Further research of single valued neutrosophic rough set model by Ioan Degău

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

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).

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]

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.

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

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

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.

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(

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.

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

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

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 .

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

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.

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

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

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

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

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

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

(x))

= =

∧

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

∧

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

= Id

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

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

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

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

e eτ = R. Thus R e R

(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.

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

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.

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