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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Zbl 139.24606.

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]

Journal of Applied Mathematics and Decision Sciences

Special Issue on

Intelligent Computational Methods for Financial Engineering

Call for Papers

As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.

However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used e

ectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)

This special issue will include (but not be limited to) the following topics:

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: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, imple- mentation

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Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

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Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;

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[email protected]

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