INTUITIONISTIC FUZZY IDEALS OF BCK-ALGEBRAS

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INTUITIONISTIC FUZZY IDEALS OF BCK-ALGEBRAS

YOUNG BAE JUN and KYUNG HO KIM (Received 16 February 2000)

Abstract.We consider the intuitionistic fuzzification of the concept of subalgebras and ideals in BCK-algebras, and investigate some of their properties. We introduce the notion of equivalence relations on the family of all intuitionistic fuzzy ideals of a BCK-algebra and investigate some related properties.

Keywords and phrases. (Intuitionistic) fuzzy subalgebra, (intuitionistic) fuzzy ideal, upper (respectively, lower)t-level cut, homomorphism.

2000 Mathematics Subject Classification. Primary 06F35, 03G25, 03E72.

1. Introduction. After the introduction of the concept of fuzzy sets by Zadeh [9]

several researches were conducted on the generalizations of the notion of fuzzy sets.

The idea of “intuitionistic fuzzy set” was first published by Atanassov [1, 2], as a generalization of the notion of fuzzy set. The first author (together with Hong, Kim, Kim, Meng, Roh, and Song) considered the fuzzification of ideals and subalgebras in BCK-algebras (cf. [3, 4, 5, 6, 7, 8]). In this paper, using the Atanassov’s idea, we establish the intuitionistic fuzzification of the concept of subalgebras and ideals in BCK-algebras, and investigate some of their properties. We introduce the notion of equivalence relations on the family of all intuitionistic fuzzy ideals of a BCK-algebra and investigate some related properties.

2. Preliminaries. First we present the fundamental definitions. By aBCK-algebra we mean a nonempty setXwith a binary operation∗and a constant 0 satisfying the following conditions:

(I) ((x∗y)∗(x∗z))∗(z∗y)=0, (II) (x∗(x∗y))∗y=0,

(III) x∗x=0, (IV) 0∗x=0,

(V) x∗y=0 andy∗x=0 imply thatx=y for allx,y,z∈X.

A partial ordering “≤” on X can be defined by x ≤y if and only if x∗y =0.

A nonempty subset S of a BCK-algebra X is called a subalgebra ofX if x∗y∈S wheneverx,y∈S. A nonempty subsetIof a BCK-algebraXis called anidealofXif

(i) 0∈I,

(ii) x∗y∈Iandy∈Iimply thatx∈Ifor allx,y∈X.

By afuzzy set µ in a nonempty set X we mean a functionµ :X →[0,1], and the complement ofµ, denoted by ¯µ, is the fuzzy set inXgiven by ¯µ(x)=1−µ(x)for all x∈X. A fuzzy setµin a BCK-algebraXis called afuzzy subalgebraofXifµ(x∗y)≥

min{µ(x),µ(y)}for allx,y∈X. A fuzzy setµin a BCK-algebraX is called afuzzy idealofXif

(i) µ(0)≥µ(x)for allx∈X,

(ii) µ(x)≥min{µ(x∗y),µ(y)}for allx,y∈X.

An intuitionistic fuzzy set (briefly, IFS)Ain a nonempty setX is an object having the form

A=

(x,αA(x),βA(x))|x∈X

, (2.1)

where the functionsαA:X→[0,1]andβA:X→[0,1]denote the degree of member- ship and the degree of nonmembership, respectively, and

0≤αA(x)+βA(x)≤1 ∀x∈X. (2.2) An intuitionistic fuzzy setA= {(x,αA(x),βA(x))|x∈X}inXcan be identified to an ordered pair(αA,βA)inI^X×I^X. For the sake of simplicity, we shall use the symbol A=(αA,βA)for the IFSA= {(x,αA(x),βA(x))|x∈X}.

3. Intuitionistic fuzzy ideals. In what follows, letXdenote a BCK-algebra unless otherwise specified.

Definition3.1. An IFSA=(αA,βA)inXis called anintuitionistic fuzzy subalgebra ofXif it satisfies:

(IS1) αA(x∗y)≥min{αA(x),αA(y)}, (IS2) βA(x∗y)≤max{βA(x),βA(y)}, for allx,y∈X.

Example3.2. Consider a BCK-algebraX={0,a,b,c}with the following Cayley table:

∗ 0 a b c

0 0 0 0 0

a a 0 0 a

b b a 0 b

c c c c 0

LetA=(αA,βA)be an IFS inXdefined by

αA(0)=αA(a)=αA(c)=0.7>0.3=αA(b),

βA(0)=βA(a)=βA(c)=0.2<0.5=βA(b). (3.1) ThenA=(αA,βA)is an intuitionistic fuzzy subalgebra ofX.

Proposition3.3. Every intuitionistic fuzzy subalgebraA=(αA,βA)ofXsatisfies the inequalitiesαA(0)≥αA(x)andβA(0)≤βA(x)for allx∈X.

Proof. For anyx∈X, we have

αA(0)=αA(x∗x)≥min

αA(x),αA(x)

=αA(x), βA(0)=βA(x∗x)≤max

βA(x),βA(x)

=βA(x). (3.2)

This completes the proof.

Definition3.4. An IFSA=(αA,βA)inXis called anintuitionistic fuzzy idealofX if it satisfies the following inequalities:

(IF1) αA(0)≥αA(x)andβA(0)≤βA(x), (IF2) αA(x)≥min{αA(x∗y),αA(y)}, (IF3) βA(x)≤max{βA(x∗y),βA(y)}, for allx,y∈X.

Example3.5. LetX= {0,1,2,3,4}be a BCK-algebra with the following Cayley table:

∗ 0 1 2 3 4

0 0 0 0 0 0

1 1 0 1 0 0

2 2 2 0 0 0

3 3 3 3 0 0

4 4 3 4 1 0

Define an IFSA=(αA,βA)inXas follows:

αA(0)=αA(2)=1, αA(1)=αA(3)=αA(4)=t,

βA(0)=βA(2)=0, βA(1)=βA(3)=βA(4)=s, (3.3) wheret∈[0,1], s∈[0,1], and t+s≤1. By routine calculation we know thatA= (αA,βA)is anintuitionistic fuzzy idealofX.

Lemma3.6. Let anIFSA=(αA,βA)inXbe an intuitionistic fuzzy ideal ofX. If the inequalityx∗y≤zholds inX, then

αA(x)≥min

αA(y),αA(z)

, βA(x)≤max

βA(y),βA(z)

. (3.4)

Proof. Letx,y,z∈Xbe such thatx∗y≤z. Then(x∗y)∗z=0, and thus αA(x)≥min

αA(x∗y),αA(y)

≥min min

αA

(x∗y)∗z

,αA(z)

,αA(y)

=min min

αA(0),αA(z)

,αA(y)

=min

αA(y),αA(z) , βA(x)≤max

βA(x∗y),βA(y)

≤max max

βA

(x∗y)∗z ,βA(z)

,βA(y)

=max max

βA(0),βA(z)},βA(y)

=max

βA(y),βA(z) ,

(3.5)

this completes the proof.

Lemma3.7. LetA=(αA,βA)be an intuitionistic fuzzy ideal ofX. Ifx≤yinX, then αA(x)≥αA(y), βA(x)≤βA(y), (3.6) that is,αAis order-reserving andβAis order-preserving.

Proof. Letx,y∈Xbe such thatx≤y. Thenx∗y=0 and so αA(x)≥min

αA(x∗y),αA(y)

=min

αA(0),αA(y)

=αA(y), βA(x)≤max

βA(x∗y),βA(y)

=max

βA(0),βA(y)

=βA(y). (3.7) This completes the proof.

Theorem3.8. If A=(αA,βA) is an intuitionistic fuzzy ideal of X, then for any x,a1,a2,...,an∈X,(···((x∗a1)∗a2)∗···)∗an=0implies

αA(x)≥min αA

a1 ,αA

a2 ,...,αA

an , βA(x)≤max

βA a1

,βA a2

,...,βA an

. (3.8)

Proof. Using induction onnand Lemmas 3.6 and 3.7, the proof is straightforward.

Theorem3.9. Every intuitionistic fuzzy ideal ofX is an intuitionistic fuzzy subal- gebra ofX.

Proof. LetA=(αA,βA)be an intuitionistic fuzzy ideal ofX. Sincex∗y≤xfor allx,y∈X, it follows from Lemma 3.7 that

αA(x∗y)≥αA(x), βA(x∗y)≤βA(x), (3.9) so by (IF2) and (IF3),

αA(x∗y)≥αA(x)≥min

αA(x∗y),αA(y)

≥min

αA(x),αA(y) , βA(x∗y)≤βA(x)≤max

βA(x∗y),βA(y)

≤max

βA(x),βA(y)

. (3.10) This shows thatA=(αA,βA)is an intuitionistic fuzzy subalgebra ofX.

The converse of Theorem 3.9 may not be true. For example, the intuitionistic fuzzy subalgebraA=(αA,βA)in Example 3.2 is not an intuitionistic fuzzy ideal ofXsince

βA(b)=0.5>0.2=min

βA(b∗a),βA(a)

. (3.11)

We now give a condition for an intuitionistic fuzzy subalgebra to be an intuitionistic fuzzy ideal.

Theorem3.10. LetA=(αA,βA)be an intuitionistic fuzzy subalgebra ofXsuch that αA(x)≥min

αA(y),αA(z)

, βA(x)≤max

βA(y),βA(z)

(3.12) for allx,y,z∈Xsatisfying the inequalityx∗y≤z. ThenA=(αA,βA)is an intuition- istic fuzzy ideal ofX.

Proof. LetA=(αA,βA)be an intuitionistic fuzzy subalgebra of X. Recall that αA(0)≥αA(x)andβA(0)≤βA(x)for allX. Sincex∗(x∗y)≤y, it follows from the hypothesis that

αA(x)≥min

αA(x∗y),αA(y)

, βA(x)≤max

βA(x∗y),βA(y)

. (3.13) HenceA=(αA,βA)is an intuitionistic fuzzy ideal ofX.

Lemma3.11. AnIFSA=(αA,βA)is an intuitionistic fuzzy ideal ofXif and only if the fuzzy setsαAandβ¯Aare fuzzy ideals ofX.

Proof. LetA=(αA,βA)be an intuitionistic fuzzy ideal ofX. Clearly,αAis a fuzzy ideal ofX. For everyx,y∈X, we have

β¯A(0)=1−βA(0)≥1−βA(x)=β¯A(x), β¯A(x)=1−βA(x)≥1−max

βA(x∗y),βA(y)

=min

1−βA(x∗y),1−βA(y)

=minβ¯A(x∗y),β¯A(y) .

(3.14)

Hence ¯βAis a fuzzy ideal ofX.

Conversely, assume thatαAand ¯βAare fuzzy ideals ofX. For everyx,y∈X, we get αA(0)≥αA(x), 1−βA(0)=β¯A(0)≥β¯A(x)=1−βA(x), (3.15) that is,βA(0)≤βA(x); αA(x)≥min{αA(x∗y),αA(y)}and

1−βA(x)=β¯A(x)≥minβ¯A(x∗y),β¯A(y)

=min

1−βA(x∗y),1−βA(y)

=1−max

βA(x∗y),βA(y) ,

(3.16)

that is,βA(x)≤max{βA(x∗y),βA(y)}. HenceA=(αA,βA)is an intuitionistic fuzzy ideal ofX.

Theorem3.12. LetA=(αA,βA)be an IFS inX. ThenA=(αA,βA)is an intuition- istic fuzzy ideal ofXif and only ifA=(αA,α¯A)and♦A=(β¯A,βA)are intuitionistic fuzzy ideals ofX.

Proof. IfA=(αA,βA)is an intuitionistic fuzzy ideal ofX, thenαA=α¯¯AandβA

are fuzzy ideals ofX from Lemma 3.11, henceA=(αA,α¯A)and♦A=(β¯A,βA)are intuitionistic fuzzy ideals ofX. Conversely, ifA=(αA,α¯A)and♦A=(β¯A,βA)are intuitionistic fuzzy ideals ofX, then the fuzzy setsαAand ¯βAare fuzzy ideals ofX, henceA=(αA,βA)is an intuitionistic fuzzy ideal ofX.

For anyt∈[0,1]and a fuzzy setµin a nonempty setX, the set U(µ;t)=

x∈X|µ(x)≥t

(3.17) is called anuppert-level cutofµand the set

L(µ;t)=

x∈X|µ(x)≤t

(3.18) is called alowert-level cut ofµ.

Theorem3.13. AnIFSA=(αA,βA)is an intuitionistic fuzzy ideal ofXif and only if for alls,t∈[0,1], the setsU(αA;t)andL(βA;s)are either empty or ideals ofX.

Proof. LetA=(αA,βA)be an intuitionistic fuzzy ideal ofXandU(αA;t)≠∅≠ L(βA;s)for any s,t∈[0,1]. It is clear that 0∈U(αA;t)∩L(βA;s) sinceαA(0)≥t and βA(0)≤s. Letx,y∈X be such thatx∗y∈U(αA;t) and y∈U(αA;t). Then αA(x∗y)≥tandαA(y)≥t. It follows that

αA(x)≥min

αA(x∗y),αA(y)

≥t (3.19)

so thatx∈U(αA;t). HenceU(αA;t)is an ideal ofX. Now letx,y∈Xbe such that x∗y∈L(βA;s)andy∈L(βA;s). ThenβA(x∗y)≤sandβA(y)≤s, which imply that

βA(x)≤max

βA(x∗y),βA(y)

≤s. (3.20)

Thus x∈L(βA;s), and therefore L(βA;s)is an ideal of X. Conversely, assume that for eacht,s∈[0,1], the setsU(αA;t)andL(βA;s)are either empty or ideals ofX.

For any x∈X, letαA(x)=t and βA(x)=s. Then x∈U(αA;t)∩L(βA;s), and so U(αA;t)≠∅≠L(βA;s). SinceU(αA;t)and L(βA;s)are ideals of X, therefore 0∈ U(αA;t)∩L(βA;s). HenceαA(0)≥t=αA(x)andβA(0)≤s=βA(x)for allx∈X. If there existx,y∈Xsuch thatαA(x) <min{αA(x∗y),αA(y)}, then by taking

t0=1 2

αA x

+min αA

x∗y ,αA

y

, (3.21)

we have

αA x

< t0<min αA

x∗y ,αA

y

. (3.22)

Hencex∈U(αA;t0),x∗y∈U(αA;t0)andy∈(αA;t0), that is,U(αA;t0)is not an ideal ofX, which is a contradiction. Finally, assume that there exista,b∈Xsuch that

βA(a) >max

βA(a∗b),βA(b)

. (3.23)

Takings0:=(1/2)(βA(a)+max{βA(a∗b),βA(b)}),then max

βA(a∗b),βA(b)

< s0< βA(a). (3.24) Thereforea∗b∈L(βA;s0)andb∈L(βA;s0), buta∈L(βA;s0), which is a contradic- tion, this completes the proof.

LetΛbe a nonempty subset of[0,1].

Theorem3.14. Let{It|t∈Λ}be a collection of ideals ofXsuch that (i) X= ∪t∈ΛIt,

(ii) s > tif and only ifIs⊂It for alls,t∈Λ.

Then anIFSA=(αA,βA)inXdefined by αA(x):=sup

t∈Λ|x∈It

, βA(x):=inf

t∈Λ|x∈It

(3.25) for allx∈Xis an intuitionistic fuzzy ideal ofX.

Proof. According to Theorem 3.13, it is sufficient to show that U(αA;t) and L(βA;s)are ideals ofX for everyt∈[0,αA(0)]ands∈[βA(0),1]. In order to prove

thatU(αA;t)is an ideal ofX, we divide the proof into the following two cases:

(i) t=sup{q∈Λ|q < t}, (ii) t≠sup{q∈Λ|q < t}.

Case (i) implies that x∈U

αA;t)⇐⇒x∈Iq ∀q < t⇐⇒x∈ ∩q<tIq, (3.26) so that U(αA;t) = ∩q<tIq, which is an ideal of X. For the case (ii), we claim that U(αA;t)= ∪q≥tIq.Ifx∈ ∪q≥tIq,thenx∈Iqfor someq≥t. It follows thatαA(x)≥ q≥ t, so that x ∈U(αA;t). This shows that∪q≥tIq ⊆U(αA;t). Now assume that x ∈ ∪q≥tIq. Thenx ∈ Iq for all q ≥t. Since t ≠sup{q∈ Λ|q < t}, there exists ε >0 such that(t−ε,t)∩Λ= ∅. Hencex∈Iq for allq > t−ε, which means that if x ∈Iq,then q≤t−ε. Thus αA(x)≤t−ε < t, and so x ∈U(αA;t). Therefore U(αA;t)⊆ ∪q≥tIq, and thusU(αA;t)= ∪q≥tIqwhich is an ideal ofX. Next we prove thatL(βA;s)is an ideal ofX. We consider the following two cases:

(iii) s=inf{r∈Λ|s < r}, (iv) s≠inf{r∈Λ|s < r}.

For the case (iii), we have x∈L

βA;s

⇐⇒x∈Ir ∀s < r⇐⇒x∈ ∩s<rIr, (3.27) and henceL(βA;s)= ∩s<rIr which is an ideal ofX. For the case (iv) there existsε >0 such that(s,s+ε)∩Λ= ∅.We will show thatL(βA;s)= ∪s≥rIr.If x∈ ∪s≥rIr, then x ∈Ir for some r ≤s. It follows that βA(x)≤r ≤s so that x ∈L(βA;s). Hence

∪s≥rIr ⊆L(βA;s). Conversely, ifx∈ ∪s≥rIr, thenx∈Ir for allr≤s, which implies thatx∈Ir for allr < s+ε, that is, ifx∈Ir,thenr≥s+ε. ThusβA(x)≥s+ε > s, that is,x∈L(βA;s). ThereforeL(βA;s)⊆ ∪s≥rIr and consequentlyL(βA;s)= ∪s≥rIr

which is an ideal ofX. This completes the proof.

A mappingf :X →Y of BCK-algebras is called a homomorphism if f (x∗y)= f (x)∗f (y) for all x,y ∈X. Note that if f :X→Y is a homomorphism of BCK- algebras, thenf (0)=0. Letf:X→Y be a homomorphism of BCK-algebras. For any IFSA=(αA,βA)inY, we define a new IFSA^f=

α^f_A,β^f_A inXby α^f_A(x):=αA

f (x)

, β^f_A(x):=βA f (x)

∀x∈X. (3.28)

Theorem3.15. Letf:X→Y be a homomorphism of BCK-algebras. If anIFSA= (αA,βA)inY is an intuitionistic fuzzy ideal ofY, then anIFSA^f =

α^f_A,β^f_A

inXis an intuitionistic fuzzy ideal ofX.

Proof. We first have that α^f_A(x)=αA

f (x)

≤αA(0)=αA f (0)

=α^f_A(0), β^f_A(x)=βA

f (x)

≥βA(0)=βA f (0)

=β^f_A(0) (3.29) for allx∈X. Letx,y∈X. Then

min

α^f_A(x∗y),α^f_A(y)

=min αA

f (x∗y) ,αA

f (y)

=min αA

f (x)∗f (y) ,αA

f (y)

≤αA f (x)

=α^f_A(x), max

β^f_A(x∗y),β^f_A(y)

=max βA

f (x∗y) ,βA

f (y)

=max βA

f (x)∗f (y) ,βA

f (y)

≥βA f (x)

=β^f_A(x).

(3.30)

HenceA^f= α^f_A,β^f_A

is an intuitionistic fuzzy ideal ofX.

If we strengthen the condition off, then we can construct the converse of Theorem 3.15 as follows.

Theorem3.16. Letf :X →Y be an epimorphism of BCK-algebras and let A= (αA,βA)be an IFS in Y. If A^f =

α^f_A,β^f_A

is an intuitionistic fuzzy ideal ofX, then A=(αA,βA)is an intuitionistic fuzzy ideal ofY.

Proof. For anyx∈Y, there existsa∈Xsuch thatf (a)=x. Then αA(x)=αA

f (a)

=α^fA(a)≤α^fA(0)=αA f (0)

=αA(0), βA(x)=βA

f (a)

=β^f_A(a)≥β^f_A(0)=βA f (0)

=βA(0). (3.31) Letx,y∈Y. Thenf (a)=xandf (b)=yfor somea,b∈X. It follows that

αA(x)=αA f (a)

=α^f_A(a)

≥min

α^f_A(a∗b),α^f_A(b)

=min αA

f (a∗b) ,αA

f (b)

=min αA

f (a)∗f (b) ,αA

f (b)

=min

αA(x∗y),αA(y) , βA(x)=βA

f (a)

=β^f_A(a)

≤max

β^f_A(a∗b),β^f_A(b)

=max βA

f (a∗b) ,βA

f (b)

=max βA

f (a)∗f (b) ,βA

f (b)

=max

βA(x∗y),βA(y) .

(3.32)

This completes the proof.

Let IF(X)be the family of all intuitionistic fuzzy ideals ofXand lett∈[0,1]. Define binary relationsU^tandL^t on IF(X)as follows:

(A,B)∈U^t⇐⇒U αA;t

=U αB;t

, (A,B)∈L^t⇐⇒L βA;t

=L βB;t

, (3.33) respectively, forA=(αA,βA)andB=(αB,βB)in IF(X). Then clearlyU^t and L^t are

equivalence relations on IF(X). For anyA=(αA,βA)∈IF(X), let[A]U^t (respectively, [A]_L^t) denote the equivalence class of A modulo U^t (respectively, L^t), and denote by IF(X)/U^t(respectively, IF(X)/L^t) the system of all equivalence classes moduloU^t (respectively,L^t); so

IF(X)/U^t:=

[A]_U^t|A= αA,βA

∈IF(X)

, (3.34)

respectively,

IF(X)/L^t:=

[A]L^t |A= αA,βA

∈IF(X)

. (3.35)

Now letI(X)denote the family of all ideals ofXand lett∈[0,1]. Define mapsftand gt from IF(X)toI(X)∪ {∅}byft(A)=U(αA;t)and gt(A)=L(βA;t),respectively, for allA=(αA,βA)∈IF(X). Thenftandgtare clearly well defined.

Theorem3.17. For anyt∈(0,1)the mapsftandgtare surjective fromIF(X)to I(X)∪{∅}.

Proof. Lett∈(0,1). Note that 0∼=(0,1)is in IF(X), where0and 1are fuzzy sets inXdefined by0(x)=0 and1(x)=1 for allx∈X. Obviouslyft(0∼)=U(0;t)=

∅ =L(1;t)=gt(0∼).LetG(≠∅)∈I(X). ForG∼=(χG,χ¯G)∈IF(X), we haveft(G∼)= U(χG;t)=Gandgt(G∼)=L(χ¯G;t)=G. Henceftandgtare surjective.

Theorem3.18. The quotient setsIF(X)/U^tandIF(X)/L^t are equipotent toI(X)∪

{∅}for everyt∈(0,1).

Proof. Fort∈(0,1)letf_t^∗ (respectively, g^∗_t) be a map from IF(X)/U^t (respec- tively, IF(X)/L^t) toI(X)∪{∅}defined byf_t^∗([A]U^t)=ft(A)(respectively,g^∗_t([A]L^t)= gt(A)) for all A= (αA,βA)∈ IF(X). If U(αA;t) = U(αB;t) and L(βA;t) = L(βB;t) forA=(αA,βA)and B=(αB,βB)in IF(X), then(A,B)∈U^t and(A,B)∈L^t; hence [A]U^t=[B]U^t and[A]L^t=[B]L^t.Therefore the mapsf_t^∗andg_t^∗are injective. Now let G(= ∅)∈I(X). ForG∼=(χG,χ¯G)∈IF(X), we have

f_t^∗ G∼

U^t

=ft G∼

=U χG;t

=G, gt^∗

G∼

L^t

=gt G∼

=L χ¯G;t

=G. (3.36)

Finally, for0∼=(0,1)∈IF(X)we get f_t^∗

0∼

U^t

=ft 0∼

=U(0;t)= ∅, g^∗_t

0∼

L^t

=gt 0∼

=L(0;t)= ∅. (3.37)

This shows thatf_t^∗andg_t^∗are surjective. This completes the proof.

For anyt∈[0,1], we define another relationR^ton IF(X)as follows:

(A,B)∈R^t⇐⇒U αA;t

∩L βA;t

=U αB;t

∩L βB;t

(3.38)

for anyA=(αA,βA),B=(αB,βB)∈IF(X). Then the relationR^tis also an equivalence relation on IF(X).

Theorem3.19. For any t∈(0,1), the map φt : IF(X)→I(X)∪ {∅}defined by φt(A)=ft(A)∩gt(A)for eachA=(αA,βA)∈IF(X)is surjective.

Proof. Lett∈(0,1). For0∼=(0,1)∈IF(X), φt

0∼

=ft 0∼

∩gt 0∼

=U(0;t)∩L(1;t)= ∅. (3.39) For anyH∈IF(X), there existsH∼=(χH,χ¯H)∈IF(X)such that

φt H∼

=ft H∼

∩gt H∼

=U χH;t

∩L

¯ χH;t

=H. (3.40)

This completes the proof.

Theorem3.20. For any t ∈ (0,1), the quotient set IF(X)/R^t is equipotent to I(X)∪{∅}.

Proof. Let t∈(0,1)and let φ^∗t : IF(X)/R^t →I(X)∪ {∅} be a map defined by φ^∗_t([A]_R^t)=φt(A)for all[A]_R^t∈IF(X)/R^t.Ifφ^∗_t([A]_R^t)=φ^∗_t([B]_R^t)for any[A]_R^t, [B]R^t ∈IF(X)/R^t, then ft(A)∩gt(A)=ft(B)∩gt(B), that is, U(αA;t)∩L(βA;t)= U(αB;t)∩L(βB;t),hence(A,B)∈R^t. It follows that[A]R^t=[B]R^t so thatφ^∗_t is injec- tive. For0∼=(0,1)∈IF(X),

φ^∗_t 0∼

R^t

=φt 0∼

=ft 0∼

∩gt 0∼

=U(0;t)∩L(1;t)= ∅. (3.41) IfH∈IF(X), then forH∼=(χH,χ¯H)∈IF(X), we have

φ^∗_t H∼

R^t

=φ H∼

=ft H∼

∩gt H∼

=U χH;t

∩L

¯ χH;t

=H. (3.42) Henceφ^∗t is surjective, this completes the proof.

Acknowledgement. The first author was supported by Korea Research Founda- tion Grant (KRF-99-005-D00003).

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Young Bae Jun: Department of Mathematics Education, Gyeongsang National Uni- versity, Chinju660-701, Korea

E-mail address:[email protected]

Kyung Ho Kim: Department of Mathematics, Chungju National University, Chungju 380-702, Korea

E-mail address:[email protected]

Mathematical Problems in Engineering

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