Topologies and approximation operators induced by binary relations

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J. Math. Comput. Sci. 7 (2017), No. 4, 642-657 ISSN: 1927-5307

TOPOLOGIES AND APPROXIMATION OPERATORS INDUCED BY BINARY RELATIONS

NURETTIN BA ˘GIRMAZ1,∗, A. FATIH ¨OZCAN2, HATICE TAS¸BOZAN3AND ˙ILHAN ˙IC¸ EN2

1_{Mardin Artuklu University, Mardin, Turkey}

2_{Department of Mathematics, Inonu University, Malatya, Turkey} 3_{Department of Mathematics, Mustafa Kemal University, Hatay, Turkey}

Copyright c2017 Ba˘girmaz, ¨Ozcan, Tas¸bozan and ˙Ic¸en. This is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. Theory of rough sets is an extension of classic set theory, which is an important mathematical tool for dealing with uncertain or vague information. The core concepts of classical rough sets are lower and upper ap-proximations based on equivalence relations. Topologies via relations have very important role in rough set theory. This paper studies some new topologies induced by a binary relation on universe with respect to neighborhood operators. Moreover, the relations among them are studied. In addition, lower and upper approximations of rough sets using the binary relation with respect to neighborhood operators are studied and examples are given.

Keywords:topological spaces; rough sets; binary relations; lower and upper approximations. 2010 AMS Subject Classification:54A99, 68P01.

1. Introduction

∗_{Corresponding author}

E-mail address: [email protected] Received March 7, 2017

The theory of rough sets was introduced by Pawlak as a mathematical tool to process informa-tion with uncertainty and vagueness [11].The theory of rough sets deals with the approximainforma-tion of sets for classification of objects through equivalence relations. Important applications of the rough set theory have been applied in many fields, for example in medical science, data analysis, knowledge discovery in database [12, 13, 15, 20]

In the original theory of rough sets, an ordered pair(U,R)is called an approximation space, whereU is a finite non- empty set called universe and R is an equivalent relation onU. But for practical use, it needs to some extensions on original rough set concept. This is to replace the equivalent relation by a general binary relation [4, 5, 6, 10, 16, 23, 24, 25, 26, 27]. We call the ordered pair (U,R) as a generalized approximation space when R is a binary relation on

U.Topology is one of the most important subjects in mathematics, which to study rough sets.

Many authors studied relationship between rough sets and topologies based on binary relations [1, 2, 7, 8, 9, 14, 17, 18, 19, 21, 22]

In this paper, we proposed and studied connections between topologies generated using successor, predecessor, successor-and-predecessor, and successor-or-predecessor neighborhood operators as a subbase by various binary relations on a universe, respectively. LetRbe a binary relation on a given universeU. Sets

Si=S{Ri(x)|x∈U},wherei:s,p,s∧p and s∨p

defined by successor, predecessor, successor-and-predecessor, and successor-or-predecessor neighborhood operators, respectively. IfSi,where i:s, p,s∧p and s∨p,forms a subbase for

some topology onU, then the topology generated Si , denoted Ti. Z. Pei at al. in [14] used

Ssas a subbase for some topology onU if and only ifRis invers serial which for eachx∈U,

there existsy∈U such that (y,x)∈R.A basic problem is: when doesSi,where i:s, p, s∧p

2. Preliminaries

In this section, we shall briefly review basic concepts and relational propositions of the rela-tion based rough sets and topology. For more details, we refer to [24, 25, 14].

2.1. Basic properties relation based rough approximations and

neighbor-hood operators

In this paper, we always assume thatU is a finite universe, i.e., a non-empty finite set of objects,Ris a binary relation onU, i.e., a subset ofU2=U×U [14].

Ris serial if for eachx∈U, there existsy∈U such that(x,y)∈R;Ris inverse serial if for eachx∈U, there existsy∈U such that(y,x)∈R;Ris reflexive if for eachx∈U,(x,x)∈R;R

is symmetric if for allx,y∈U,(x,y)∈Rimplies(y,x)∈R;Ris transitive if for allx,y,z∈U,

(x,y)∈Rand(y,z)∈Rimply(x,z)∈R[9].

Ris called a pre-order relation ifRis both reflexive and transitive;Ris called a similarity (or, tolerance) relation ifRis both reflexive and symmetric;Ris called an equivalence relation ifR

is reflexive, symmetric and transitive [14].

Given a universeU and a binary relationRonU,x,y∈U,the sets

Rs(x) ={y∈U|(x,y)∈R},

Rp(x) ={y∈U|(y,x)∈R},

R_s∧p(x) ={y∈U|(x,y)∈R and (y,x)∈R}=R_s(x)∩R_p(x),

R_s∨p(x) ={y∈U|(x,y)∈R or (y,x)∈R}=R_s(x)∪R_p(x)

Rs:x7→Rs(x),

R_p:x7→R_p(x),

R_s∧p:x7→R_s∧p(x),

R_s∨p:x7→R_s∨p(x)

are called the successor, predecessor, successor-and-predecessor, and successor-or-predecessor neighborhood operators, respectively. Relationships between these neighborhood systems can be expressed as:

R_s∧p(x)⊆R_s(x)⊆R_s∨p(x),

R_s∧p(x)⊆Rp(x)⊆Rs∨p(x)

[25, 14].

Definition 2.1 [25] Let Rbe a binary relation onU. The ordered pair (U,R)is called a gen-eralized approximation space based on the relation R. For X ⊆U, the lower and upper ap-proximations of X with respect to R_s(x),R_p(x),R_s∧p(x),R_s∨p(x) are respectively defined as follows:

apr

Rs(X) ={x∈U|Rs(x)⊆X},

apr

Rp(X) =

x∈U|R_p(x)⊆X ,

apr

Rs∧p(X) =

x∈U|R_s∧p(x)⊆X ,

apr

Rs∨p(X) =

x∈U|R_s∨p(x)⊆X ,

apr_Rs(X) ={x∈U|Rs(x)∩X 6= /0},

apr_Rp(X) =x∈U|R_p(x)∩X 6= /0 ,

apr_Rs

∧p(X) =

x∈U|R_s∧p(x)∩X 6= /0 ,

apr_Rs

∨p(X) =

In Pawlak’s classical rough set theory for lower and upper approximations operators, an equivalence relation R is used. In this case, four neighborhood operators become the same, i.e.,Rs(x) =Rp(x) =Rs∧p(x) =Rs∨p(x) = [x]R, where [x]Ris the equivalence class containing

x.

Proposition 2.2[24] For an arbitrary neighborhood operator in an approximation space(U,R), the pair of approximation operators satisfy the following properties:

(L0)apr(X) = (apr(Xc))c,

(U0)apr(X) = apr(Xc)c,

(L1)apr(U) =U,

(U1)apr(/0) =/0,

(L2)apr(X∩Y) =apr(X)∩apr(Y),

(U2)apr(X∪Y) =apr(X)∪apr(Y).

whereXcis the complement ofX with respect toU. Moreover, ifRis reflexive, then

(L3)apr(X)⊆X,

(U3)X ⊆apr(X).

IfRis symmetric, then

(L4)X ⊆apr(apr(X)),

(U4)apr(apr(X))⊆X.

IfRis transitive, then

(L5)apr(X)⊆apr(apr(X)),

2.2. The concept of a topological space

In this section, we give some basic information about the topology [3, 14].

Definition 2.3 [3] A topological space is a pair (U,T) consisting of a setU and a set T of subsets ofU (called ”open sets”), such that the following axioms hold:

(A1)Any union of open sets is open.

(A2)The intersection of any two open sets is open.

(A3) /0 andU are open.

The pair(U,T)speaks simply of a topological spaceU.

Definition 2.4[14] LetU be a topological space. (1) X ⊆U is called closed whenXcis open.

(2) X ⊆U is called a neighborhood ofx∈X if there is an open setV withx∈V ⊆X. (3) A pointxof a setX is an interior point ofX ifX is a neighborhood ofx, and the set of

all interior points ofX is called the interior ofX. The interior ofX is denoted by

o

X. (4) The closure of a subsetX of a topological spaceU is the intersection of the family of all

closed sets containingX. The closure ofX is denoted byX−. In topological spaceU, the operator

o_{:P(U)→P(U), X7→X}o

is an interior operator onU and for allX,Y ⊆U the following properties hold:

I1)

o

U =U,

I2)Xo ⊆X,

I3)

o _o X

=X,

I4)

_o

X∩Y = _o X ∩ _o Y .

In topological spaceU, the operator

−_{:P(U)→P(U), X7→X}−

C1) −

/0= /0, C2)X ⊆X

C3)X =X,

C4)X∪Y =X∪Y.

In a topological space (U,T) a familyB ⊆T of sets is called a base for the topology T if for each pointx of the space, and each neighborhoodX ofx, there is a memberV ofB such that x∈V ⊆X. We know that a subfamilyB of a topology T is a base for T if and only if each member ofT is the union of members ofB. Moreover,B⊆P(U)forms a base for some topologies onU if and only ifB satisfies the following conditions:

B1)U =S

{B|B∈B},

B2) For every two membersX andY ofBand each pointx∈X∩Y, there isZ∈Bsuch that

x∈Z⊆X∩Y.

Also, a family S ⊆T of sets is a subbase for the topology T if the family of all finite intersections of members of S is a base forT. Moreover,S ⊆P(U) is a subbase for some topology onU if and only ifS satisfies the following condition:

(S0)U =S

{S|S∈S}.

3. Generating topologies by relations

In this section, we show the subbase generated using successor, predecessor, successor-and-predecessor, and successor-or-predecessor neighborhood operators by various binary relations on a universe, respectively. Then, we introduce the topologies generated by these subbases and compare these topologies.

LetRbe a binary relation on a given universeU. Sets

Si=S{Ri(x)|x∈U},wherei:s,p,s∧p and s∨p

defined by successor, predecessor, successor-and-predecessor, and successor-or-predecessor neighborhood operators, respectively. IfSi,forms a subbase for some topology onU, then the

The following theorem states that a inverse serial is sufficient for theSsforms a subbase for

some topologies onU.

Theorem 3.1[14] IfRis a binary relation onU, thenSs forms a subbase for some topologies

onU if and only ifRis inverse serial.

Remark 3.2It is clear that ifRis inverse serial onU, then

∀x∈U,Rp(x)6=∅and U =

[

x∈U

Rs(x).

Thus the familySs is covering ofU.

Theorem 3.3IfRis a binary relation onU, thenSpforms a subbase for some topologies onU

if and only ifRis serial.

Proof. IfRis serial, then

∀x∈U,R_s(x)6=_∅and U = [

x∈U

R_p(x).

Thus the familyS_pprovides the condition (S0). ThereforeSpforms a subbase for some

topolo-gies onU.

Example 3.4LetU={a,b,c,d}and

R={(a,a),(a,c),(b,b),(b,c),(c,a),(d,c)}.

ThenRis a serial relation onU,and

R_s(a) ={a,c}, R_s(b) ={c}, R_s(c) ={a}, R_s(d) ={c},

Rp(a) ={a,c}, Rp(b) ={b}, Rp(c) ={b,d}, Rp(d) =∅.

Thus

Sp=

[

R_p(x)|x∈U ={{a,c},{b},{b,d}}.

Hence

Tp={∅,U,{b},{b,d},{a,c},{a,b,c}}.

Lemma 3.5LetU be the universe andRis a symmetric relation. Then

Proof. Suppose thatRis a symmetric relation onU.

R is a serial relation⇐⇒ ∀x∃y[(x,y)∈R]

⇐⇒ ∀x∃y[(y,x)∈R]

⇐⇒R is an inverse serial relation

Theorem 3.6IfRis a binary relation onU, thenSs∧pforms a subbase for some topologies on

U if and only ifRis symmetric and serial or inverse serial.

Proof. From Lemma 3.5 and Theorem 3.3 we get

Rs(x) =Rp(x)6=∅.

Thus

Ss∧p=

[

R_s∧p(x)|x∈U =[{R_s(x)|x∈U}=[R_p(x)|x∈U 6=_∅

, thereforeSs∧pforms a subbase for some topologies onU.

Theorem 3.7IfRis a binary relation onU, thenSs∨pforms a subbase for some topologies on

U if and only ifRis serial or inverse serial.

Proof. IfRis serial or inverse serial onU, then

∀x∈U,R_s∨p(x)6=_∅and U = [

x∈U

R_s∨p(x).

Thus the familySs∨p provides the condition (S0). ThereforeSs∨p forms a subbase for some

topologies onU.

LetS1andS2be covering ofU.A partitionS1is finner thanS2, or is coarser thanS1, for

each neighborhood operator inS1produced byx, is subset the neighborhood operator inS2by x. This relation is denoted asS1S2.

S1S2⇐⇒if every set ofS1is contained in some sets ofS2, for allx∈U.

Moreover,T₁ andT₂ are two topologies onU andT₁⊆T₂, then we say thatT₂is finer than

T₁.

In effect, the following proposition holds.

R_i(x)⊆R_j(x)⇔SiSj,f or all x∈U,where i,j:s,p,s∧pands∨p.

Proposition 3.9LetU be the universe andRis a serial relation. Then the following conditions are provided:

(1)SpSs∨p,

(2)T_pT_s∨p.

Proof. Clearly,(2)is a direct corollary of(1)We only prove(1).

SinceR_p(x)⊆R_s∨p(x)for allx∈U,then from Proposition 3.8 we getSpSs∨p.

Proposition 3.10LetU be the universe andRis an inverse serial relation. Then the following conditions are provided:

(1)SsSs∨p,

(2)TsTs∨p.

Proof. Clearly,(2)is a direct corollary of(1)We only prove(1).

SinceR_s(x)⊆R_s∨p(x)for allx∈U,then from Proposition 3.8 we getSsSs∨p.

The next proposition presents equality betweenSs∧p,Ss,SpandSs∨psubbases, and

topolo-gies generated by them that are equality each other.

Proposition 3.11 LetU be the universe and R is a symmetric and a serial (or inverse serial) relation. Then the following conditions are provided:

(1)Ss∧p=Ss=Sp=Ss∨p,

(2)T_s∧p=T_s=T_p=T_s∨p.

Proof. (1) If R is a symmetric and a serial (or inverse serial) relation on U, then R_s∧p(x) =

Rs(x) =Rp(x) =Rs∨p(x).TherforeSs∧p=Ss=Sp=Ss∨p.

(2)is a direct corolary of(1).

Corollary 3.12 LetU be the universe andR is a tolerance (symmetric and reflexive) relation. Then, the following conditions are provided:

(1)Ss∧p=Ss=Sp=Ss∨p,

Proposition 3.13 Let U be the universe and R is a reflexive relation. Then, the following conditions are provided:

(1)Ss∧pSs,SpSs∨p,

(2)T_s∧pT_s,T_pT_s∨p.

Proof. (1) Let R is a reflexive relation on U. Then, R are serial and invers serial relation. Since R_s∧p(x) =R_s(x)∩R_p(x) and R_s∨p(x) = R_s(x)∪R_p(x), then R_s∧p(x)⊆R_s(x),R_p(x) ⊆

Rs∨p(x).ThusSs∧pSs,SpSs∨p.

(2)is a direct corollary of(1).

Remark 3.14Notice that the reflexive relationRdoes not need to holdR_s(x) =R_p(x)for each

x∈U.This is evident by following example.

Example 3.15LetU={a,b,c,d}and

R={(a,a),(a,c),(b,b),(b,c),(c,c),(d,d)}.

ThenRis a reflexive relation onU,and

R_s(a) ={a,c}, R_s(b) ={b,c}, R_s(c) ={c}, R_s(d) ={d},

R_p(a) ={a},R_p(b) ={b},R_p(c) ={b,c},R_p(d) ={d},

R_p∧s(a) ={a}, R_p∧s(b) ={b}, R_p∧s(c) ={c}, R_p∧s(d) ={d}

R_p∨s(a) ={a,c}, R_p∨s(b) ={b,c}, R_p∨s(c) ={b,c}, R_p∨s(d) ={d}.

ThusR_s(a)6=R_p(a),fora∈U.Let us note thatR_p∧s(a)⊆R_s(a),R_p(a)⊆R_p∨s(a). Therfore

Ss∧pSs,SpSs∨pandTs∧pTs,TpTs∨p.

Corollary 3.16 LetU be the universe and R is a preorder (reflexive and transitive) relation. Then the following conditions are provided:

(1)S_s∧pS_s,S_pS_s∨p,

(2)T_s∧pT_s,T_pT_s∨p.

Remark 3.17 If R is a preorder relation on U , Ss∧p (Ss,Sp,Ss∨p) form a base for Ts∧p

Corollary 3.18 Let U be the universe and R is an equivalent relation. Then, the following conditions are provided:

(1)Ss∧p=Ss=Sp=Ss∨p,

(2)T_s∧p=T_s=T_p=T_s∨p

Remark 3.19In the case whenRis an equivalent relation onU, i.e,(U,R)is a Pawlak approx-imation space, then the setSs∧p(Ss,Sp,Ss∨p)is a base forTs∧p(Ts,Tp,Ts∨p)topology onU,

respectively. In these topologies, each neighborhood operator is one equivalence class for all

x∈U.

4.Rough approximation operators induced by relations

In this section, we investigate connection between lower and upper approximation operators using successor, predecessor, successor-and-predecessor, and successor-or-predecessor neigh-borhood operators by various binary relations on a universe, respectively.

Proposition 4.1LetU be the universe and R is a binary relation. Then, for lower and upper approximation operators ofX⊆U, the following conditions are provided:

1)apr

p∨s(X)⊆aprs(X),aprp(X)⊆aprp∧s(X),

2)apr_p∧s(X)⊆apr_s(X),apr_p(X)⊆apr_p∨s(X).

Proof. (1) Let x∈apr

p∨s(X) for anyx∈U. Since Rp∨s(x)⊆X and Rs(x)⊆Rp∨s(x) then

Rs(x)⊆X and sox∈apr_s(X). Now, since x∈apr_s(X) and Rp∧s(x)⊆Rs(x)⊆X then x∈

apr

p∧s(X).Thereforeaprp∨s(X)⊆aprs(X)⊆aprp∧s(X).Similarly,aprp∨s(X)⊆aprp(X)⊆

apr

p∧s(X).

(2)Let x∈apr_p∧s(X)for anyx∈U.SinceR_p∧s(x)∩X 6=/0 andR_p∧s(x)⊆R_s(x),R_p(x)⊆

Rp∨s(x)thenRp∨s(x)∩X6=/0 an sox∈aprp∨s(X).Therefore,aprp∧s(X)⊆aprs(X),aprp(X)⊆

apr_p∨s(X).

Example 4.2LetU={a,b,c,d}and

be a binary relation onU.Then,

R_s(a) ={a,c}, R_s(b) ={c}, R_s(c) ={a,d}, R_s(d) = /0,

R_p(a) ={a,c}, R_p(b) = /0, R_p(c) ={a,b}, R_p(d) ={c},

R_p∧s(a) ={a,c}, Rp∧s(b) = /0, Rp∧s(c) ={a}, Rp∧s(d) = /0,

Rp∨s(a) ={a,c}, Rp∨s(b) ={c}, Rp∨s(c) ={a,b,d}, Rp∨s(d) ={c}.

LetX ={a,c,d}.Then,

apr

s(X) ={a,b,c,d},

apr

p(X) ={a,b,d},

apr

p∧s(X) ={a,b,c,d},

apr

p∨s(X) ={a,b,d},

apr_s(X) ={a,b,c},

apr_p(X) ={a,c,d},

apr_p∧s(X) ={a,c},

apr_p∨s(X) ={a,b,c,d}.

Hence, note thatapr

s(X)⊃aprs(X).

In the original rough set theory, lower approximation ofXis a subset of its upper approxima-tion. In order to provide this condition, we need some properties to add binary relations.

The next propositions presents this conditions.

Proposition 4.3[4] LetU be the universe andRis a binary relation. Then, for allX⊆U

Ris serial ⇒apr

s(X)⊆aprs(X)

Corollary 4.4LetU be the universe andRis a binary relation. Then, for allX ⊆U

Ris serial ⇒apr

p∨s(X)⊆aprs(X)⊆aprs(X)⊆aprp∨s(X)

Proposition 4.5LetU be the universe andRis a binary relation. Then, for allX ⊆U

Ris inverse serial ⇒apr

p(X)⊆aprp(X).

Proof. Let x∈apr

p(X). ThenRp(x)⊆X, which givesRp(x)∩X =Rp(x)6= /0, that is, x∈

apr_p(X).

Corollary 4.6LetU be the universe andRis a binary relation. Then, for allX ⊆U

Ris inverse serial ⇒apr

p∨s(X)⊆aprp(X)⊆aprp(X)⊆aprp∨s(X).

Proof. Proof is clear from Proposition 4.1 and Proposition 4.5.

Proposition 4.7LetU be the universe andRis a binary relation. Then, for allX ⊆U

Ris symmetric and serial (or inverse serial)⇒apr

p∧s(X)⊆aprp∧s(X).

Proof. Let x∈apr

p∧s(X).Then, from Lemma 3.5Rp∧s(x) =Rp(x)which givesaprp∧s(X) =

apr

p(X). So, from Proposition 4.3x∈aprp∧s(X).

The next propositions present a set between its lower approximation and its upper approxi-mation conditions.

Proposition 4.8LetU be the universe andRis a binary relation. Then, for allX ⊆U

Ris reflexive⇒apr

i(X)⊆X ⊆apri(X), wherei: s, p, p∧sand p∨s, respectively.

Proof. Proof is clear from Proposition 2.2.

Corollary 4.9LetU be the universe andRis a reflexive or preorder binary relation. Then, for allX ⊆U

(1)apr

p∨s(X)⊆aprp(X)⊆aprp∧s(X)⊆X⊆aprp∧s(X)⊆aprp(X)⊆aprp∨s(X)

(2)apr

p∨s(X)⊆aprs(X)⊆aprp∧s(X)⊆X ⊆aprp∧s(X)⊆aprs(X)⊆aprp∨s(X) .

Proof. Proof is clear from Proposition 4.1 and Proposition 4.8.

apr

i(X) =aprj(X)⊆X⊆apri(X) =aprj(X)

, wherei,j:s,p,s∧p and s∨p.

Proof. IfRis a tolerance or equivalent binary relation, then for allx∈U,

R_s∧p(x) =R_s(x) =R_px) =R_s∨p(x)

.Thus, for allX ⊆U

apr

s(X) =aprp(X) =aprp∧s(X) =aprp∨s(X)

and

apr_s(X) =apr_p(X) =apr_p∧s(X) =apr_p∨s(X) . Therefore, from Proposition

apr

i(X) =aprj(X)⊆X ⊆apri(X) =aprj(X),

wherei,j:s,p,s∧p and s∨p.

Conflict of Interests

The authors declare that there is no conflict of interests.

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