl ={A e and U ec = e e) = (T1 ) By Definition 2.5, we have e ∅c = U ∅. Then by (Cl1), cl(e ∅c ) = cl(U e =e e c ) = cl(e e c , which means that e e ∈ τ cl .

U ∅c and cl(U ∅) = e ∅=U ∅, U eB e ∈ τ cl , cl(A ec ) = A ec and cl(B ec) = B e c . By Proposition 2.7 and (Cl4), (T2 ) For any A, e B) e c ) = cl(A ec d B e c ) = cl(A ec )dcl(B ec) = A ec d B e c = (Ae e B) e c . So Ae e B e ∈ τ cl . we have cl((Ae ei ∈ τ cl , then cl(A ei c ) = A ei c for any i ∈ I. By (Cl2), we have (T3 ) Suppose that A ei )c b cl((di∈I A ei )c ). (di∈I A ei )c ) = cl(ei∈I A ei c ) b cl(A ei c ) for any i ∈ I. Conversely, by (Cl5), we have cl((di∈I A ei )c ) b ei∈I cl(A ei c ) = ei∈I A ei c = (di∈I A ei )c .

Thus cl((di∈I A ei )c ) = (di∈I A ei )c . So di∈I A ei ∈ τ cl .

Thus cl((di∈I A Hence τ cl is a single valued neutrosophic topology on U . It is obvious that clτcl = cl.

Theorem 3.7. (1) Let int : SVNS(U ) −→ SVNS(U ) be a single valued neutrosophic operator satisfying the properties (Int1)–(Int4). Define e |A e ∈ SVNS(U )}, τ int = {int(A) ′

′

then τ int =τ int .

(2) Let cl : SVNS(U ) −→ SVNS(U ) be a single valued neutrosophic operator satisfying the properties (Cl1)–(Cl4). Define ′ e c|A e ∈ SVNS(U )}, τ cl = {(cl(A)) ′

then τ cl =τ cl . e ∈ SVNS(U ) | int(A) e = A} e b τ ′int . Conversely, for any Proof. (1) Obviously, τ int = {A e ∈ SVNS(U ), by (Int3), int(int(A)) e = int(A), e from which we have int(A) e ∈ τ int . Hence A ′

′

τ int b τ int . So τ int = τ int . e ∈ SVNS(U ), (cl(A)) e c ∈ τ ′ , we have cl(((cl(A)) e c )c ) = cl(cl(A)) e = (2) For any A cl e = ((cl(A)) e c )c , which means that (cl(A)) e c ∈ τ cl . Then τ b τcl . Conversely, for any cl(A) cl ′

e ∈ τ cl , A e = (cl(A ec ))c . Since A ec ∈ SVNS(U ), we have A e = (cl(A ec ))c ∈ τ ′ , which means A cl ′

′

that τ cl b τ cl . So τ cl = τ cl .

Theorem 3.8. Let int : SVNS(U ) −→ SVNS(U ) be a single valued neutrosophic operator satisfying the properties (Int0)–(Int4) and cl : SVNS(U ) −→ SVNS(U ) be a single valued neutrosophic operator satisfying the properties (Cl0)–(Cl4). Then the following result

10

holds: ′

′

τ int = τ int = τ cl = τ cl . ′

′

Proof. By Theorem 3.7, we have τ int = τ int and τ cl = τ cl . Thus, we only need to prove ′

′

that τ int = τ cl .

By (Int0) and (Cl0), we have e |A e ∈ SVNS(U )} τ int = {int(A) ′

ec ))c | A e ∈ SVNS(U )} = {(cl(A e c|A ec ∈ SVNS(U )} = {(cl(A)) e c|A e ∈ SVNS(U )} = {(cl(A)) ′

= τ cl .

4

Relationships between single valued neutrosophic approximation spaces and single valued neutrosophic topological spaces In this section, we will discuss the relationships between single valued neutrosophic

approximation spaces and single valued neutrosophic topological spaces.

4.1. F rom single valued neutrosophic approximation spaces to single valued neutrosophic topological spaces e be a single valued neutrosophic approximation space. Define Let (U, R) e ∈ SVNS(U ) | R( e A) e = A} e τ Re = {A

(1)

e in U can The following Theorem 4.1 shows that, by Equation (1), a reflexive SVNR R induce a single valued neutrosophic topology τ Re on U . e is a reflexive SVNR in U , then τ e is a single valued neutrosophic Theorem 4.1. If R R topology on U .

Proof. We verify τ Re satisfies (T1 )–(T3 ). e is a reflexive relation, R e is serial. By Propositions 2.11 and 2.12, we have (T1 ) Since R e U e ) = (U e ). Hence e e ∈ τ e. e e ∅) = (e ∅) and R( ∅, U R( R

eB e ∈ τ e , we have R( e A) e =A e and (T2 ) According to the definition of τ Re , for any A, R e B) e = B. e By Proposition 2.11, we have R( e A e e B) e = R( e A) e e R( e B) e = A e e B, e which R( eeB e ∈ τ e. implies that A R

11

e Table 1: SVNR R e R x1 x2 x3

x1 (0,0,1) (0,0.2,0.4) (1,0,1)

x2 (0.3,0.1,0.6) (0.6,0.5,1) (1,0.5,1)

x3 (1,0,0.4) (0.6,0,1) (1,0,0)

e A ei ∈ τ e , we have R( ei ) = A ei for any i ∈ I. By Proposition 2.12 and (T3 ) For any A R e we have R(d e i∈I A ei ) b di∈I A ei . Conversely, by Proposition 2.11, A ei = the reflexivity of R, eA fi ) b R(d e i∈I A ei ), which implies that di∈I A ei b R(d e i∈I A ei ). Thus di∈I A ei = R(d e i∈I A ei ) R( ei ∈ τ e . and di∈I A R So τ Re is a single valued neutrosophic topology on U . e is not reflexive, then τ e defined by Equations (1) may not Remark 4.2. If the SVNR R R be a single valued neutrosophic topology, as shown in the following example. e be a single valued neutrosophic approximation space, where Example 4.3. Let (U, R) e ∈ SVNR(U × U ) is given in Table 1.

U = {x1 , x2 , x3 } and R e is not reflexive. According to Definition 2.10, we have Obviously, R ∧ (y)) = 0.4, TR( (x ) = e (x1 , y) ∨ Te 1 e e y∈U (FR ∅ ∅) ∨ IR( (x ) = y∈U ((1 − IRe (x1 , y)) ∧ Ie∅ (y)) = 1, e e ∅) 1 ∨ FR( (x ) = y∈U (TRe (x1 , y) ∧ Fe∅ (y)) = 1, e e ∅) 1 ∨ TR( e (y)) = 1, e (x1 , y) ∧ TU e U e ) (x1 ) = y∈U (TR ∧ IR( e (x1 , y) ∨ IU e (y)) = 0, e U e ) (x1 ) = y∈U (IR ∧ FR( e (y)) = 0.4. e (x1 , y) ∨ FU e U e ) (x1 ) = y∈U (FR ee ee Hence R( ∅)(x1 ) = (0.4, 1, 1). Similarly, we can obtain R( ∅)(x2 ) = (0.4, 1, 0.6) and ee ee R( ∅)(x3 ) = (0, 1, 1). Thus R( ∅) = {⟨x1 , 0.4, 1, 1⟩, ⟨x2 , 0.4, 1, 0.6⟩, ⟨x3 , 0, 1, 1⟩} ̸= e ∅, which means that e ∅∈ / τRe . So τ Re do not form a single valued neutrosophic topology. e is a reflexive and transitive SVNR in U , for any A e ∈ SVNS(U ), we Lemma 4.4. If R have e R( e A)) e = R( e A), e R( e R( e A)) e = R( e A). e R( Proof. It can be easily verified by Proposition 2.12.

12

Theorem 4.5. Then the following conclusions hold: e is a reflexive and transitive SVNR in U , then τ e = {R( e A) e |A e ∈ SVNS(U )}; (1) If R R eA ec ) = A ec }. e ∈ SVNS(U ) | R( (2) τ Re = {A e A) e |A e ∈ SVNS(U )}. By Lemma 4.4, for any A e∈ Proof. (1) It is obvious that τ Re b {R( e A) e R( e A)) e = R( e A), e which implies that R( e A) e ∈ τ e and {R( e |A e∈ SVNS(U ), we have R( R e A) e |A e ∈ SVNS(U )}.

SVNS(U )} b τRe . So τ Re = {R( e and R. e (2) It follows immediately from the duality of R The above Theorem 4.5 (1) states a reflexive and transitive single valued neutrosophic relation can generate a single valued neutrosophic topology and this topology is just the family of all single valued neutrosophic lower approximations induced by the given single valued neutrosophic relation. ei ∈ SVNS(U ) for any i ∈ I. If R e is a Proposition 4.6. Let I be an index set, and A e A ei )) = di∈I R( e A ei ). e i∈I R( reflexive and transitive SVNR in U , then R(d e we have R(d e i∈I R( e A ei )) b di∈I R( e A ei ).

Proof. By the reflexivity of R, e A ei ) c R( e A ei ), R(d e i∈I R( e A ei )) c R( e R( e A ei )). By Lemma On the other hand, since di∈I R( e i∈I R( e A ei )) c R( e A ei ), which means that R(d e i∈I R( e A ei )) c di∈I R( e A ei ).

4.4, we have R(d e i∈I R( e A ei )) = di∈I R( e A ei ).

Thus R(d The following Theorem 4.7 shows that the single valued neutrosophic lower and upper approximation operators are the interior and closure operators of a single valued topological space induced by a reflexive and transitive SVNR, respectively.

Theorem 4.7. Let (U, τ Re ) be a single valued neutrosophic topological space induced by e i.e., τ e = {R( e A) e |A e ∈ SVNS(U )}, then a reflexive and transitive SVNR w.r.t. (U, R), R e ∈ SVNS(U ), ∀A e A) e = intτ (A) (1) R( e R e B) e | R( e B) e b A, eB e ∈ SVNS(U )}, = d{R( e A) e = clτ (A) e (2) R( e R e B)) e c | (R( e B)) e c c A, eB e ∈ SVNS(U )} = e{(R( eB e c ) | R( eB e c ) c A, eB e ∈ SVNS(U )}. = e{R( e b A. e So R( e A) e b e and Proposition 2.12, we have R( e A) Proof. (1) By reflexivity of R

13

e B) e | R( e B) e b A, eB e ∈ SVNS(U )}. On the other hand, we have d{R( e B) e | R( e B) e b d{R( eB e ∈ SVNS(U )} b A, e from which it follows that R(d{ e e B) e | R( e B) e b A, eB e ∈ A, R( e A). e e B) e | R( e B) e b A, eB e ∈ SVNS(U )} = SVNS(U )}) b R(

By Proposition 4.6, d{R( e e B) e | R( e B)) e b R( e A). e Hence we can conclude that R( e A) e = d{R( e B) e | R( e B) e b R(d{ R( eB e ∈ SVNS(U )}.

A, e and R. e (2) It follows immediately from the duality of R e ∈ SVNR(U × U ), then ∀x, y ∈ U , we have Proposition 4.8. Let R e(1 ) (x) = (1, 0, 0) and R e(1y ) (x) = R(x, e y).

R y Proof. It can easily be verified by Definition 2.10. e be a single valued neutrosophic approximation space and Theorem 4.9. Let (U, R) e where R e is a (U, τ Re ) be a single valued neutrosophic topological space induced by (U, R), reflexive and transitive SVNR in U , then e = {⟨(x, y), T e (x, y), I e (x, y), F e (x, y)⟩ | (x, y) ∈ U × U }, R R R R ∨ ∧ I e (x), FRe (x, y) = T (x), I (x, y) = where ∀x, y ∈ U , TRe (x, y) = B∈(y) e e e e B R B∈(y) τe B τe R R ∨ e ∈SVNS(U ) | B e c ∈ τ e , T e (y) = 1, I e (y) = 0, F e (y) = 0}.

FBe (x) and (y)τRe = {B e R B B B B∈(y) τ e R

e y ) = clτ (1y ). Moreover, by PropoProof. For any x, y ∈ U , by Theorem 4.7, we have R(1 e R sition 4.8, we have TRe (x, y) = TR(1 (x) = Tclτ e (1y ) (x), IRe (x, y) = IR(1 (x) = Iclτ e (1y ) (x) e e ) ) y

R

y

R

and FRe (x, y) = FR(1 (x) = Fclτ e (1y ) (x). e ) y

R

e ∈SVNS(U ) | B e c ∈ τ e , 1y b B}. e Then for any x, y ∈ U , Notice that clτRe (1y ) = e{B R we have TRe (x, y) = Tclτ e (1y ) (x) R

= Te{B∈SV e e c ∈τ e ,1y bB} e (x) N S(U )|B R ∧ e c ∈ τ e , T1y (t) ≤ T e (t) for any t ∈ U } = {TBe (x) | B R B ∧ c e = {TBe (x) | B ∈ τ Re , TBe (y) = 1} ∧ = B∈(y) TBe (x), e τ e R

IRe (x, y) = Iclτ e (1y ) (x) R

= Ie{B∈SV e e c ∈τ e ,1y bB} e (x) N S(U )|B R ∨ e c ∈ τ e , I1y (t) ≥ I e (t) for any t ∈ U } = {IBe (x) | B R B ∨ c e = {IBe (x) | B ∈ τ Re , IBe (y) = 0} ∨ = B∈(y) IBe (x), e τ e R

14

FRe (x, y) = Fclτ e (1y ) (x) R

= Fe{B∈SV e e c ∈τ e ,1y bB} e (x) N S(U )|B R ∨ e c ∈ τ e , F1y (t) ≥ F e (t) for any t ∈ U } = {FBe (x) | B R B ∨ e c ∈ τ e , F e (y) = 0} = {FBe (x) | B R B ∨ = B∈(y) FBe (x). e τ e R

This completes the proof.

Theorem 4.9 above shows that a reflexive and transitive SVNR can be represented by its induced single valued neutrosophic topology.

4.2. F rom single valued neutrosophic topological spaces to single valued neutrosophic approximation spaces In subsection 4.1, we obtain a reflexive single valued neutrosophic relation derived from a single valued neutrosophic approximation space can generate a single valued neutrosophic topology. Furthermore, a reflexive and transitive single valued neutrosophic relation can induce a single valued neutrosophic topological space such that its single valued neutrosophic interior and closure operators are the lower and upper approximation operators induced by this single valued neutrosophic relation, respectively. In this subsection, we discuss the reverse problem—if a single valued neutrosophic topological space can induce a single valued neutrosophic approximation space under some specifical conditions?

Theorem 4.10. Let (U, τ ) be a single valued neutrosophic topological space and int, cl : SVNS(U ) −→ SVNS(U ) be its single valued neutrosophic interior and closure operafτ in U such tors, respectively. Then there exists a reflexive and transitive SVNR R e = int(A) e and R fτ (A) e = cl(A) e for all A e ∈ SVNS(U ) iff int satisfies the fτ (A) that R eB e ∈ axioms (I1) and (I2), or equivalently, cl satisfies the axioms (C1) and (C2): ∀A, SVNS(U ), ∀α1 , α2 , α3 ∈ [0, 1], e d α1\ e d α1\ (I1) int(A , α2 , α3 ) = int(A) , α2 , α3 , e e B) e = int(A) e e int(B), e (I2) int(A e e α1\ e e α1\ (C1) cl(A , α2 , α3 ) = cl(A) , α2 , α3 , e d B) e = cl(A) e d cl(B). e (C2) cl(A fτ in U such Proof. “ =⇒ ” Suppose that there exists a reflexive and transitive SVNR R fτ (A) e = int(A) e and R fτ (A) e = cl(A) e for all A e ∈ SVNS(U ), then it can be easily that R 15

observed that the axioms (I1), (I2), (C1) and (C2) hold. “ ⇐= ” Assume that the operator cl satisfies the axioms (C1) and (C2). By cl, we fτ = {⟨(x, y), T f (x, y), I f (x, y), F f (x, y)⟩ | define a single valued neutrosophic relation R Rτ Rτ Rτ (x, y) ∈ U × U } in U × U as follows: TRfτ (x, y) = Tcl(1y ) (x), IRfτ (x, y) = Icl(1y ) (x), FRfτ (x, y) = Fcl(1y ) (x)

(2)

e ∈ SVNS(U ), Moreover, we can prove that for any A e = dy∈U (1y e T e(y), I\ A e(y), FA e(y)).

A A In fact, for any x ∈ U , by Definition 2.6, we have ∨ (x) (x) = Td (1y eT (y),I\ y∈U T1y eT e (y),I\ y∈U e (y)) e (y),FA e (y),FA e (y) e A A A A ∨ (x)) = y∈U (T1y (x) ∧ TT (y),I\ e (y) e e (y),FA A A ∨ = (1 ∧ TAe(x)) y∈U,y̸=x (0 ∧ TAe(y)) = TAe(x). ∧ (x) Id (1y eT (y),I\ (x) = y∈U I1y eT (y),I\ (y),F (y)) y∈U e e (y),FA e (y)) e e e A A A A A ∧ (x)) = y∈U (I1y (x) ∨ IT (y),I\ e e (y),FA e (y) ∧ A A = (0 ∨ IAe(x)) y∈U,y̸=x (1 ∨ IAe(y)) = IAe(x). ∧ (x) Fd (1y eT (y),I\ (x) = y∈U F1y eT (y),I\ (y),F (y)) y∈U e e (y),FA e (y)) e e e A A A A A ∧ (x)) = y∈U (F1y (x) ∨ FT (y),I\ e e (y),FA e (y) A A ∧ = (0 ∨ FAe(x)) y∈U,y̸=x (1 ∨ FAe(y)) = FAe(x). e = dy∈U (1y e T e(y), I\ e So A e(y), FA e(y)) for any A ∈ SVNS(U ).

A A e =R fτ (A) e and int(A) e =R fτ (A). e Next, we prove cl(A) For any x ∈ U , by definition 2.10 and the axioms (I1), (I2), (C1) and (C2), we have ∨ TRf (A) (x) = e(y)) fτ (x, y) ∧ TA y∈U (TR e τ ∨ = y∈U (Tcl(1y ) (x) ∧ TT (y),I\ (x)) e e (y),FA e (y) A A ∨ = y∈U (Tcl(1y )eT (y),I\ (x)) e e (y),FA e (y) A A ∨ = y∈U (Tcl(1y eT (y),I\ (x)) (y),F (y)) e A

= Td

e A

e A

\

y∈U cl(1y eTA e (y),IA e (y),FA e (y))

= Tcl(d

(x)

\

y∈U (1y eTA e (y),IA e (y),FA e (y)))

(x)

= Tcl(A) e (x). ∧ IRf (A) e(y)) fτ (x, y) ∨ IA y∈U (IR e (x) = τ ∧ = y∈U (Icl(1y ) (x) ∨ IT (y),I\ (y),F e A

e A

e (y) A

16

(x))

= =

∧

(x)) y∈U (Icl(1y )eT e (y),I\ e (y),F e (y)

∧

A

A

A

(x)) y∈U (Icl(1y eT e (y),I\ e (y),F e (y)) A

= Id

A

A

\

y∈U cl(1y eTA e (y),IA e (y),FA e (y))

= Icl(d

(x)

\

y∈U (1y eTA e (y),IA e (y),FA e (y)))

(x)

= Icl(A) e (x). ∧ FRf (A) e(y)) fτ (x, y) ∨ FA y∈U (FR e (x) = τ ∧ (x)) = y∈U (Fcl(1y ) (x) ∨ FT (y),I\ e (y) e (y),FA e A A ∧ (x)) = y∈U (Fcl(1y )eT (y),I\ e (y) e (y),FA e A A ∧ (x)) = y∈U (Fcl(1y eT (y),I\ (y),F (y)) e A

= Fd

e A

\

e A

y∈U cl(1y eTA e (y),IA e (y),FA e (y))

= Fcl(d

\

(x)

y∈U (1y eTA e (y))) e (y),IA e (y),FA

(x)

= Fcl(A) e (x). e =R fτ (A). e Note that cl and int are dual to each other, we have int(A) e = Thus cl(A) fτ (A). e By Theorem 3.5, we have R fτ (A) e b A. e Then, by Proposition 2.12, R fτ is reflexR fτ (R fτ (A)) e =R fτ (A), e which means that ive. Moreover, by Theorem 3.5 again, we have R fτ (A) e bR fτ (R fτ (A)). e fτ is transitive. Hence R fτ is a reflexive and R

By Proposition 2.12, R transitive SVNR. The above Theorem 4.10 gives the sufficient and necessary conditions that a single valued neutrosophic interior (closure, respectively) operator derived from a single valued neutrosophic topological space is just the single valued neutrosophic lower (upper, respectively) approximation operator induced by a reflexive and transitive single valued neutrosophic relation. Based on Theorem 4.10, we give the following definition.

Definition 4.11. Let (U, τ ) be a single valued neutrosophic topological space and int, cl: SVNS(U ) −→ SVNS(U ) be the single valued neutrosophic interior and closure operators of τ , respectively. If int satisfies the axioms (I1) and (I2) (or equivalently, cl satisfies the axioms (C1) and (C2)), then we call (U, τ ) a single valued neutrosophic rough topological space and τ a single valued neutrosophic rough topology. e be the set of all reflexive and transitive SVNRs in U and T be the set of all Let R single valued neutrosophic rough topologies. We can obtain the following results. e ∈ R, e τ e is defined by Equation (1) and R eτ is defined by Theorem 4.12. (1) If R e R R eτ = R. e Equation (2), then R e R 17

eτ is defined by Equation (2) and τ e is defined by Equation (1), then (2) If τ ∈ T , R Rτ τ Reτ = τ . e we have R e = intτ Proof. (1) By Theorem 4.7, and the reflexivity and transitivity of R, e R e = clτ . According to Proposition 4.8, for any x, y ∈ U , we have and R e R TReτ (x, y) = Tclτ e (1y ) (x) = TR(1 (x) = TRe (x, y), e ) e R

y

R

IReτ (x, y) = Iclτ e (1y ) (x) = IR(1 (x) = IRe (x, y), e ) e R

R

y

FReτ (x, y) = Fclτ e (1y ) (x) = FR(1 (x) = FRe (x, y). e ) e R

e eτ = R. Thus R e R

R

y

(2) By Equation (1) and Theorem 4.10, we have e ∈ SVNS(U ) | R fτ (A) e = A} e = {A e ∈ SVNS(U ) | int(A) e = A} e = τ. τReτ = {A e and T .

Theorem 4.13. There exists a one-to-one correspondence between R e −→ T as ∀R e ∈ R, f (R) e = τ e.

Proof. Define a mapping f : R R e as ∀τ ∈ T , g(τ ) =R fτ . Then, On the other hand, define a mapping g : T −→ R

by Theorem 4.12, it is easy to verify that both f and g are one-to-one correspondences between R and T .

Theorem 4.13 shows that there exists a one-to-one correspondence between the set of all reflexive and transitive single valued neutrosophic relations and the set of all single valued neutrosophic rough topologies such that the single valued neutrosophic lower and upper approximation operators induced by the reflexive and transitive single valued neutrosophic relations are the single valued neutrosophic interior and closure operators of single valued neutrosophic rough topologies, respectively.

5

Conclusion In this paper, we study the topological structures of single valued neutrosophic rough

sets. Firstly, we prove that a reflexive and transition single valued neutrosophic relation can induce a single valued neutrosophic topological space such that its single valued neutrosophic interior and closure operators are the lower and upper approximation operators induced by this single valued neutrosophic relation, respectively. Then, we investigate the sufficient and necessary conditions that a single valued neutrosophic interior (closure, respectively) operator derived from a single valued neutrosophic topological space is just

18

the single valued neutrosophic lower (upper, respectively) approximation operator derived from a single valued neutrosophic approximation space. Finally, we show there exists a one-to-one correspondence between the set of all reflexive and transitive single valued neutrosophic relations and the set of all single valued neutrosophic rough topologies.

Acknowledgements This work is partially supported by the National Natural Science Foundation of China (No. 61473181) and the Fundamental Research Funds For the Central Universities (GK201702008).

Compliance with ethical standards Conflict of interest Authors declares that they have no conflict of interest.

Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.

References [1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1) (1986) 87–96.

[2] K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (1989) 343–349.

[3] Z. Bonikowski, E. Bryniarski, U.W. Skardowska, Extensions and intentions in the rough set theory, Information Sciences 107 (1998) 149–167.

[4] D. Boixader, J. Jacas, J. Recasens, Upper and lower approximation of fuzzy sets, International Journal of General Systems 29 (2000) 555–568.

[5] E. Bryniaski, A calculus of rough sets of the first order, Bulletin of the Polish Academy of Sciences Mathematics 36(16) (1989) 71–77.

[6] S. Broumi, F. Smarandace, Single valued neutrosophic information systems based on rough set theory, http://www.researchgate.net/publication/280742564.

[7] M. Chuchro, A certain conception of rough sets in topological Boolean algebras, Bulletin of the Section of Logic 22 (1993) 9–12.

[8] M. Chuchro, On rough sets in topological Boolean algebras, In: W.Ziarko(ed), Rough sets, fuzzy sets and knowledge discovery. Spring, Berlin, 1994, pp. 157–160.

[9] D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems 17 (1990) 191–209.

[10] J. Kelley, General Topology, Van Nostrand Company, 1955.

19

[11] T.J. Li, W.X. Zhang, Rough fuzzy approximation on two universes of discourse, Information Sciences 178 (2008) 892–906.

[12] Z. Li, R. Cui, Similarity of fuzzy relations based on fuzzy topologies induced by fuzzy rough approximation operators, Information Sciences 305 (2015) 219–233.

[13] P. Majudar, S.K. Samant, On similarity and entropy of neutrosophic sets, Journal of Intelligent and Fuzzy Systems 26 (2014) 1245–1252.

[14] Z.M. Ma, B.Q. Hu, Topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets, Information Sciences 218 (2013) 194–204.

[15] Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (1982) 341–356.

[16] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, Boston, 1991.

[17] K.Y. Qin, Z. Pei, On the topological properties of fuzzy rough sets, Fuzzy Sets and Systems 151 (2005) 601–613.

[18] K.Y. Qin, J.L. Yang, Z. Pei, Generalized rough sets based on reflexive and transitive relation, Information Sciences 178 (2008) 4138–4141.

[19] A.M. Radzikowska, E.E. Kerre, A comparative study of fuzzy rough sets, Fuzzy Sets and Systems 126 (2001) 137–155.

[20] R. Slowinski, D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE Transaction on Knowledge and Data Engineering 12 (2000) 331–336.

[21] F. Smarandache, Neutrosophy. Neutrosophic Probability, Set, and Logic, American Research Press, Rehoboth, USA, 1998.

[22] F. Smarandache, A unifying field in logics. Neutrosophy: neutrosophic probability, set and logic, American Research Press, Rehoboth, USA, 1999.

[23] H. Thiele, On aximatic characterisation of fuzzy approximation operators: The rough fuzzy sets case, in: Proceeding of the 31st IEEE Internation Symposium on Multiple-Valued Logic, 2001, pp. 330–335.

[24] I. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20 (1986) 191–210.

[25] H. Wang, F. Smarandache, Y.Q. Zhang, R. Sunderraman, Single valued neutrosophic sets, Multispace and Multistruct 4 (2010) 410–413.

[26] R. Wiweger, On topological rough sets, Bulletin of Polish Academy of Science: Mathematic 37 (1989) 89–93.

[27] W.Z. Wu, L. Zhou, On intuitionistic fuzzy topologies based on intuitionistic fuzzy reflexive and transitive relation, Soft Computing 15 (2011) 1183–1194.

[28] H.L. Yang, C.L. Zhang, Z.L. Guo, Y.L. Yiu, X.W. Liao, A hybrid model of single valued neutrosophic sets and rough sets: single valued neutrosophic rough set model, Soft Computing, 2016, DOI:10.1007/s00500-016-2356-y.

20

[29] H.L. Yang, Z.L. Guo, Y.H. She, X.W. Liao, On single valued neutrosophic relations, Journal of Intelligent and Fuzzy Systems 30 (2016) 1045–1056.

[30] T. Yang, Q. Li, Reduction about approximation spaces of covering generalized rough sets, International Journal of Approximation Reasoning 51(3) (2010) 335–345.

[31] Y.Y. Yao, Constructive and algebraic methods of the theory of rough sets, Information Sciences 109 (1998) 21–47.

[32] Y.Y. Yao, On generalizing pawlak approximation operators, LNAI, 1424 (1998) 298–307.

[33] Y.Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences 111 (1998) 239–259.

[34] J. Ye, Multicriteria decision-making method using the correlation coefficient under single valued neutrosophic enviroment, Intelligent Journal of General Systems 42 (2013) 386–394.

[35] J. Ye, Improved correlation coefficients of single valued neutrosophic sets and interval neutrosophic sets for multiple attributedecision making, Journal of Intelligent and Fuzzy Systems 27 (2014) 2453–2462.

[36] J. Ye, A multicriteria decision-making method using aggregation operators for simplied neutrosophic sets, Journal of Intelligent and Fuzzy Systems 26(5) (2014) 2459–2466.

[37] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353.

[38] H.D. Zhang, L. Shu, S.L. Liao, Topological structures of interval-valued hesitant fuzzy rough set and its application, Journal of Intelligent and Fuzzy Systems 30 (2016) 1029–1043.

21