KUL HUR, SU YOUN JANG, AND HEE WON KANG
Received 14 February 2005 and in revised form 19 June 2005
We introduce the categoryIRel(H) consisting of intuitionistic fuzzy relational spaces on sets and we study structures of the categoryIRel(H) in the viewpoint of the topological universe introduced by Nel. Thus we show thatIRel(H) satisfies all the conditions of a topological universe overSet except the terminal separator property andIRel(H) is cartesian closed overSet.
1. Introduction
In 1965, Zadeh [30] introduced a concept of a fuzzy set as the generalization of a crisp set. Also, in 1971, he introduced a fuzzy relation naturally, as a generalization of a crisp relation in [31].
Nel [27] introduced the notion of a topological universe which implies concrete qu-asitopos [1]. Every topological universe satisfies all the properties of a topos except one condition on the subobject classifier. The notion of a topological universe has already been put to effective use in several areas of mathematics in [24,25,28]. In 1980, Cerruti [8] introduced the category ofL-fuzzy relations and investigated some of its properties. After that time, Hur [14] introduced the category Rel(H) of the fuzzy relational spaces with a complete Heyting algebraHas a codomain and he studied the categoryRel(H) in the sense of a topological universe.
In 1983, Atanassov [2] introduced the concept of an intuitionistic fuzzy set as the gen-eralization of fuzzy sets and he also investigated many properties of intuitionistic fuzzy sets (cf. [3]). After that time, Banerjee and Basnet [4], Biswas [6], and Hur and his col-leagues [15,16,17,20] applied the concept of intuitionistic fuzzy sets to algebra. Also, C¸oker [9], Hur and his colleagues [21], and S. J. Lee and E. P. Lee [26] applied one to topology. In particular, Hur and his colleagues [18] applied the notion of intuitionistic fuzzy sets to topological group.
In this paper, we introduce the categoryIRel(H) of intuitionisticH-fuzzy relational spaces and study the categoryIRel(H) in a topological universe viewpoint. In particular, we show thatIRel(H) satisfies all the conditions of a topological universe overSetexcept
Copyright©2005 Hindawi Publishing Corporation
the terminal separator property. AlsoIRel(H) is shown to be cartesian closed overSet. For general categorical background, we refer to Herrlich and Strecker [12].
2. Preliminaries
In this section, we will introduce some basic definitions and well-known results which are needed in the next sections.
LetXbe a set, let (Xi)i∈Ibe a family of sets indexed by a classI, and let fibe a mapping
with domainX for eachi∈I. Then a pair (X, (fi)I) (simply, (fi)I) is called asource of
mappings. Asink of mappingsis the dual notion of a source of mappings. Definition 2.1[12]. LetAbe a concrete category and letIbe a class.
(1) Asource inAis a pair (X, (fi)I) (simply, (X,fi) or (fi)I), whereXis anA-object and
(fi:X→Xi)Iis a family ofA-morphisms each with domainX. In this case,X is called
thedomain of the sourceand the family (Xi)Iis called thecodomain of the source.
(2) A source (X,fi) is called amonosourceprovided that the fican be simutaneously
canceled from the left; that is, provided that for any pair Y sr X of morphisms such that fi◦r= fi◦sfor eachi∈I, it follows thatr=s.
Dual notions:sink inAandepisink.
Definition 2.2[23]. LetAbe a concrete category and let ((Yi,ξi))Ibe a family of objects inAindexed by a classI. For any setX, let (fi:X→Yi)Ibe a source of mappings indexed
byI. AnA-structureξonXis said to beinitial with respect to(X, (fi), ((Yi,ξi))) provided
that the following conditions hold.
(1) For eachi∈I, fi: (X,ξ)→(Yi,ξi) is anA-morphism.
(2) If (Z,ρ) is an A-object andg:Z→X is mapping such that for eachi∈Z, the mapping fi◦g: (Z,ρ)→(Yi,ξi) is anA-morphism, theng: (Z,ρ)→(X,ξ) is an A-morphism. In this case, (fi: (X,ξ)→(Yi,ξi))Iis called aninitial source inA.
Dual notions:final structureandfinal sink.
Definition 2.3[23]. A concrete categoryAis said to betopological overSetprovided that for each setX, for any family ((Yi,ξi))I of A-objects, and for any source (fi:X→Yi)I
of mappings, there exists a uniqueA-structureξ onX which is initial with respect to (X, (fi), ((Yi,ξi))).
Dual notions:cotopological category.
Result2.4 [23, Theorem 1.5]. A concrete categoryAis topological if and only ifAis co-topological.
Result2.5 [23, Theorem 1.6]. LetAbe a topological category overSet. ThenAis complete and cocomplete.
Definition 2.6[11]. A categoryAis called cartesian closedprovided that the following conditions hold.
(1) For anyA-objectsAandB, there exists a productA×BinA.
and anyA-morphism f :A×C→B, there exists a uniqueA-morphism f :C→BA such that the diagram
A×BA eA,B
B
A×C
f
∃1A×f
(2.1)
commutes.
Definition 2.7[23]. LetAbe a concrete category.
(1) TheA-fiberof a setXis the class of allA-structures onX.
(2)Ais calledproperly fibered overSetprovided that the following conditions hold. (i)Fiber-smallness. For each setX, theA-fiber ofXis a set.
(ii)Terminal separator property. For each singleton setX, theA-fiber ofXhas pre-cisely one element.
(iii) Ifξ andηareA-structures on a setX such that 1X: (X,ξ)→(X,η) and 1X: (X,η)→(X,ξ) areA-morphisms, thenξ=η.
Definition 2.8[27]. A categoryAis called atopological universe overSetprovided that the following conditions hold.
(1)Ais well structured overSet, that is, (i)Ais a concrete category; (ii)Ahas the fiber-smallness condition; (iii)Ahas the terminal separator property.
(2)Ais cotopological overSet.
(3) Final episinks inAare preserved by pullbacks, that is, for any final episink (gλ:
X→Y)Λ and anyA-morphism f :W→Y, the family (eλ:Uλ→W)Λ, obtained
by taking the pullback of f andgλfor eachλ, is again a final episink.
Definition 2.9[29]. A categoryAis called atoposprovided that the following conditions hold.
(1) There is a terminal objectUinA, that is, for eachA-objectA, there exists one and only oneA-morphism fromAtoU.
(2)Ahas equalizers, that is, for anyA-objectsAandBandA-morphisms
A g
f
B (2.2)
there exist anA-objectCand anA-morphismh:C→Asuch that (a) f◦h=g◦h,
(b) for eachA-objectCandA-morphismh:C→Awith f ◦h=g◦h, there exists a uniqueA-morphismh:C→Csuch thath=h◦h, that is, the dia-gram
C h A gf B
C ∃h
h (2.3)
(3)Ais cartesian closed;
(4) there is a subobject classifier inA, that is, there is anA-objectΩandA-morphism v:U→Ωsuch that for eachA-monomorphismm:A→A, there exists a unique A-morphismφm:A→Ωsuch that the following diagram is a pullback:
A
m
U
v
A φm Ω
(2.4)
Remark 2.10. LetAbe any category with a subobject classifier. If f is any bimorphism in A, then f is an isomorphism inA(cf. [7]).
3. The category IRel(H)
First we will list some concepts and one result which are needed in this section and the next section. Next, we introduce the categoryIRel(H)of intuitionisticH-fuzzy relational spaces and show that it has similar structures as those ofISet(H).
Definition 3.1[5,22]. A latticeH is called acomplete Heyting algebraifH satisfies the following conditions:
(1)His a complete lattice;
(2) for anya,b∈H, the set{x∈H:x∧a≤b}has a greatest element denoted bya→ b(calledpseudocomplement ofaandb), that is,x∧a≤bif and only ifx≤(a→b). In particular, for eacha∈H,N(a)=a→0 is called thenegationor the pseudocomple-mentofa.
Result3.2 [5, Example 6, page 46]. LetHbe a complete Heyting algebra and leta,b∈H. Then
(1)ifa≤b, thenN(b)≤N(a), that is,N:H→His an involutive order-reversing oper-ation in(H,≤);
(2)a≤NN(a); (3)N(a)=NNN(a);
(4)N(a∨b)=N(a)∧N(b)andN(a∧b)=N(a)∧N(b).
Throughout this paper, we useHas a complete Heyting algebra.
Definition 3.3[19]. LetXbe a set. A triple (X,µ,ν) is called anintuitionisticH-fuzzy set (in short,IHFS) onXif the following conditions holds:
(i)µ,ν∈HX, that is,µandνareH-fuzzy sets;
(ii)µ≤N(ν), that is,µ(x)≤N(ν(x)) for eachx∈X, whereN:H→His an involutive order-reversing operation in (H,≤).
Definition 3.4[19]. Let (X,µX,νX) and (Y,µY,νY) be IHFSs. A mapping f :X→Y is
called amorphismifµX≤µY◦f andνX≥νY◦f.
It is clear that iff : (X,µX,νX)→(Y,µY,νY) is anISet(H)-mapping, thenf : (X,µX)→ (Y,µY) is aSet(H)-mapping (cf. [13]).
Definition 3.5[14]. (1) LetXbe a set.Ris called anH-fuzzy relation(or simply, afuzzy relation) onX ifµR:X×X→H is a mapping. In this case, (X,R) is called anH-fuzzy
relational space(or simply, afuzzy relational space).
(2) Let (X,RX) and (Y,RY) be any fuzzy relational spaces. A map f :X→Yis called a
relation-preserving mapprovided thatµR≤µR◦f2, where f2=f×f.
FromDefinition 3.5, we can form a concrete categoryRel(H) consisting of all rela-tional spaces and relation preserving mappings between them. EveryRel(H)-morphism will be called aRel(H)-mapping.
Definition 3.6. LetXbe a set. A pairR=(µR,νR) is called anintuitionisticH-fuzzy relation (in short,IHFR) onXif it satisfies the following conditions:
(i)µR:X×X→H andνR:X×X→H are mappings, whereµR andνR denote the
degree of membership(namely,µR(x,y)) and thedegree of nonmembership(namely, νR(x,y)) of each (x,y)∈X×XtoR;
(ii)µR≤N(νR), that is,µR(x,y)≤N(νR(x,y)) for each (x,y)∈X×X.
In this case, (X,R) or (X,µR,νR) is called anintuitionisticH-fuzzy relatinal space (in short,IHFRS).
Definition 3.7. Let (X,RX) and (Y,RY) be an IHFRSs. A mapping f :X→Y is called a
relation-preserving mappingifµRX≤µRY◦f2andνRX≥νRY◦f2, where f2= f×f.
The following is the immediate result ofDefinition 3.7. Proposition3.8. Let(X,RX),(Y,RY), and(Z,RZ)be IHFRSs.
(1) 1X: (X,RX)→(X,RX)is a relation-preserving mapping.
(2)If f : (X,RX)→(Y,RY)andg: (Y,RY)→(Z,RZ)are relation-preserving mappings,
theng◦f : (X,RX)→(Z,RZ)is a relation-preserving mapping.
From Definitions3.6and3.7, andProposition 3.8, we can form a concrete category IRel(H) consisting of all IHFRSs and relation-preserving mappings between them. Ev-eryIRel(H)-morphism will be called anIRel(H)-mapping. Moreover, it is clear that if f : (X,RX)→(Y,RY) is anIRel(H)-mapping, then f : (X,µRX)→(Y,µRY) is aRel(H
)-mapping.
Theorem3.9. IRel(H)is topological overSet.
Proof. LetXbe any set and let ((Xα,Rα))Γ be any family of IHFRSs indexed by a classΓ.
Let (fα:X→Xα)Γbe any source of mappings. We define two mappingsµR:X×X→H
andνR:X×X→H byµR=ΓµRα◦fα2andνR=
ΓνRα◦fα2. Then, by the definition of
R=(µR,νR),µR≤N(νR). Thus (X,R)∈IRel(H). Moreover, fα: (X,R)→(Xα,Rα) is an IRel(H)-mapping for eachα∈Γ.
For any (Y,RY)∈IRel(H), letg:Y→Xbe any mapping for which fα◦g: (Y,RY)→
(Xα,Rα) is anIRel(H)-mapping for eachα∈Γ. Then we can easily check thatg: (Y,RY)→
(X,R) is anIRel(H)-mapping. HenceR=(µR,νR) is the initial structure onXwith respect
Example 3.10. (1)Inverse image of an IHFR. LetX be a set, let (Y,RY) be an IHFRS, and let f :X→Y be any mapping. Then there exists the initial IHFRRonX for which f : (X,R)→(Y,RY) is anIRel(H)-mapping. In this case,Ris called theinverse imageof
RYunder f. In particular, ifX⊂Y and f :X→Yis the canonical mapping, then (X,R) is called anintuitionisticH-fuzzy relational subspaceof (Y,RY), whereR=(µR,νR) is the
inverse image ofRYunderf. In fact,µR=µRY|X×XandνR=νRY|X×X.
(2)Intuitionistic fuzzy product structure. Let ((Xα,Rα))Γbe any family of IHFRSs and
letX=Xαbe the product set of (Xα)Γ. Then there exists the initial IHFRRonXfor
which each projectionπα: (X,R)→(Xα,Rα) is anIRel(H)-mapping. In this case, R is
called theproduct of (Rα)Γ and is denoted by R=Rα and (Xα,Rα) is called the
intuitionisticH-fuzzy product relational spaceof ((Xα,Rα))Γ. In fact,µΠR=ΓµRα◦πα2and
νΠR=ΓνRα◦πα2.
In particular, ifH= {1, 2}, thenµR1×R2((x1,y1), (x2,y2))=µR1(x1,x2)∧µR2(y1,y2) and
νR1×R2((x1,y1), (x2,y2))=νR1(x1,x2)∨νR2(y1,y2) for any (x1,y1), (x2,y2)∈X1×X2.
Corollary3.11. IRel(H)is complete and cocomplete. Moreover, by definition, it is easy to show thatIRel(H)is well powered and co-well-powered.
FromResult 2.4andTheorem 3.9, it is clear thatIRel(H)is cotopological. However, we show directly thatIRel(H)is cotopological.
Theorem3.12. IRel(H)is cotopological overSet.
Proof. LetXbe any set and let ((Xα,Rα))Γ be any family of IHFRSs indexed by a classΓ. Let (fα:Xα→X)Γbe any sink of mappings. We define two mappingsµR:X×X→Hand
νR:X×X→Hby, for each (x,y)∈X×X,
µR(x,y)=
Γ
(xα,yα)∈f−12 α (x,y)
µRα
xα,yα,
νR(x,y)=
Γ
(xα,yα)∈fα−12(x,y)
νRα
xα,yα,
(3.1)
where f−12
α =fα−1×fα−1. Then clearly (X,R)∈IRel(H). Moreover, fα: (Xα,Rα)→(X,R)
is anIRel(H)-mapping for eachα∈Γ.
For any (Y,RY)∈IRel(H), letg:X→Ybe any mapping for whichg◦fα: (Xα,Rα)→
(Y,RY) is anIRel(H)-mapping for eachα∈Γ. Then we can easily check thatg: (X,R)→
(Y,RY) is anIRel(H)-mapping. HenceR=(µR,νR) is the final structure onXwith respect
to (((Xα,Rα)), (fα),X). This completes the proof.
Example 3.13. (1)IntuitionisticH-fuzzy quotient relation. Let (X,R)∈IRel(H), let∼be an equivalence relation onX, and letϕ:X→X/Rthe canonical mapping. Then there exists the final intuitionisticH-fuzzy relation (µX/∼,νX/∼) onX/∼for whichϕ: (X,R)→ (X/∼,µX/∼,νX/∼) is anIRel(H)-mapping. In this case, (µX/∼,νX/∼) is called the intuition-isticH-fuzzy quotient relationofXbyR.
eachα∈Γ. Then there exists the final IHFRRonX. In fact, for each ((xα,α), (yβ,β))∈
X×X, µR((xα,α), (yβ,β))=ΓµRα(x,y) and νR((xα,α), (yβ,β))=
ΓνRα(x,y). In this
case,Ris called thesumof (Rα)Γand (X,R) is called thesumof ((Xα,Rα))Γ.
Theorem3.14. Final episinks inIRel(H)are preserved by pullbacks.
Proof. Let (gα: (Xα,Rα)→(Y,RY))Γbe any final episink inIRel(H) and letf : (W,RW)→
(Y,RY) be anyIRel(H)-mapping. For eachα∈Γ, letUα= {(w,xα)∈W×Xα:f(w)=
gα(xα)}and let us define two mappingsµRUα:Uα×Uα→H andνRUα :Uα×Uα→H by
for each ((w,xα), (w,xα))∈Uα×Uα,
µRUα
w,xα,w,xα=µRW(w,w)∧µRα
xα,xα,
νRUα
w,xα,w,xα=νRW(w,w)∨νRα
xα,xα.
(3.2)
Leteα:Uα→Wandpα:Uα→Xαdenote the usual projections ofUα. Then clearly (Uα,
RUα)∈IRel(H) for eachα∈Γ. Moreover,eα: (Uα,RUα)→(W,RW) andpα: (Uα,RUα)→
(Xα,Rα) areIRel(H)-mappings for eachα∈Γ. And the following diagram is a pullback square inIRel(H):
Uα,Rµα
pα
eα
X
α,Rα gα
W,RW f Y,RY
(3.3)
We will show that (eα: (Uα,RUα)→(W,Rw))Γis a final episink inIRel(H). By the process
of the proof of [14, Theorem 2.5], (eα)Γ is an episink inIRel(H). SupposeR=(µR,νR)
is another final IHFR on W with respect to (eα)Γ. By the process of the proof of [14,
Theorem 2.5],µR=µRW. Thus it is sufficient to show thatνR=νRW. Let (w,w)∈W×W.
Then
νRW(w,w)=νRW(w,w)∨νRW(w,w)
≥νRW(w,w)∨
νRY◦f2(w,w)
since f :W,Rw−→Y,RYis anIRel(H)-mapping
=νRW(w,w)∨νRY
f(w),f(w)
=νRW(w,w)∨
Γ
(xα,xα)∈g−12
α (f(w),f(w))
νRα
xα,xα
sincegαΓis final
=
Γ
(xα,xα)∈g−12
α (f(w),f(w))
νRW(w,w)∨νRα
xα,xα
=
Γ
((w,xα),(w,xα))∈e−α1(w,w)
νRUα
w,x
α,w,xα.
ThusνRW(w,w)≥νR(w,w) for each (w,w)∈W×W. SoνRW ≥νR. On the other hand,
since (eα: (Uα,RUα)→(W,R))Γis final, 1W : (W,R)→(W,RW) is anIRel(H)-mapping.
ThusνR≥νRW. SoνR=νRW. HenceR=RW. This completes the proof.
For any singleton set{a}, since the IHFRRon{a}is not unique, the categoryIRel(H) is not properly fibered overSet. Hence, by Theorems3.12and3.14, we obtain the follow-ing result.
Theorem3.15. IRel(H)satisfies all the conditions of a topological universe overSetexcept the terminal separator property.
Theorem3.16. IRel(H)is cartesian closed overSet.
Proof. It is clear thatIRel(H) has products byCorollary 3.11. We will show thatIRel(H) has exponential objects.
For any IHFRSsX=(X,RX) andY=(Y,RY), letYXbe the set of all mappings from XintoY. We define two mappingsµR:YX×YX→H andνR:YX×YX→H as follows:
for each (f,g)∈YX×YX,
µR(f,g)=h∈H:µRX(x,y)∧h≤µRY
f(x),g(y)for each (x,y)
∈X×X, νR(f,g)=h∈H:νRX(x,y)∨h≥νRY
f(x),g(y)for each (x,y)∈X×X. (3.5)
Then clearly (YX,R)∈IRel(H). LetYX=(YX,R). Then, by the definition ofR,
µRX(x,y)∧µR(f,g)≤µRY
f(x),g(y), νRX(x,y)∨νR(f,g)≥νRY
f(x),g(y) (3.6)
for each (f,g)∈YXand (x,y)∈X×X.
DefineeX,Y:X×YX→Y byeX,Y(x,f)= f(x) for each (x,f)∈X×YX. Let ((x,f), (y,g))∈(X×YX)×(X×YX). Then, by the process of the proof of [14, Theorem
2.7],µRX×R((x,f), (y,g))≤µRY◦e2X,Y((x,f), (y,g)). SoµRX×R≤µRY◦e2X,Y. On the other hand,
νRX×R
(x,f), (y,g)=νRX(x,y)∨νR(f,g)
≥νRY
f(x),g(y) =νRY
e
X,Y(x,f),eX,Y(y,g)
=νRY◦e2X,Y
(x,f), (y,g).
(3.7)
ThusνRX×R≥νRY◦e2X,Y. HenceeX,Y: X×YX→ Yis anIRel(H)-mapping.
For any Z=(Z,RZ)∈IRel(H), let h:X×Z→Y be an IRel(H)-mapping. We
letx,x∈X. Then, by the process of the proof of [14, Theorem 2.7],µRZ(z,z)≤µR◦
h2(z,z). SoµR
Z≤µR◦h
2
. On the other hand,
νRX×RZ
(x,z), (x,z)=νR
X(x,x)∨νRZ(z,z)
≥νRY◦h2
(x,z), (x,z )
sinceh:X×Z−→Yis anIRel(H)-mapping =νRY
h(x,z),h(x,z)
=νRY
h(z) (x),h(z) (x).
(3.8)
Thus, by the definition ofR,νRZ(z,z)≥νR(h(z),h(z))=νRY◦h
2
(z,z). SoνR
Z≥νR◦h
2 . Henceh:Z→YX is anIRel(H)-mapping. Moreover,his the uniqueIRel(H)-mapping such thateX,Y◦(1X×h)=h. This completes the proof.
Remark 3.17. IRel(H) has no subobject classifier. HenceIRel(H) is not topos.
Example 3.18. LetH= {0, 1}be the two points chain and letX= {a}. LetR1andR2be the IHFRs onXgiven byµR1(a,a)=0,νR1(a,a)=1 andµR2(a,a)=1,νR2(a,a)=0. Let 1X:
(X,R1)→(X,R2) be the identity mapping. Then clearly, 1Xis both a monomorphism and an epimorphism inIRel(H). But, 1X is not an isomorphism inIRel(H). HenceIRel(H)
has no subobject classifier (see [7]).
4. The relations between IRel(H)and Rel(H) Lemma4.1. DefineG1,G2:IRel(H)→Rel(H)by
G1
X,µR,νR=X,µR,
G2
X,µR,νR=X,NνR,
G1(f)=G2(f)= f .
(4.1)
ThenG1andG2are functors.
Proof. ClearlyG1(X,µRX,νRX)=(X,µRX)∈Rel(H) for each (X,µR,νR)∈IRel(H). Let (X,
µRX,νRX), (Y,µRY,νRY)∈IRel(H) and let f : (X,µRX,νRX)→(Y,µRY,νRY) be anIRel(H
)-mapping. ThenµRX≤µRY◦f2. ThusG1(f)=f : (X,µRX)→(Y,µRY) is aRel(H)-mapping.
HenceG1:IRel(H)→ Rel(H) is a functor. AlsoG2(X,µRX,νRX)=(X,N(νRX))∈ Rel(H)
for each (X,µRX,νRX)∈IRel(H). Now let (X,µRX,νRX), (Y,µRY,νRY)∈IRel(H) and let f :
(X,µRX,νRX)→(Y,µRY,νRY) be anIRel(H)-mapping. ThenνRX≥νRY◦f2. ThusN(νRX)≤
N(νRY)◦f2. SoG2(f)=f : (X,N(νRX))→(Y,N(νRY)) is aRel(H)-mapping. HenceG2:
IRel(H)→Rel(H) is a functor.
Proof. For each (X,µRX)∈Rel(H),µ≤NN(µRX). ThusF1(X,µRX)=(X,µRX,N(µRX))∈
IRel(H). Let (X,µRX), (Y,µRY)∈Rel(H) and let f : (X,µRX)→(Y,µRY) be an Rel(H
)-mapping. ThenµRX≤µRY◦f. Consider the mappingF1(f)=f : (X,µRX,N(µRX))→(Y,
µRY,N(µRY)). SinceµRX ≤µRY◦f,N(µRX)≥N(µRY)◦f. So f : (X,µRX,N(µRX))→(Y,
µRY,N(µRY)) is anIRel(H)-mapping. HenceF1is a functor. Lemma4.3. DefineF2:Rel(H)→IRel(H)byF2(X,µR)=(X,NN(µR),N(µR))andF2(f)=
f. ThenF2is a functor.
Proof. It is clear that F2(X,µRX)∈IRel(H) for each (X,µRX)∈Rel(H). Let (X,µRX),
(Y,µRY)∈Rel(H) and let f : (X,µRX)→(Y,µRY) be anRel(H)-mapping. Consider the
mapping F2(f)= f : F2(X,µRX) →(Y,NN(µRY),N(µRY)), where F2(X,µRX) = (X,
NN(µRX),N(µRX)) andF2(Y,µRY)=(Y,NN(µRY),N(µRY)). Since f : (X,µRX)→(Y,µRY)
is a Rel(H)-mapping, µRX ≤µRY ◦ f2. Thus NN(µRX)≤NN(µRY)◦ f2. Moreover
N(µRX)≥N(µRY)◦f2. SoF2(f)= f :F2(X,µRX)→F2(Y,µRY) is anIRel(H)-mapping.
HenceF2is a functor.
Theorem 4.4. The functor F1:Rel(H)→IRel(H) is a left adjoint of the functor G1: IRel(H)→Rel(H).
Proof. For each (X,µR)∈Rel(H), 1X : (X,µR)→G1F1(X,µR)=(X,µR) is a Rel(H
)-mapping. Let (Y,µRY,νRY)∈IRel(H) and let f : (X,µR)→G1(Y,µRY,νRY) be anIRel(H
)-mapping. We will show that f :F1(X,µR)=(X,µR,N(µR))→(Y,µRY,νRY) is anIRel(H
)-mapping. Since f : (X,µR)=G1(Y,µRY,µRY)→(Y,µRY) is aRel(H)-mapping,µR≤µRY◦
f2. Then N(µR)≥N(µR
Y)◦f2. Since µRY ≤N(νRY), νRY ≤NN(νRY)≤N(µRY). Thus
N(µR)≥νRY◦f2. So f :F1(X,µR)=(Y,µRY,νRY) is anIRel(H)-mapping. Hence 1X is
aG1-universal map for (X,µR) in Rel(H). This completes the proof.
For each (X,µR)∈ Rel(H), F1(X,µR)=(X,µR,N(µR)) is called an intuitionistic
H-fuzzy set inXinducedby (X,µR). Let us denote the category of all induced intuitionis-ticH-fuzzy sets andIRel(H)-mappings asIRel∗(H). Then it is clearIRel∗(H) is a full subcategory ofIRel(H).
Theorem4.5. Two categoriesRel(H)andIRel∗(H)are isomorphic.
Proof. It is clear thatF1:Rel(H)→IRel∗(H) is a functor byLemma 4.2. Consider the re-strictionG1:IRel∗(H)→Rel(H) of the functorG1inLemma 4.1. Let (X,µR)∈Rel(H).
Then, byLemma 4.2,F1(X,µR)=(X,µR,N(µR)). ThusG1F1(X,µR)=G1(X,µR,N(µR))= (X,µR). So G1◦F=1Rel(H). Now let (X,µR,N(µR))∈ISet∗(H). Then, byLemma 4.1,
G1(X,µR,N(µR)) =(X,µR). Thus FG1(X,µR,N(µR)) = (X,µR,N(µR)). So F ◦G1 = 1ISet∗(H). HenceF:Rel(H)→ISet∗(H) is an isomorphism. This completes the proof. Remark 4.6. We are going to investigate “intuitionisticH-fuzzy reflexive relations,” “some subcategories of the category IRelk (H),” and “intuitionisticH-fuzzy relations on intu-itionisticH-fuzzy sets” in the viewpoint of topological universe.
Acknowledgment
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Kul Hur: Division of Mathematics and Informational Statistics and Institute of Basic Natural Sci-ence, Wonkwang University, Iksan, Chonbuk 579-792, Korea
E-mail address:[email protected]
Su Youn Jang: Division of Mathematics and Informational Statistics and Institute of Basic Natural Science, Wonkwang University, Iksan, Chonbuk 579-792, Korea
E-mail address:[email protected]
Hee Won Kang: Department of Mathematics Education, Woosuk University, Hujong-Ri, Samrae-Eup, Wanju-kun, Chonbuk 565-701, Korea
