Fuzzy BCI subalgebras with interval valued membership functions

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FUZZY BCI-SUBALGEBRAS WITH INTERVAL-VALUED

MEMBERSHIP FUNCTIONS

SUNG MIN HONG, YOUNG BAE JUN, SEON JEONG KIM, and GWANG IL KIM

(Received 3 May 2000)

Abstract.The purpose of this paper is to define the notion of an interval-valued fuzzy BCI-subalgebra (briefly, an i-v fuzzy BCI-subalgebra) of a BCI-algebra. Necessary and suf-ficient conditions for an i-v fuzzy set to be an i-v fuzzy BCI-subalgebra are stated. A way to make a new i-v fuzzy BCI-subalgebra from old one is given. The images and inverse images of i-v fuzzy BCI-subalgebras are defined, and how the images or inverse images of i-v fuzzy BCI-subalgebras become i-v fuzzy BCI-subalgebras is studied.

2000 Mathematics Subject Classiﬁcation. Primary 06F35, 03B52.

1. Introduction. The notion of BCK-algebras was proposed by Iami and Iséki in 1966. In the same year, Iséki [2] introduced the notion of a BCI-algebra which is a gen-eralization of a BCK-algebra. Since then numerous mathematical papers have been written investigating the algebraicproperties of the BCK/BCI-algebras and their re-lationship with other universal structures including lattices and Boolean algebras. Fuzzy sets were initiated by Zadeh [3]. In [4], Zadeh made an extension of the concept of a fuzzy set by an interval-valued fuzzy set (i.e., a fuzzy set with an interval-valued membership function). This interval-valued fuzzy set is referred to as an i-v fuzzy set. In [4], Zadeh also constructed a method of approximate inference using his i-v fuzzy sets. In [1], Biswas deﬁned interval-valued fuzzy subgroups (i.e., i-v fuzzy sub-groups) of Rosenfeld’s nature, and investigated some elementary properties. In this paper, using the notion of interval-valued fuzzy set by Zadeh, we introduce the con-cept of an interval-valued fuzzy BCI-subalgebra (brieﬂy, i-v fuzzy BCI-subalgebra) of a BCI-algebra, and study some of their properties. Using an i-v level set of an i-v fuzzy set, we state a characterization of an i-v fuzzy BCI-subalgebra. We prove that every BCI-subalgebra of a BCI-algebraX can be realized as an i-v level BCI-subalgebra of an i-v fuzzy BCI-subalgebra ofX. In connection with the notion of homomorphism, we study how the images and inverse images of i-v fuzzy BCI-subalgebras become i-v fuzzy BCI-subalgebras.

2. Preliminaries. In this section, we include some elementary aspects that are nec-essary for this paper.

Recall that aBCI-algebrais an algebra(X,∗,0)of type(2,0)satisfying the following axioms:

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(III) x∗x=0,and

(IV) x∗y=0 andy∗x=0 implyx=y, for everyx,y,z∈X.

Note that the equality 0∗(x∗y)=(0∗x)∗(0∗y) holds in a BCI-algebra. A non-empty subsetS of a BCI-algebraXis called aBCI-subalgebra ofX ifx∗y∈S wheneverx,y∈S. A mappingf:X→Y of BCI-algebras is called ahomomorphismif f (x∗y)=f (x)∗f (y)for allx,y∈X.

We now review some fuzzy logic concepts. Let X be a set. Afuzzy setinX is a functionµ:X→[0,1]. Letfbe a mapping from a setXinto a setY. Letνbe a fuzzy set inY. Then theinverse imageofν, denoted byf−1_[ν]_{, is the fuzzy set in}_X_deﬁned byf−1_[ν](x)₌_{ν(f (x))}_{for all}_x_∈_X_{. Conversely, let}_µ_{be a fuzzy set in}_X_{. The}_image ofµ, written asf [µ], is a fuzzy set inY deﬁned by

f [µ](y)=         

sup

z∈f−1_(y)µ(z) iff

−1_(y)_{= ∅}_,

0 otherwise,

(2.1)

for ally∈Y, wheref−1_(y)_{= {}_x_|_{f (x)}₌_y_}_.

Aninterval-valued fuzzy set(brieﬂy,i-v fuzzy set)Adeﬁned onXis given by

A=x,µL

A(x),µUA(x) , ∀x∈X

brieﬂy, denoted byA=µL

A,µAU , (2.2)

whereµL

AandµUA are two fuzzy sets inXsuch thatµLA(x)≤µUA(x)for allx∈X. Let ¯µA(x)=[µLA(x),µUA(x)],∀x∈Xand letD[0,1]denotes the family of all closed subintervals of[0,1]. IfµL

A(x)=µUA(x)=c, say, where 0≤c≤1, then we have ¯µA(x)= [c,c]which we also assume, for the sake of convenience, to belong toD[0,1]. Thus ¯

µA(x)∈D[0,1],∀x∈X, and therefore the i-v fuzzy setAis given by

A=x,µ¯A(x) , ∀x∈X, where ¯µA:X →D[0,1]. (2.3)

Now let us define what is known asrefined minimum(briefly, rmin) of two elements inD[0,1]. We also define the symbols “≥”, “≤”, and “=” in case of two elements in D[0,1]. Consider two elementsD1:=[a1,b1]andD2:=[a2,b2]∈D[0,1]. Then

rminD1,D2 =mina1,a2,minb1,b2;

D1≥D2 if and only ifa1≥a2, b1≥b2; (2.4)

and similarly we may haveD1≤D2andD1=D2.

Deﬁnition2.1. A fuzzy setµin a BCI-algebraXis called afuzzy BCI-subalgebra ofXifµ(x∗y)≥minµ(x),µ(y)for allx,y∈X.

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Proposition 3.1. Let f be a homomorphism from a BCI-algebra X into a BCI-algebraY. If ν is a fuzzy BCI-subalgebra of Y, then the inverse image f−1_[ν]_of_ν is a fuzzy BCI-subalgebra ofX.

Proof. For anyx,y∈X, we have

f−1_[ν](x_∗_y)₌_ν_{f (x}_∗_y) ₌_ν_{f (x)}_∗_{f (y)} ≥minνf (x) ,νf (y)

=minf−1_[ν](x),f−1_[ν](y)_.

(3.1)

Hencef−1_[ν]_{is a fuzzy BCI-subalgebra of}_X_.

Proposition3.2. Letf:X→Y be a homomorphism between BCI-algebrasXand Y. For every fuzzy BCI-subalgebraµofX, the imagef [µ]ofµis a fuzzy BCI-subalgebra ofY.

Proof. We ﬁrst prove that

f−1_y

1 ∗f−1y2 ⊆f−1y1∗y2 (3.2)

for all y1,y2∈Y. For, if x∈f−1(y1)∗f−1(y2),then x =x1∗x2 for somex1∈

f−1_(y

1) and x2 ∈ f−1(y2). Since f is a homomorphism, it follows that f (x) =

f (x1∗x2)=f (x1)∗f (x2)=y1∗y2so thatx∈f−1(y1∗y2). Hence (3.2) holds. Now lety1,y2∈Ybe arbitrarily given. Assume thaty1∗y2∉Im(f ). Thenf [µ](y1∗y2)= 0. But ify1∗y2∉Im(f ),that is,f−1(y1∗y2)= ∅, thenf−1(y1)= ∅orf−1(y2)= ∅ by (3.2). Thusf [µ](y1)=0 orf [µ](y2)=0,and so

f [µ]y1∗y2 =0=minf [µ]y1 ,f [µ]y2 . (3.3)

Suppose thatf−1_y

1∗y2 = ∅. Then we should consider the two cases:

f−1_y

1 = ∅ or f−1y2 = ∅, (3.4)

f−1_y

1 = ∅ and f−1y2 = ∅. (3.5)

For the case (3.4), we havef [µ]y1 =0 orf [µ]y2 =0,and so

f [µ]y1∗y2 ≥0=minf [µ]y1 ,f [µ]y2 . (3.6)

Case (3.5) implies, from (3.2), that

f [µ]y1∗y2 = sup

z∈f−1_(y₁_∗_y₂₎µ(z)≥_z_∈_f−1_(ysup₁₎_∗_f−1_(y₂₎µ(z)

= sup

x1∈f−1(y1), x2∈f−1(y2)

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Sinceµis a fuzzy BCI-subalgebra ofX, it follows from the deﬁnition of a fuzzy BCI-subalgebra that

f [µ]y1∗y2 ≥ sup x1∈f−1(y1), x2∈f−1(y2)

minµx1 ,µx2

= sup x1∈f−1(y1)

 _min

  _x sup

2∈f−1(y2)

µx1 ,µx2      = sup x1∈f−1(y1)

 _min

 

µx1 , sup x2∈f−1(y2)

µx2      = sup x1∈f−1(y1)

minµx1 ,f [µ]y2

=min   _x sup

1∈f−1(y1)

µx1 ,f [µ]y2   

=minf [µ]y1 ,f [µ]y2 .

(3.8)

Hencef [µ](y1∗y2)≥minf [µ](y1),f [µ](y2)for ally1,y2∈Y. This completes the proof.

Deﬁnition 3.3. An i-v fuzzy set A in X is called aninterval-valued fuzzy BCI-subalgebra(brieﬂy,i-v fuzzy BCI-subalgebra) ofXif

¯

µA(x∗y)≥rminµ¯A(x),µ¯A(y) ∀x,y∈X. (3.9)

Example3.4. LetX=0,a,b,cbe a BCI-algebra with the following Cayley table:

Table3.1.

∗ 0 a b c

0 0 c 0 a

a a 0 a c

b b c 0 a

c c a c 0

let an i-v fuzzy setAdeﬁned onXbe given by

¯ µA(x)=

      

[0.2,0.8] ifx∈ {0,b},

[0.1,0.7] otherwise. (3.10)

It is easy to check thatAis an i-v fuzzy BCI-subalgebra ofX.

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Proof. For everyx∈X, we have

¯

µA(0)=µ¯A(x∗x)≥rminµ¯A(x),µ¯A(x) =rminµL

A(x),µAU(x)

,µL

A(x),µUA(x)

=µL

A(x),µAU(x)

=µ¯A(x),

(3.11)

this completes the proof.

Theorem3.6. LetAbe an i-v fuzzy BCI-subalgebra ofX. If there is a sequence{xn} inXsuch that

lim n→∞µ¯A

xn =[1,1], (3.12)

thenµ¯A(0)=[1,1].

Proof. Since ¯µA(0)≥µ¯A(x)for allx∈X, we have ¯µA(0)≥µ¯A(xn)for every posi-tive integern. Note that

[1,1]≥µ¯A(0)≥_nlim_→∞µ¯Axn =[1,1]. (3.13)

Hence ¯µA(0)=[1,1].

Theorem3.7. An i-v fuzzy setA=µL A,µAU

inXis an i-v fuzzy BCI-subalgebra of Xif and only ifµL

AandµAUare fuzzy BCI-subalgebras ofX.

Proof. Suppose thatµL

AandµUAare fuzzy BCI-subalgebras ofX. Letx,y∈X. Then

¯

µA(x∗y)=µAL(x∗y),µUA(x∗y)

≥minµL

A(x),µAL(y)

,minµU

A(x),µAU(y)

=rminµL

A(x),µUA(x),µLA(y),µUA(y) =rminµ¯A(x),µ¯A(y).

(3.14)

HenceAis an i-v fuzzy BCI-subalgebra ofX.

Conversely, assume thatAis an i-v fuzzy BCI-subalgebra ofX. For anyx,y∈X, we have

µL

A(x∗y),µUA(x∗y)

=µ¯A(x∗y)≥rminµ¯A(x),µ¯A(y) =rminµL

A(x),µAU(x),µAL(y),µUA(y) =minµL

A(x),µAL(y),minµUA(x),µUA(y).

(3.15)

It follows thatµL

A(x∗y)≥minµAL(x),µAL(y)andµUA(x∗y)≥minµUA(x),µAU(y). HenceµL

AandµAUare fuzzy BCI-subalgebras ofX.

Theorem3.8. LetAbe an i-v fuzzy set inX. ThenAis an i-v fuzzy BCI-subalgebra ofXif and only if the nonempty set

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We then call ¯UA;[δ1,δ2] thei-v level BCI-subalgebraofA.

Proof. Assume thatAis an i-v fuzzy BCI-subalgebra ofXand let[δ1,δ2]∈D[0,1] be such thatx,y∈U(A¯ ;[δ1,δ2]). Then

¯

µA(x∗y)≥rminµ¯A(x),µ¯A(y)≥rminδ1,δ2,δ1,δ2=δ1,δ2, (3.17) and sox∗y∈U(A¯ ;[δ1,δ2]). Thus ¯U(A;[δ1,δ2])is a BCI-subalgebra ofX.

Conversely, assume that ¯U(A;[δ1,δ2]) (= ∅)is a BCI-subalgebra of X for every

[δ1,δ2]∈D[0,1]. Suppose there existx0,y0∈Xsuch that ¯

µAx0∗y0 <rminµ¯Ax0 ,µ¯Ay0 . (3.18) Let ¯µA(x0)=[γ1,γ2], ¯µA(y0)=[γ3,γ4],and ¯µA(x0∗y0)=[δ1,δ2].Then

δ1,δ2<rminγ1,γ2,γ3,γ4=minγ1,γ3,minγ2,γ4. (3.19) Henceδ1<min{γ1,γ3}andδ2<min{γ2,γ4}. Taking

λ1,λ2=1₂µ¯Ax0∗y0 +rminµ¯Ax0 ,µ¯Ay0 , (3.20) we obtain

λ1,λ2=1₂δ1,δ2+minγ1,γ3,minγ2,γ4 =

₁ 2

δ1+minγ1,γ3 ,₂1δ2+minγ2,γ4 .

(3.21)

It follows that

minγ1,γ3> λ1=1₂δ1+minγ1,γ3 > δ1, minγ2,γ4> λ2=1₂δ2+minγ2,γ4 > δ2

(3.22)

so that min{γ1,γ3},min{γ2,γ4} > [λ1,λ2] > [δ1,δ2] = µ¯A(x0∗y0). Therefore,

x0∗y0∉U¯A;[λ1,λ2] . On the other hand, ¯

µAx0 =γ1,γ2≥minγ1,γ3,minγ2,γ4>λ1,λ2, ¯

µAy0 =γ3,γ4≥minγ1,γ3,minγ2,γ4>λ1,λ2, (3.23) and so x0,y0∈U(A¯ ;[λ1,λ2]). It contradicts that ¯U(A;[λ1,λ2])is a BCI-subalgebra of X. Hence ¯µA(x∗y)≥rmin{µ¯A(x),µ¯A(y)}for all x,y ∈X. This completes the proof.

Theorem3.9. Every BCI-subalgebra ofX can be realized as an i-v level BCI-sub-algebra of an i-v fuzzy BCI-subBCI-sub-algebra ofX.

Proof. LetYbe a BCI-subalgebra ofXand letAbe an i-v fuzzy set onXdeﬁned by

¯ µA(x)=

      

α1,α2 ifx∈Y ,

[0,0] otherwise, (3.24)

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is an i-v fuzzy BCI-subalgebra ofX. Letx,y∈X. Ifx,y∈Y, thenx∗y∈Y and so

¯

µA(x∗y)=α1,α2=rminα1,α2,α1,α2=rminµ¯A(x),µ¯A(y). (3.25) Ifx,y∉Y, then ¯µA(x)=[0,0]=µ¯A(y)and thus

¯

µA(x∗y)≥[0,0]=rmin{[0,0],[0,0]} =rminµ¯A(x),µ¯A(y). (3.26)

Ifx∈Y andy∉Y, then ¯µA(x)=[α1,α2]and ¯µA(y)=[0,0]. It follows that ¯

µA(x∗y)≥[0,0]=rminα1,α2,[0,0]=rminµ¯A(x),µ¯A(y). (3.27) Similarly for the case x ∉Y and y ∈Y, we get ¯µA(x∗y)≥rmin{µ¯A(x),µ¯A(y)}. ThereforeAis an i-v fuzzy BCI-subalgebra ofX, and the proof is complete.

Theorem3.10. LetY be a subset ofX and letAbe an i-v fuzzy set onXwhich is given in the proof of Theorem 3.9. IfAis an i-v fuzzy BCI-subalgebra ofX, thenY is a BCI-subalgebra ofX.

Proof. Assume thatA is an i-v fuzzy BCI-subalgebra ofX. Let x,y∈Y. Then ¯

µA(x)=[α1,α2]=µ¯A(y), and so ¯

µA(x∗y)≥rminµ¯A(x),µ¯A(y)=rminα1,α2,α1,α2=α1,α2. (3.28) This implies thatx∗y∈Y. HenceY is a BCI-subalgebra ofX.

Theorem3.11. IfAis an i-v fuzzy BCI-subalgebra ofX, then the set

Xµ¯A:=

x∈X|µ¯A(x)=µ¯A(0) (3.29)

is a BCI-subalgebra ofX.

Proof. Letx,y∈Xµ¯A. Then ¯µA(x)=µ¯A(0)=µ¯A(y), and so ¯

µA(x∗y)≥rminµ¯A(x),µ¯A(y)=rminµ¯A(0),µ¯A(0)=µ¯A(0). (3.30)

Combining this and Lemma 3.5, we get ¯µA(x∗y)=µ¯A(0), that is,x∗y∈Xµ¯A. Hence

Xµ¯Ais a BCI-subalgebra ofX.

The following is a way to make a new i-v fuzzy BCI-subalgebra from old one.

Theorem3.12. For an i-v fuzzy BCI-subalgebraAofX, the i-v fuzzy setA∗ _in_X deﬁned byµ¯A∗(x)=µ¯_A(0∗x)for allx∈Xis an i-v fuzzy BCI-subalgebra ofX.

Proof. Since the equality 0∗(x∗y)=(0∗x)∗(0∗y)holds for all x,y∈X, we have

¯

µA∗(x∗y)=µ¯_A0∗(x∗y) =µ¯_A(0∗x)∗(0∗y) ≥rminµ¯A(0∗x),µ¯A(0∗y)

=rminµ¯A∗(x),µ¯_A∗(y)

(3.31)

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Deﬁnition3.13(Biswas [1]). Letfbe a mapping from a setXinto a setY. LetB be an i-v fuzzy set inY. Then theinverse imageofB, denoted byf−1_[B]_{, is the i-v} fuzzy set inXwith the membership function given by ¯µf−1_[B](x)=µ¯B(f (x))for all x∈X.

Lemma3.14(Biswas [1]). Letf be a mapping from a setXinto a set Y. Letm=

[mL_,mU_]_and_n₌_[nL_,nU_]_{be i-v fuzzy sets in}_X_and_Y_{, respectively. Then} (i) f−1_(n)₌_f−1_(nL_),f−1_(nU₎_,

(ii) f (m)=f (mL_{),f (m}U₎_.

Theorem3.15. Letfbe a homomorphism from a BCI-algebraXinto a BCI-algebra Y. IfBis an i-v fuzzy BCI-subalgebra ofY, then the inverse imagef−1_[B]_of_B_{is an i-v} fuzzy BCI-subalgebra ofX.

Proof. SinceB = [µL

B,µBU] is an i-v fuzzy BCI-subalgebra of Y, it follows from Theorem 3.7 thatµL

BandµBUare fuzzy BCI-subalgebras ofY. Using Proposition 3.1, we know thatf−1_[µL

B]andf−1[µUB]are fuzzy BCI-subalgebras ofX. Hence, by Lemma 3.14 and Theorem 3.7, we conclude thatf−1_[B]₌_f−1_[µL

B],f−1[µUB]

is an i-v fuzzy BCI-subalgebra ofX.

Deﬁnition3.16(Biswas [1]). Letf be a mapping from a setXinto a setY. LetA be an i-v fuzzy set inX. Then theimageofA, denoted byf [A], is the i-v fuzzy set in Y with the membership function deﬁned by

¯

µf [A](y)=         

rsup

z∈f−1_(y)µ¯A(z) iff

−1_(y)_{= ∅}_,_∀_y_∈_{Y ,}

[0,0] otherwise,

(3.32)

wheref−1_(y)_{= {}_x_|_{f (x)}₌_y_}_.

Theorem3.17. Letfbe a homomorphism from a BCI-algebraXinto a BCI-algebra Y. IfAis an i-v fuzzy BCI-subalgebra ofX, then the imagef [A]ofAis an i-v fuzzy BCI-subalgebra ofY.

Proof. Assume thatAis an i-v fuzzy BCI-subalgebra ofX. Note thatA=µL A,µUA is an i-v fuzzy BCI-subalgebra ofXif and only ifµL

AandµAUare fuzzy BCI-subalgebras ofX. It follows from Proposition 3.2 that the imagesf [µL

A]andf [µUA]are fuzzy BCI-subalgebras ofY. Combining Theorem 3.7 and Lemma 3.14, we conclude thatf [A]=

f [µL

A],f [µUA]

is an i-v fuzzy BCI-subalgebra ofY.

Acknowledgement. This work was supported by Korea Research Foundation Grant KRF-99-005-D00003.

References

[1] R. Biswas,Rosenfeld’s fuzzy subgroups with interval-valued membership functions, Fuzzy Sets and Systems63(1994), no. 1, 87–90. CMP 1 273 001. Zbl 844.20060.

[2] K. Iséki,An algebra related with a propositional calculus, Proc. Japan Acad.42 (1966), 26–29. MR 34#2433. Zbl 207.29304.

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[4] ,The concept of a linguistic variable and its application to approximate reasoning. I, Information Sci.8(1975), 199–249. MR 52#7225a. Zbl 397.68071.

Sung Min Hong: Department of Mathematics, Gyeongsang National University, Chinju660-701, Korea

E-mail address:smhong@nongae.gsnu.ac.kr

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

E-mail address:ybjun@nongae.gsnu.ac.kr

Seon Jeong Kim: Department of Mathematics, Gyeongsang National University, Chinju660-701, Korea

E-mail address:skim@nongae.gsnu.ac.kr

Gwang Il Kim: Department of Mathematics, Gyeongsang National University, Chinju 660-701, Korea