fuzzy ideals in bci-algebras

Fuzzy Sets and Systems 123 (2001) 227–237www.elsevier.com/locate/fss

Fuzzy ideals in BCI-algebras�

Yong Lin Liua ; ∗, Jie MengbaDepartment of Mathematics, Nanping Teachers College, Nanping 353000, Fujian, People’s Republic of China

bDepartment of Mathematics, Northwest University, Xian 710069, Shannxi, People’s Republic of China

Received 10 May 1999; received in revised form 3 November 1999; accepted 24 January 2000

Abstract

The aim of this paper is to introduce the notions of fuzzy BCI-positive implicative ideals and fuzzy BCI-implicative idealsof BCI-algebras and to investigate their properties. We give several characterizations of fuzzy BCI-positive implicative idealsand fuzzy BCI-implicative ideals. The extension theorems of fuzzy BCI-positive implicative ideals and fuzzy BCI-implicativeideals are obtained. We discuss the relations among various fuzzy ideals and show that a fuzzy set � of a BCI-algebra is fuzzyBCI-implicative ideal if and only if � is both fuzzy BCI-positive implicative ideal and fuzzy BCI-commutative ideal. Finally,we characterize implicative (positive implicative) BCI-algebras via fuzzy BCI-implicative (fuzzy BCI-positive implicative)ideals. c© 2001 Elsevier Science B.V. All rights reserved.

Keywords: Fuzzy (BCI-implicative; BCI-positive implicative; BCI-commutative) ideal of BCI-algebra; (implicative; positiveimplicative; commutative) BCI-algebra; (BCI-implicative; BCI-positive implicative; BCI-commutative) ideal of BCI-algebra

1. Introduction

The notions of BCK=BCI-algebras were introducedby Is8eki [4,5] and were extensively investigated bymany researchers. They are two important classesof logical algebras. The notion of fuzzy sets wasintroduced by Zadeh [21]. Chang [1] applied it tothe topological spaces. Rosenfeld [19] applied it to theelementary theory of groups. In 1991, Xi [20] appliedthe concept to BCK-algebras. From then on Jun, Menget al. [10,15] applied the concept to the ideals theoryof BCK=BCI-algebras. In this paper, we introduce

� Supported by the FJEC Foundation.∗ Corresponding author.E-mail addresses: [email protected] (Y.L. Liu),

[email protected] (J. Meng).

the notions of fuzzy BCI-positive implicative idealsand fuzzy BCI-implicative ideals of BCI-algebras,which are naturally generalizations for fuzzy posi-tive implicative ideals and fuzzy implicative idealsof BCK-algebras, respectively, and investigate theirproperties.

2. Preliminaries

An algebra (X ; ∗; 0) of type (2; 0) is called a BCI-algebra if it satisBes the following axioms:(1) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y)= 0,(2) (x ∗ (x ∗ y)) ∗ y=0,(3) x ∗ x=0,(4) x ∗ y=0 and y ∗ x=0 imply x=y,

0165-0114/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(00)00047 -6

228 Y.L. Liu, J. Meng / Fuzzy Sets and Systems 123 (2001) 227–237

for all x; y; z ∈X . In a BCI-algebra X , we can deBnea partial ordering 6 by putting x6y if and only ifx ∗ y=0.If a BCI-algebra X satisBes the identity:(5) 0 ∗ x=0,for all x∈X , then X is called a BCK-algebra.In any BCI-algebra X , the following hold:(6) (x ∗ z) ∗ (y ∗ z)6x ∗ y,(7) x ∗ (x ∗ (x ∗ y))= x ∗ y,(8) 0 ∗ (x ∗ y)= (0 ∗ x) ∗ (0 ∗ y),(9) x ∗ 0= x,(10) (x ∗ y) ∗ z=(x ∗ z) ∗ y,(11) x6y implies x ∗ z6y ∗ z and z ∗ y6z ∗ x,for all x; y; z ∈X .For more details of BCI-algebras we refer the reader

to Is8eki [4]. For no confusion, we shall add the term“BCI-” to some names of ideals of BCI-algebras.A nonempty subset I of a BCI-algebra X is called

an ideal of X if (I1) : 0∈ I; (I2) : x ∗ y∈ I and y∈ Iimply x∈ I . A nonempty subset I of a BCI-algebra Xis called a BCI-commutative ideal (i.e., commutativeideal) if it satisBes (I1) and (I3): (x∗y)∗z ∈ I and z ∈ Iimply x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))∈ I [13];a BCI-positive implicative ideal (i.e., weakly positiveimplicative ideal) if it satisBes (I1) and (I4) : ((x ∗ z) ∗z) ∗ (y ∗ z)∈ I and y∈ I imply x ∗ z ∈ I [12]; a BCI-implicative ideal (i.e., implicative ideal) if it satisBes(I1) and (I5): (((x ∗ y) ∗ y) ∗ (0 ∗ y)) ∗ z ∈ I and z ∈ Iimply x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))∈ I [11].

De�nition 2.1 (Zadeh [21]). Let S be a set. A fuzzyset in S is a function � : S→ [0; 1].

De�nition 2.2 (Das [3]). Let � be a fuzzy set ina set S. For t ∈ [0; 1], the set �t = {s∈ S | �(s)¿t} is called a level subset of �.

De�nition 2.3 (Xi [20]). Let X be a BCI-algebra. Afuzzy set � in X is said to be a fuzzy ideal of X if itsatisBes

(F1) �(0)¿�(x) for all x∈X;(F2) �(x)¿min{�(x ∗ y); �(y)} for all x; y∈X:

Proposition 2.4. Every fuzzy ideal � of a BCI-algebra X is order reversing.

Proof. It is similar to the proof of Lemma in [7] andomitted.

Proposition 2.5 (Jun and Meng [9]). Let � be afuzzy ideal of a BCI-algebra X . Then x ∗ y6zimplies �(x)¿min{�(y); �(z)} for all x; y; z ∈X .

3. Fuzzy BCI-positive implicative ideals ofBCI-algebras

De�nition 3.1. Let X be a BCI-algebra. A fuzzy set� in X is called a fuzzy BCI-positive implicative idealof X if it satisBes (F1) and

(F3) �(x ∗ z)¿min{�(((x ∗ z) ∗ z) ∗ (y ∗ z)); �(y)}

for all x; y; z ∈X:

Example 3.2. Let X = {0; 1; 2; 3} be a BCI-algebrawith Cayley table as follows:

∗ 0 1 2 3

0 0 0 0 31 1 0 0 32 2 2 0 33 3 3 3 0

DeBne � :X → [0; 1] by �(0)= 1; �(1)= �(2)= �(3)= t, where t ∈ [0; 1). By routine calculations give that� is a fuzzy BCI-positive implicative ideal of X .The following theorem gives a relation between

fuzzy ideals and fuzzy BCI-positive implicativeideals.

Theorem 3.3. Any fuzzy BCI-positive implicativeideal of a BCI-algebra X is a fuzzy ideal of X; butthe converse does not hold.

Proof. Assume that � is a fuzzy BCI-positive im-plicative ideal of X and put z=0 in (F3), we get�(x)¿min{�(x ∗ y); �(y)}. This means that � satis-Bes (F2). Combining (F1) � is a fuzzy ideal of X .To show the last half part, we see the following

example.

Y.L. Liu, J. Meng / Fuzzy Sets and Systems 123 (2001) 227–237 229

Example 3.4. Let X = {0; 1; 2} be a BCI-algebra withCayley table given by

∗ 0 1 2

0 0 0 01 1 0 02 2 1 0

DeBne � :X → [0; 1] by �(0)= t0 and �(1)= �(2)= t1, where t0; t1 ∈ [0; 1] and t0¿t1. It is easy toverify that � is a fuzzy ideal of X . But it is nota fuzzy BCI-positive implicative ideal of X since:�(2 ∗ 1)� min{�(((2 ∗ 1) ∗ 1) ∗ (0 ∗ 1)); �(0)}.

Next, we give characterizations of fuzzy BCI-positive implicative ideals of BCI-algebras.

Theorem 3.5. Let � be a fuzzy ideal of a BCI-algebra X . Then the following are equivalent:(i) � is a fuzzy BCI-positive implicative ideal of X;(ii) �((x ∗ y) ∗ z)¿�(((x ∗ z) ∗ z) ∗ (y ∗ z)) for all

x; y; z ∈X;(iii) �((x ∗ y) ∗ z)= �(((x ∗ z) ∗ z) ∗ (y ∗ z)) for all

x; y; z ∈X;(iv) �(x∗y)= �(((x∗y)∗y)∗(0∗y)) for all x; y∈X;(v) �(x∗y)¿�(((x∗y)∗y)∗(0∗y)) for all x; y∈X .

Proof. (i) ⇒ (ii) Suppose that � is a fuzzy BCI-positive implicative ideal of X . By (F3) and (F1)we have �((x ∗ y) ∗ z)¿min{�((((x ∗ y) ∗ z) ∗ z) ∗(0 ∗ z)); �(0)}= �((((x ∗ y) ∗ z) ∗ z) ∗ (0 ∗ z)). Since(((x ∗ y) ∗ z) ∗ z) ∗ (0 ∗ z)

= (((x ∗ y) ∗ z) ∗ z) ∗ ((y ∗ y) ∗ z)

= (((x ∗ z) ∗ z) ∗ y) ∗ ((y ∗ z) ∗ y)

6((x ∗ z) ∗ z) ∗ (y ∗ z);by Proposition 2.4, we get �((((x ∗ y) ∗ z) ∗ z) ∗(0 ∗ z)¿�(((x ∗ z) ∗ z) ∗ (y ∗ z)). Hence �((x ∗ y) ∗ z)¿�(((x ∗ z) ∗ z) ∗ (y ∗ z)) and (ii) holds.(ii) ⇒ (iii) Since ((x ∗ z) ∗ z) ∗ (y ∗ z)6(x ∗ z) ∗

y=(x ∗y) ∗ z; �((x ∗y) ∗ z)6�(((x ∗ z) ∗ z) ∗ (y ∗ z))by applying Proposition 2.4. Combining (ii), we get�((x ∗ y) ∗ z)= �(((x ∗ z) ∗ z) ∗ (y ∗ z)).(iii)⇒ (iv) Substituting 0 for y and y for z in (iii),

respectively, we have (iv).

(iv) ⇒ (v) Obviously.(v) ⇒ (i) Since

(((x ∗ y) ∗ y) ∗ (0 ∗ y)) ∗ (((x ∗ y) ∗ y) ∗ (z ∗ y))

6(z ∗ y) ∗ (0 ∗ y)6z;by Proposition 2.5, we obtain �(((x ∗ y) ∗ y) ∗(0 ∗ y))¿min{�(((x ∗ y) ∗ y) ∗ (z ∗ y)); �(z)}. From(v) �(x ∗ y)¿min{�(((x ∗ y) ∗ y) ∗ (z ∗ y)); �(z)}.Hence, � is a fuzzy BCI-positive implicative idealof X .

Proposition 3.6 (Jun et al. [7]). A fuzzy ideal � of aBCK-algebra is a fuzzy positive implicative if andonly if it satis>es �(x ∗ y)¿�((x ∗ y) ∗ y).Theorem 3.7. For a BCK-algebra X; a fuzzy ideal �is a fuzzy positive implicative ideal of X if and onlyif � satis>es one of Theorem 3:5.

Proof. Assume that � satisBes one of Theorem 3.5.Since X is a BCK-algebra, 0 ∗ y=0. By Theorem3.5(v), �(x∗y)¿�(((x∗y)∗y)∗(0∗y))= �((x∗y)∗y).Hence � is a fuzzy positive implicative ideal of a BCK-algebra X by Proposition 3.6.Conversely, if � is a fuzzy positive implica-

tive ideal of a BCK-algebra X; by Proposition 3.6,�(x ∗ y)¿�((x ∗ y) ∗ y)= �(((x ∗ y) ∗ y) ∗ (0 ∗ y))since 0 ∗ y=0 in a BCK-algebra X . Hence � is afuzzy BCI-positive implicative ideal of a BCI-algebraX by Theorem 3.5(v). This completes the proof.

Theorem 3.7 tells us that fuzzy BCI-positive im-plicative ideals of BCI-algebras is a generalization offuzzy positive implicative ideals of BCK-algebras.

Notation. Let � and � be fuzzy sets of a BCI-algebra X . We denote by �6� if �(x)6�(x) for allx∈X .Now, we give an extension theorem about fuzzy

BCI-positive implicative ideals.

Theorem 3.8. Let � and � be fuzzy ideals of a BCI-algebra X such that �6� and �(0)= �(0). If � is afuzzy BCI-positive implicative ideal of X; then so is �.

Proof. For any x; y∈X , by Theorem 3.5(v) we wantto show that �(x ∗ y)¿�(((x ∗ y) ∗ y) ∗ (0 ∗ y)).


Putting s=((x ∗ y) ∗ y) ∗ (0 ∗ y), then (((x ∗ s) ∗y) ∗ y) ∗ (0 ∗ y)= (((x ∗ y) ∗ y) ∗ (0 ∗ y)) ∗ s=0.Hence �((((x ∗ s) ∗ y) ∗ y) ∗ (0 ∗ y))= �(0)= �(0).Since � is a fuzzy BCI-positive implicative ideal ofX , and using Theorem 3.5(v), we get �((x ∗ s) ∗y)¿�((((x∗s)∗y)∗y)∗(0∗y))= �(0). Thus, �((x∗s)∗y)¿�((x∗s)∗y)¿�(0)¿�(s), i.e., �((x∗y)∗s)¿�(s).Since � is a fuzzy ideal, we have �(x∗y)¿min{�((x∗y)∗s); �(s)}= �(s)= �(((x∗y)∗y)∗(0∗y)). It meansthat � is a fuzzy BCI-positive implicative ideal of Xand completing the proof.

The following two theorems give the relationsbetween fuzzy BCI-positive implicative ideals andBCI-positive implicative ideals of BCI-algebras.

Theorem 3.9. A fuzzy set � of a BCI-algebra X isa fuzzy BCI-positive implicative ideal of X if andonly if; for all t ∈ [0; 1]; �t is either empty or a BCI-positive implicative ideal of X .

Proof. Let � be a fuzzy BCI-positive implicative idealin X and �t �= ∅ for t ∈ [0; 1]. Since �(0)¿�(x)¿t forsome x∈ �t , we get 0∈ �t . If ((x ∗ z) ∗ z) ∗ (y ∗ z)∈ �tand y∈ �t , then �(((x∗z)∗z)∗(y∗z))¿t and �(y)¿t.It follows from (F3) that �(x ∗ z)¿min{�(((x ∗z) ∗ z) ∗ (y ∗ z)); �(y)}¿t. Hence x ∗ z ∈ �t . Thismeans that �t is a BCI-positive implicative ideal ofX by (I4).Conversely, suppose that for each t ∈ [0; 1]; �t

is either empty or a BCI-positive implicative idealof X . For any x∈X , setting �(x)= t, then x∈ �t .Since �t(�= ∅) is a BCI-positive implicative ideal ofX , we have 0∈ �t and hence �(0)¿t= �(x). Thus�(0)¿�(x) for all x∈X . Now we prove that � satis-Bes (F3). If not, then there exist x′; y′; z′ ∈X such that�(x′ ∗ z′)¡ min{�(((x′ ∗ z′) ∗ z′) ∗ (y′ ∗ z′)); �(y′)}.Putting t0 = [�(x′ ∗ z′) + min{�(((x′ ∗ z′) ∗ z′) ∗(y′ ∗ z′)); �(y′)}]=2, then �(x′ ∗ z′) ¡ t0 ¡min{�(((x′ ∗ z′) ∗ z′) ∗ (y′ ∗ z′)); �(y′)}. Hence((x′ ∗ z′) ∗ z′) ∗ (y′ ∗ z′)∈ �t0 and y′ ∈ �t0 , butx′ ∗ z′ =∈ �t0 . Thus �t0 is not a BCI-positive implicativeideal of X . This is a contradiction with the hypothe-sis. Therefore, � is a fuzzy BCI-positive implicativeideal completing the proof.

Theorem 3.10. Let I be a BCI-positive implicativeideal of a BCI-algebra X . Then there exists a fuzzy

BCI-positive implicative ideal � of X such that �t = Ifor some t ∈ (0; 1].

Proof. DeBne � :X → [0; 1] by

�(x) =

{t if x∈ I;0 if x =∈ I;

where t is a Bxed number in (0; 1]. We show that � is afuzzy BCI-positive implicative ideal of X . Since I is apositive implicative ideal of X , if ((x∗z)∗z)∗(y∗z)∈ Iand y∈ I then x ∗ z ∈ I . Hence �(((x ∗ z) ∗ z) ∗ (y ∗z))= �(y)= �(x ∗ z)= t, thus �(x ∗ z)= min{�(((x ∗z) ∗ z) ∗ (y ∗ z)); �(y)}. If at least one of ((x ∗ z) ∗z) ∗ (y ∗ z) and y is not in I , then at least one of�(((x ∗ z) ∗ z) ∗ (y ∗ z)) and �(y) is 0. Therefore,�(x ∗ z)¿min{�(((x ∗ z) ∗ z) ∗ (y ∗ z)); �(y)}. Thismeans that � satisBes (F3). Since 0∈ I; �(0)= t¿�(x)for all x∈X and so � satisBes (F1). Thus, � is a fuzzyBCI-positive implicative ideal of X . It is clear that�t = I . We complete the proof.

4. Fuzzy BCI-implicative ideals of BCI-algebras

De�nition 4.1. Let X be a BCI-algebra. A fuzzy set� in X is called a fuzzy BCI-implicative ideal of X ifit satisBes (F1) and

(F4) �(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))))¿min{�((((x ∗ y) ∗ y) ∗ (0 ∗ y)) ∗ z); �(z)}for all x; y; z ∈X:

Example 4.2. Let X = {0; 1; 2; 3} be a BCI-algebrawith Cayley table as follows:

∗ 0 1 2 3

0 0 0 3 21 1 0 3 22 2 2 0 33 3 3 2 0

DeBne � :X → [0; 1] by �(0)= t0 and �(1)= �(2)=�(3)= t1, where t0; t1 ∈ [0; 1] and t0¿t1. Routine cal-culations give that � is a fuzzy implicative ideal of X .

Theorem 4.3. Any fuzzy BCI-implicative ideal of aBCI-algebra X is a fuzzy ideal of X; but the conversedoes not hold.


Proof. Suppose that � is a fuzzy BCI-implicativeideal of X and let y=0 in (F4). The we have

�(x) = �(x ∗ ((0 ∗ (0 ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ 0)))))

¿min{�((((x ∗ 0) ∗ 0) ∗ (0 ∗ 0)) ∗ z); �(z)}

= min{�(x ∗ z); �(z)}:Hence � satisBes (F2). Combining (F1), � is a fuzzyideal.The last part is shown by Example 3.2. We have

known that � is a fuzzy BCI-positive implicative idealof X . So � is a fuzzy ideal of X by Theorem 3.3.But it is not a fuzzy BCI-implicative ideal of X; since�(1 ∗ ((2 ∗ (2 ∗ 1)) ∗ (0 ∗ (0 ∗ (1 ∗ 2))))) = �(1) ��(0) = min{�((((1 ∗ 2) ∗ 2) ∗ (0 ∗ 2)) ∗ 0); �(0)}. Theproof is completed.

The characterizations of fuzzy BCI-implicativeideals of BCI-algebras are given by the followingtheorem.

Theorem 4.4. Let � be a fuzzy ideal of a BCI-algebra X. Then the following are equivalent:(i) � is a fuzzy BCI-implicative ideal of X;(ii) �(x ∗ ((y ∗ (y ∗x))∗ (0∗ (0∗ (x ∗y)))))¿�(((x ∗

y) ∗ y) ∗ (0 ∗ y)) for all x; y∈X .(iii) �(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))) =

�(((x ∗ y) ∗ y) ∗ (0 ∗ y)) for all x; y∈X .

Proof. (i)⇒ (ii) Letting z = 0 in (F4).(ii)⇒ (iii) Since

(((x ∗ y) ∗ y) ∗ (0 ∗ y)) ∗ (x ∗ ((y ∗ (y ∗ x))

∗ (0 ∗ (0 ∗ (x ∗ y)))))

= (((x ∗ (x ∗ ((y ∗ (y ∗ x))

∗ (0 ∗ (0 ∗ (x ∗ y)))))) ∗ y) ∗ y) ∗ (0 ∗ y)

6((((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∗ y)

∗y) ∗ (0 ∗ y) (by (2))

= (((0 ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∗ y)

∗ (0 ∗ y)

= ((((0 ∗ y) ∗ (0 ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))

∗y) ∗ (0 ∗ y)

= ((0 ∗ (0 ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∗ y

6((0 ∗ (x ∗ y)) ∗ (0 ∗ x)) ∗ y (by (1))

6(x ∗ (x ∗ y)) ∗ y = 0;

by Proposition 2.5, we get

�(((x ∗ y) ∗ y) ∗ (0 ∗ y))

¿min{�(x ∗ ((y ∗ (y ∗ x))

∗ (0 ∗ (0 ∗ (x ∗ y))))); �(0)}

= �(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))):Combining (ii) we get (iii).(iii)⇒ (i) Since � is a fuzzy ideal, we have

�(((x ∗ y) ∗ y) ∗ (0 ∗ y))

¿min{�((((x ∗ y) ∗ y) ∗ (0 ∗ y)) ∗ z); �(z)}:Combining (iii) we get (F4). Hence � is a fuzzy BCI-implicative ideal of X . The proof is complete.

De�nition 4.5 (Jun [6]). A fuzzy ideal � of a BCI-algebra X is said to be closed if �(0 ∗ x)¿�(x) for allx∈X .

De�nition 4.6 (Xi [20]). Let X be a BCK-algebra. Afuzzy set in X is said to be a fuzzy subalgebra of X if�(x ∗ y)¿min{�(x); �(y)} for all x; y in X .

Theorem 4.7. A closed fuzzy ideal � of a BCI-algebra X is a fuzzy subalgebra of X.

Proof. If � is a closed fuzzy ideal, then �(0 ∗ y)¿�(y). By (F2), we have

�(x ∗ y)¿min{�((x ∗ y) ∗ x); �(x)}

= min{�(0 ∗ y); �(x)}

¿min{�(y); �(x)}:This shows that � is a subalgebra of X and completesthe proof.


The following shows that the characterization offuzzy BCI-implicative ideals in Theorem 4.4 has asimple form if � is a closed fuzzy ideal.

Theorem 4.8. Let � be a closed fuzzy ideal of a BCI-algebra X. Then � is a fuzzy BCI-implicative idealof X if and only if

(a) �(x ∗ (y ∗ (y ∗ x)))

¿�(((x ∗ y) ∗ y) ∗ (0 ∗ y)) for all x; y∈X:

Proof. Suppose that � is a closed fuzzy BCI-implicative ideal of X . By DeBnition 4.5

�(0 ∗ (((x ∗ y) ∗ y) ∗ (0 ∗ y)))

¿�(((x ∗ y) ∗ y) ∗ (0 ∗ y)):Since

(x ∗ (y ∗ (y ∗ x))) ∗ (x ∗ ((y ∗ (y ∗ x))

∗ (0 ∗ (0 ∗ (x ∗ y)))))

6((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))

∗ (y ∗ (y ∗ x))

= 0 ∗ (0 ∗ (0 ∗ (x ∗ y)))

= 0 ∗ (x ∗ y)and

(b) 0 ∗ (((x ∗ y) ∗ y) ∗ (0 ∗ y))

= ((0 ∗ (x ∗ y)) ∗ (0 ∗ y)) ∗ (0 ∗ (0 ∗ y))

= ((0 ∗ (0 ∗ (0 ∗ y))) ∗ (x ∗ y)) ∗ (0 ∗ y)

= ((0 ∗ y) ∗ (x ∗ y)) ∗ (0 ∗ y)

= 0 ∗ (x ∗ y);we get

(x ∗ (y ∗ (y ∗ x))) ∗ (x ∗ ((y ∗ (y ∗ x))

∗ (0 ∗ (0 ∗ (x ∗ y)))))

60 ∗ (((x ∗ y) ∗ y) ∗ (0 ∗ y)):

By Proposition 2.5. We have

�(x ∗ (y ∗ (y ∗ x)))

¿min{�(x ∗ ((y ∗ (y ∗ x))

∗ (0 ∗ (0 ∗ (x ∗ y)))));

�(0 ∗ (((x ∗ y) ∗ y) ∗ (0 ∗ y))}:Using Theorem 4.4(ii) �(x ∗ (y ∗ (y ∗ x)))¿min{�(((x ∗ y)∗y)∗(0∗y)), �(0∗(((x∗y)∗y)∗(0∗y))} = �(((x∗y)∗y)∗(0∗y)). That is � satisBes (a).Conversely, assume that � is a closed fuzzy ideal

satisfying (a). We now prove that Theorem 4.4(ii)holds. Because

(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))))

∗ (x ∗ (y ∗ (y ∗ x)))

6(y ∗ (y ∗ x)) ∗ ((y ∗ (y ∗ x))

∗ (0 ∗ (0 ∗ (x ∗ y))))

60 ∗ (0 ∗ (x ∗ y))

= 0 ∗ (0 ∗ (((x ∗ y) ∗ y) ∗ (0 ∗ y))) (by (b));

it follows from Proposition 2.5 and (a) that

�(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))))

¿min{�(x ∗ (y ∗ (y ∗ x)));

�(0 ∗ (0 ∗ (((x ∗ y) ∗ y) ∗ (0 ∗ y))))}

¿min{�(((x ∗ y) ∗ y) ∗ (0 ∗ y));

�(0 ∗ (0 ∗ (((x ∗ y) ∗ y) ∗ (0 ∗ y))))}

= �(((x ∗ y) ∗ y) ∗ (0 ∗ y)):Hence, � is a fuzzy BCI-implicative ideal completingthe proof.

Similar to Theorems 3.9 and 3.10, we have the fol-lowing two theorems, which display close relationsbetween the BCI-implicative ideals and the fuzzy BCI-implicative ideals.


Theorem 4.9. A fuzzy set � of a BCI-algebra Xis a fuzzy BCI-implicative ideal of X if and onlyif; for every t ∈ [0; 1]; �t is either empty or a BCI-implicative ideal of X.

Proof. It is similar to the proof of Theorem 3.9 andis omitted.

Theorem 4.10. Let I be a BCI-implicative ideal ofa BCI-algebra X. Then there exists a fuzzy BCI-implicative ideal � of X such that �t = I for somet ∈ (0; 1].


Proposition 4.11 (Liu and Meng [11]). If I is aBCI-implicative ideal of a BCI-algebra X; then everyclosed ideal A of X; A⊇ I; is BCI-implicative.

Proposition 4.12 (Jun [6]). A fuzzy set � of a BCI-algebra X is a closed fuzzy ideal of X if and only if;for every t ∈ [0; 1] �t is either empty or a closed idealof X.

Next we give an extension theorem of fuzzy BCI-implicative ideals.

Theorem 4.13. If � is a fuzzy BCI-implicative idealof a BCI-algebra X; then every closed fuzzy ideal� of X; �6� and �(0)= �(0); is also fuzzy BCI-implicative.

Proof. By Theorem 4.9 it suOces to show thatfor any t ∈ [0; 1], �t is either empty or a BCI-implicative ideal of X . If �t �= ∅, then �t �= ∅ and�t ⊆ �t as �6� and �(0)= �(0). Since � is afuzzy BCI-implicative ideal of X , it follows fromTheorem 4.9 that �t is a BCI-implicative ideal of X .By the hypothesis � is a closed fuzzy ideal of X , itfollows from Proposition 4.12 that �t is a closed idealof X . Applying Proposition 4.11 we have that �t is aBCI-implicative ideal of X completing the proof.

In the following, we discuss the relations betweenfuzzy BCI-implicative ideals and other fuzzy ideals inBCI-algebras.

Proposition 4.14 (Jun and Meng [9]). A fuzzy ideal� of a BCI-algebra is fuzzy BCI-commutative(i.e.; fuzzy commutative) if and only if it satis>es�(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗y)))))¿�(x ∗y).

Theorem 4.15. Any fuzzy BCI-implicative ideal of aBCI-algebra X is a fuzzy BCI-commutative ideal ofX; but the converse does not hold.

Proof. Suppose � is a fuzzy BCI-implicative ideal ofa BCI-algebra X . By Theorem 4.3, � is a fuzzy ideal.For any x; y∈X , from Theorem 4.4 (ii) we have�(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗y)))))¿ �(((x ∗y)∗y) ∗ (0 ∗y)). Since ((x ∗y) ∗y) ∗ (0 ∗y)6x ∗y,then �(((x ∗y) ∗y) ∗ (0 ∗y))¿�(x ∗y) as Proposi-tion 2.4. Hence we get �(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗(x ∗y)))))¿�(x ∗y). By Proposition 4.14, � is afuzzy BCI-commutative ideal. To show the last halfpart, we consider the following example.

Example 4.16. Let X = {0; 1; 2; 3} be a BCI-algebrawith Cayley table given by

∗ 0 1 2 3

0 0 0 0 01 1 0 0 12 2 1 0 23 3 3 3 0

DeBne � :X → [0; 1] by �(0)= 1; �(1)= �(2)= 12

and �(3)= 0. Routine calculations give that � is afuzzy BCI-commutative ideal, but it is not a fuzzyBCI-implicative ideal because �(2 ∗ ((1 ∗ (1 ∗ 2)) ∗(0 ∗ (0 ∗ (2 ∗ 1)))))= �(1)� �(0)= min{�((((2 ∗ 1)∗ 1) ∗ (0 ∗ 1)) ∗ 0); �(0)}. The proof is completed.

Theorem 4.17. Any fuzzy BCI-implicative ideal ofa BCI-algebra X is a fuzzy BCI-positive implicativeideal of X; but the converse need not be true.

Proof. Assume that � is a fuzzy BCI-implicativeideal of a BCI-algebra X . By Theorem 4.3, � is a fuzzyideal. For any x; y∈X by Theorem 4.4(ii) we obtain�(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗y)))))¿ �(((x ∗y)∗y) ∗ (0 ∗y)). Since((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))) ∗ y

= (0 ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y)))


6(0 ∗ (x ∗ y)) ∗ (y ∗ x)

= ((0 ∗ x) ∗ (0 ∗ y)) ∗ (y ∗ x)

= ((0 ∗ (y ∗ x)) ∗ x) ∗ (0 ∗ y)

= (((0 ∗ y) ∗ (0 ∗ x)) ∗ x) ∗ (0 ∗ y)

= (0 ∗ (0 ∗ x)) ∗ x = 0;

We have (y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗y)))6y and sox ∗ ((y ∗ (y ∗ x)) ∗ (0∗(0 ∗ (x ∗y))))¿x∗y. Hence�(x ∗y)¿�(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗y))))) asProposition 2.4. Thus, we get �(x ∗y)¿�(((x ∗y) ∗y)∗ (0 ∗y)). It follows from Theorem 3.5(v) that � is afuzzy BCI-positive implicative ideal.The last part is shown by Example 3.2. We have

known that � is a fuzzy BCI-positive implicativeideal. But it is not a fuzzy BCI-implicative idealas �(1 ∗ ((2 ∗ (2 ∗ 1)) ∗ (0 ∗ (0 ∗ (1 ∗ 2)))))= �(1)��(0)= min{�((((1 ∗ 2) ∗ 2)∗(0∗2))∗0); �(0)}. Thiscompletes the proof.

The following theorem gives another necessary andsuOcient condition for fuzzy BCI-implicative ideals.

Theorem 4.18. A fuzzy set � of a BCI-algebra is afuzzy BCI-implicative ideal if and only if it is botha fuzzy BCI-positive implicative ideal and a fuzzyBCI-commutative ideal.

Proof. Necessity: By Theorems 4.15 and 4.17.Su?ciency: Assume that � is both a fuzzy

BCI-positive implicative ideal and a fuzzy BCI-commutative ideal of a BCI-algebra X . ByTheorem 3.3, � is a fuzzy ideal. For any x; y∈X ,by Proposition 4.14 �(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x∗y)))))¿�(x ∗y) and by Theorem 3.5(v) �(x ∗y)¿�(((x ∗y) ∗y) ∗ (0 ∗y)). Hence �(x ∗ ((y ∗ (y ∗ x))∗ (0 ∗ (0 ∗ (x ∗y)))))¿�(((x ∗y) ∗y) ∗ (0 ∗y)). Itfollows from Theorem 4.4(ii) that � is a fuzzy BCI-implicative ideal. The proof is completed.

Any BCK-algebra is a BCI-algebra. Next, we showthat in a BCK-algebra X , the concept of fuzzy implica-tive ideals of a BCK-algebra is equivalent to the con-cept of fuzzy BCI-implicative ideals of a BCI-algebraX . We Brst state some conclusions.

Proposition 4.19 (Liu and Meng [11]). In a BCK-algebra X; a nonempty subset I is an implicativeideal of a BCK-algebra X if and only if it is anBCI-implicative ideal of a BCI-algebra X .

Proposition 4.20 (Meng et al. [15]). A fuzzy set � ofa BCK-algebra X is a fuzzy implicative ideal of X ifand only if; for all t ∈ [0; 1] �t is either empty or animplicative ideal of X .

Theorem 4.21. Let X be a BCK-algebra. A fuzzy set� is a fuzzy implicative ideal of a BCK-algebra Xif and only if it is fuzzy BCI-implicative ideal of aBCI-algebra X .

Proof. � is a fuzzy implicative ideal of a BCK-algebraX , by Proposition 4.20, if and only if for all t ∈ [0; 1] �tis either empty or an implicative ideal of a BCK-algebra X , if and only if for all t ∈ [0; 1] �t is eitherempty or a BCI-implicative ideal of a BCI-algebra Xby Proposition 4.19, if and only if � is a fuzzy BCI-implicative ideal of a BCI-algebra X by Theorem 4.9.This completes the proof.

Proposition 4.22 (Jun and Meng [8]). A fuzzy ideal� of a BCI-algebra is a fuzzy p-ideal if and only if�(x)¿�(0 ∗ (0 ∗ x)) for all x∈X .

Theorem 4.23. Any fuzzy p-ideal is a fuzzy BCI-implicative ideal; but the converse is not true.

Proof. Let � be a fuzzy p-ideal of a BCI-algebraX . Then � is a fuzzy ideal [8]. In order toprove that � is a fuzzy BCI-implicative ideal,from Theorem 4.4(ii) it suOces to show that�(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗y)))))¿ �(((x ∗y)∗y) ∗ (0 ∗y)). Since

(0 ∗ (0 ∗ (x ∗ y))) ∗ (((x ∗ y) ∗ y) ∗ (0 ∗ y))

= (0 ∗ (((x ∗ y) ∗ y) ∗ (0 ∗ y))) ∗ (0 ∗ (x ∗ y))

= (((0 ∗ (x ∗ y)) ∗ (0 ∗ y)) ∗ (0 ∗ (0 ∗ y)))

∗ (0 ∗ (x ∗ y))

= (0 ∗ (0 ∗ y)) ∗ (0 ∗ (0 ∗ y)) = 0;


we get 0 ∗ (0 ∗ (x ∗y))6((x ∗y) ∗y) ∗ (0 ∗y). ByProposition 2.4 �(0 ∗ (0 ∗ (x ∗y)))¿�(((x ∗y) ∗y)∗ (0 ∗y)). Since

0 ∗ (0 ∗ (x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗ y))))))

= (0 ∗ (0 ∗ x)) ∗ ((0 ∗ (0 ∗ (y ∗ (y ∗ x))))

∗ (0 ∗ (0 ∗ (x ∗ y))))

6(0 ∗ (0 ∗ x)) ∗ ((0 ∗ (0 ∗ (y ∗ (y ∗ x))))

∗ (x ∗ y)) (by (2) and (11))

= (0 ∗ (0 ∗ x)) ∗ ((0 ∗ (x ∗ y))

∗ (0 ∗ (y ∗ (y ∗ x))))

= (0 ∗ (0 ∗ x)) ∗ (((0 ∗ x) ∗ (0 ∗ y)) ∗ ((0 ∗ y)

∗ ((0 ∗ y) ∗ (0 ∗ x))))

6(0 ∗ (0 ∗ x)) ∗ (((0 ∗ x) ∗ (0 ∗ y)) ∗ (0 ∗ x))

= (0 ∗ (0 ∗ x)) ∗ (0 ∗ (0 ∗ y))

= 0 ∗ (0 ∗ (x ∗ y));by Propositions 4.22 and 2.4 we obtain

�(x ∗ ((y ∗ (y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗y)))))

¿�(0 ∗ (0 ∗ (x ∗ ((y ∗ x)) ∗ (0 ∗ (0 ∗ (x ∗y)))))))

¿�(0 ∗ (0 ∗ (x ∗y)))

¿�(((x ∗y) ∗y) ∗ (0 ∗y)):Therefore � is a fuzzy BCI-implicative ideal. Theproof is complete.

Combining Theorems 4.23, 4.15 and 4.17, we havethe following corollaries.

Corollary 4.24. Any fuzzy p-ideal is a fuzzy BCI-commutative ideal; but the converse is not true.

Corollary 4.25. Any fuzzy p-ideal is a fuzzy BCI-positive implicative ideal; but the converse is not true.

Proposition 4.26 (Meng et al. [15]). A fuzzy ideal �of a BCK-algebra is fuzzy implicative if and onlyif it is both fuzzy commutative and fuzzy positiveimplicative.

Summary of Theorems 4.18, 4.21, 4.23, 3.7 andProposition 4.26, the relations of various fuzzy idealscan be described by a diagram as follows (The mark“A→B” denotes that fuzzy ideal A must be the fuzzyideal B):

5. Characterization of BCI-algebras by fuzzy ideals

Proposition 5.1 (Meng [14]). A fuzzy set � in aBCI-algebra X is a fuzzy ideal of X if and only if forevery t ∈ [0; 1]; �t is either empty or an ideal of X.

Proposition 5.2 (Liu and Zhang [12]). A BCI-alge-bra X is positive implicative (i.e.; weakly positiveimplicative) if and only if every ideal of X is BCI-positive implicative.

Now, we describe positive implicative BCI-algebrasby fuzzy BCI-positive implicative ideals.

Theorem 5.3. A BCI-algebra X is positive implica-tive if and only if every fuzzy ideal of X is fuzzyBCI-positive implicative.

Proof. Let X be a positive implicative BCI-algebra.If � is a fuzzy ideal, using Proposition 5.1, for eacht ∈ [0; 1], �t is either empty or an ideal of X . ByProposition 5.2, �t is either empty or a BCI-positiveimplicative ideal of X . Hence � is a fuzzy BCI-positiveimplicative ideal of X by Theorem 3.9.


Conversely, assume that every fuzzy ideal of X isfuzzy BCI-positive implicative. Let I be an ideal ofX . DeBne � :X → [0; 1] by

�(x) =

{1=2 if x∈ I;0 if x =∈ I:

It is easy to check that � is a fuzzy ideal of X . Hence �is a fuzzy BCI-positive implicative ideal of X . By The-orem 3.9, I = �1=2 is a BCI-positive implicative idealof X . This shows that every ideal of X is BCI-positiveimplicative. Applying Proposition 5.2, X is a positiveimplicative BCI-algebra. The proof is complete.

Proposition 5.4 (Liu and Meng [11]). A BCI-alge-bra X is implicative if and only if every closed idealof X is implicative.

Using Propositions 5.4, 4.12 and Theorem 4.9,we get the following theorem, which describes theimplicative BCI-algebras by the closed fuzzy BCI-implicative ideal. The proof is similar to that ofTheorem 5.3 and is omitted.

Theorem 5.5. A BCI-algebra X is implicative if andonly if every closed fuzzy ideal of X is fuzzy BCI-implicative.

Summarizing Proposition 5.4, Theorems 5.5 and4.18, we have:

Corollary 5.6. Let X be a BCI-algebra. The follow-ing are equivalent:(i) X is an implicative BCI-algebra.(ii) Every closed ideal of X is BCI-implicative.(iii) Every closed fuzzy ideal of X is fuzzy BCI-

implicative.(iv) Every closed fuzzy ideal of X is both fuzzy

BCI-commutative and fuzzy BCI-positiveimplicative.

Recently, Meng et al. [18] gave the construction ofa quotient BCK-algebra (X=�; ∗ ; �0) via a fuzzy ideal� of a BCK-algebra X . By the same way, we constructa quotient BCI-algebra (X=�; ∗ ; �0) via a fuzzy ideal �of a BCI-algebra X , where � �= 0, X=�= {�x | x∈X },�x = {y∈X |y∼ x} and x∼y if and only if�(x ∗y)¿0 and �(y ∗ x)¿0. We omit the details ofveriBcation.

Next, we investigate quotient positive implicative(commutative, implicative) BCI-algebras by fuzzyideals.

Proposition 5.7 (Chaudhry [2]). A BCI-algebra ispositive implicative (i.e.; weakly positive implicative)if and only if it satis>es x ∗y=((x ∗y) ∗y) ∗ (0 ∗y).

Theorem 5.8. If � is a fuzzy BCI-positive implica-tive ideal of a BCI-algebra X; then (X=�; ∗ ; �0) is apositive implicative BCI-algebra.

Proof. For any x; y∈X we put s= x ∗ (((x ∗y) ∗y) ∗(0 ∗y)). Then ((s ∗y) ∗y) ∗ (0 ∗y)= 0. Since � is afuzzy BCI-positive implicative ideal and by Theorem3.5(iv), we have �(s ∗y)= �(((s ∗y) ∗y) ∗ (0 ∗y))=�(0)¿0. That is �((x ∗y) ∗ (((x ∗y) ∗y) ∗ (0 ∗y)))¿0. On the other hand (((x ∗y) ∗y) ∗ (0 ∗y)) ∗ (x ∗y)= 0 implies �((((x ∗y) ∗y) ∗ (0 ∗y)) ∗ (x ∗y))=�(0)¿0. Hence �x ∗ y = �((x ∗ y) ∗ y) ∗ (0 ∗ y), i.e.,

�x ∗ �y =((�x ∗ �y) ∗ �y) ∗ (�0 ∗ �y):

By Proposition 5.7, X=� is a positive implicative BCI-algebra completing the proof.

Proposition 5.9 (Meng [13]). ABCI-algebra is com-mutative if and only if it satis>es x ∗ (x ∗y)=y ∗(y ∗ (x ∗ (x ∗y))).

Theorem 5.10. If � is a fuzzy BCI-commutativeideal of a BCI-algebra X; then (X=�; ∗ ; �0) is acommutative BCI-algebra.


De�nition 5.11 (Meng and Xin [16]). A BCI-algebraX is said to be implicative if (x ∗ (x ∗y)) ∗ (y ∗x)=y ∗ (y ∗ x) for all x; y∈X .

Proposition 5.12 (Meng and Xin [17]).A BCI-alge-bra is implicative if and only if it is both positiveimplicative and commutative.

Theorem 5.13. If � is a fuzzy BCI-implicative idealof a BCI-algebra X; then (X=�; ∗ ; �0) is an implica-tive BCI-algebra.


Proof. It is an immediate consequence of Theo-rems 4.8, 5.8, 5.10 and Proposition 5.12.

Acknowledgements

The authors would like to thank the referees fortheir valuable suggestions and comments.

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