14

Fuzzy implicative ideals of BCK-algebras

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SOOCHOW JOURNAL OF MATHEMATICS

Volume ��� No� �� pp� ������ January ����

ON FUZZY IMPLICATIVE IDEALS OF BCK�ALGEBRAS

BY

YOUNG BAE JUN�� EUN HWAN ROH� AND SAMY M� MOSTAFA�

Abstract� In this paper we study further properties of fuzzy implicative ideals�

and construct a fuzzy characteristic implicative ideal in BCK�algebras�

�� Introduction

A BCK�algebra is an important class of logical algebras introduced by K�

Is�eki �see ���� ��� and �� and was extensively investigated by several researchers�

L� A� Zadeh ���� introduced the notion of fuzzy sets� At present this concept has

been applied to many mathematical branches� such as group� functional analysis�

probability theory� topology� and so on� In ��� O� G� Xi ���� applied this

concept to BCK�algebras� and since then Y� B� Jun and E� H� Roh ���� and Y�

B� Jun� S� M� Hong� J� Meng and X� L� Xin � � studied fuzzy commutative ideals

and fuzzy positive implicative ideals� respectively� J� Meng ��� introduced the

concept of implicative ideals in BCK�algebras and investigated the relationship

of it with the concepts of positive implicative ideals and commutative ideals� J�

Meng� Y� B� Jun and H� S� Kim ���� and S� M� Mostafa ���� fuzzi�ed the concept

of implicative ideals in BCK�algebras� independently� In this paper we study

further properties of fuzzy implicative ideals� and construct a fuzzy characteristic

implicative ideal in BCK�algebras�

Received March �� ����AMS Subject Classication� ��G��� ��F��� � A���Key words� �fuzzy� implicative ideal� level implicative ideal� �fuzzy� characteristic implicative

ideal�The �rst author was supported by the Basic Science Research Institute Program� Ministry of Edu�

cation� Korea� ����� Project No� BSRI��������

� YOUNG BAE JUN� EUN HWAN ROH AND SAMY M� MOSTAFA

�� Preliminaries

An algebra �X� �� � of type ��� � is said to be a BCK�algebra if it satis�es�

for all x� y� z � X�

�I ��x � y � �x � z � �z � y � ��

�II �x � �x � y � y � ��

�III x � x � ��

�IV � � x � ��

�V x � y � � and y � x � � imply x � y�

De�ne a binary relation � on X by letting x � y if and only if x�y � �� Then

�X�� is a partially ordered set with the least element �� In any BCK�algebra

X� the followings hold�

�� �x � y � z � �x � z � y�

�� x � y � x�

�� x � � � x�

�� �x � �y � z � z � x � y�

�� x � �x � �x � y � x � y�

� x � y implies x � z � y � z and z � y � z � x�

for all x� y� z � X� Throughout this paper X always means a BCK�algebra

without any speci�cation�

A mapping � � X � Y of BCK�algebras is called a homomorphism if ��x �

y � ��x � ��y for all x� y � X� Note that ��� � �� Let Aut�X denote

the set of all automorphisms of X� where automorphisms mean one�one and onto

homomorphisms�

A nonempty subset I of X is called an ideal of X� denoted by I C X� if it

satis�es

�I� � � I�

�I� x � y � I and y � I imply x � I for all x� y � X�

Any ideal I has the property� y � I and x � y imply x � I�

A non�empty subset I of X is called an implicative ideal of X� denoted by

I CI X� if it satis�es �I� and

�I� x � I whenever �x � �y � x � z � I and z � I� for all x� y� z � X�

ON FUZZY IMPLICATIVE IDEALS OF BCK�ALGEBRAS �

We now review fuzzy logic concepts� A fuzzy set in X is a function � � X �

��� ��� Given a fuzzy set � in X and � � ��� ��� let �� �� fx � Xj��x � �g� and

Im�� denote the image set of �� Let � be any mapping from a set X to a set

X � and let � be a fuzzy set in X� Then� the image ��� of � under �� is the

fuzzy set de�ned as follows�

����y �

�����

supx���� y�

��x if ����y �� ��

� if ����y � ��

If � is a homomorphism of BCK�algebras then ��� is called the homomorphic

image of � under �� A fuzzy set � in X has sup property if� for any subset

T � X� there exists x� � T such that

��x� � supt�T

��t�

A fuzzy set � in X is called a fuzzy ideal of X ����� De�nition ��� denoted

by � CF X� if

�F� ��� � ��x� �x � X�

�F� ��x � minf��x � y� ��yg� �x� y � X�

�� Fuzzy Implicative Ideals

We begin with the following de�nition�

De�nition ��������� De�nition ���� and ���� De�nition ���� A fuzzy set �

in X is called a fuzzy implicative ideal of X� written by � CFI X� if it satis�es

�F� and

�F� ��x � minf���x � �y � x � z� ��zg� �x� y� z � X�

Example ���� Let I CI X and let � be a fuzzy set in X de�ned by

��x �

�� if x � I�

� otherwise�

where � is a �xed number in ��� ��� We can easily check that �CFIX and �� � I�

Example ���� Let X �� f�� a� b� cg be a BCK�algebra with Cayley table

�Table � and Hasse diagram �Figure � as follows�

� YOUNG BAE JUN� EUN HWAN ROH AND SAMY M� MOSTAFA

a

b

c

� a b c

� � � �

a � a �

b b � �

c b a �

Table �

a

c

b

������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

Figure �

Let ��� ��� �� � ��� �� be such that �� � �� � ��� De�ne a fuzzy set � � X � ��� ��

by ��� � ��� ��a � ��� and ��b � ��c � ��� Then � CFI X �see ���� Example

�����

Remark ���� The example ��� shows that any implicative ideal of X can

be realized as a level implicative ideal of some fuzzy implicative ideal of X�

The following proposition will be used in the proof of Theorem ����

Proposition �������� Proposition �� Let � be a mapping from a set X to a

set X � and let � be a fuzzy set in X� Then for every � � ��� ���

���� ��

�����

�������

Lemma ��������� Theorem ���� and ���� Theorem ���� Let � be a fuzzy set

in X� Then � CFI X if and only if � for all � � ��� ��� either �� � � or �� CI X�

We then call �� the level implicative ideal of X�

S� M� Mostafa ���� proved the following result�

Result ��������� Theorem ��� Let � be a homomorphism from a BCK�

algebra X onto a BCK�algebra X� and let � CFI X with sup property� Then the

homomorphic image ��� of � under � is a fuzzy implicative ideal of X ��

Now we give another proof of Result ��� without using the sup property�

Theorem ��� Let � be as in Result ���� If � CFI X� then the homomorphic

image ��� of � under � is a fuzzy implicative ideal of X ��

ON FUZZY IMPLICATIVE IDEALS OF BCK�ALGEBRAS �

Proof� In view of Lemma �� it is su�cient to show that each nonempty

level subset of ��� is an implicative ideal of X �� Let ���� be a nonempty level

subset of ��� for every � � ��� ��� If � � � then ���� � X �� If � � ��� �� then

by Proposition ���

���� ��

�����

�������

Hence ������ is nonempty for each � � � � �� and so ���� is a nonempty

level subset of � for every � � � � �� Since � CFI X� it follows from Lemma ��

that ���� CI X� so that ������ CI X� because � is onto� Hence ���� being

an intersection of a family of implicative ideals is also an implicative ideal of X ��

ending the proof�

Theorem ��� Let I � X and let �I be a fuzzy set in X de�ned by� for all

x � X�

�I�x �

�� if x � I�

� otherwise�

Then �I CFI X if and only if I CI X�

Proof� Assume that �I CFI X� Since �I�� � �I�x for all x � X� clearly

�I�� � � and � � I� Let x� y� z � X be such that �x � �y � x � z � I and z � I�

Then �I��x � �y � x � z � � � ��z� It follows from �F� that

�I�x � minf�I��x � �y � x � z� ��zg � ��

and so �I�x � � or x � I� Hence I CI X�

Conversely assume that I CI X� Since � � I� we have �I�� � � � �I�x for

all x � X� Let x� y� z � X� We will verify that �I satis�es the condition �F�� If

�x � �y � x � z � I and z � I� then x � I because I CI X� Thus

�I��x � �y � x � z � �I�z � �I�x � ��

and so

�I�x � minf�I��x � �y � x � z� ��zg�

If �x � �y � x � z �� I and z �� I� then

�I��x � �y � x � z � �I�z � ��

� YOUNG BAE JUN� EUN HWAN ROH AND SAMY M� MOSTAFA

Hence

�I�x � minf�I��x � �y � x � z� ��zg�

If exactly one of �x � �y � x � z and z belongs to I� then exactly one of �I��x �

�y � x � z and �I�z is equal to �� Therefore

�I�x � minf�I��x � �y � x � z� ��zg�

This completes the proof�

Let � be a nonempty subset of ��� ���

Theorem ����� Let fI�j� � �g be a collection of implicative ideals of X

such that X �S���

I� and for all �� � � �� � � � if and only if I� I�� De�ne

a fuzzy set � in X by ��x �� supf� � �jx � I�g for all x � X� Then � CFI X�

Proof� Since � � I� for all � � �� clearly ��� � ��x for all x � X� Let

x� y� z � X be such that ���x� �y �x�z � m and ��z � n� where m�n � ��� ���

Without loss of generality we may assume that m � n� To Prove that � satis�es

the condition �F�� we consider the following three cases�

�i � � m� �ii m � � � n and �iii � � n�

Case �i implies that �x � �y � x � z � I� and z � I�� Since I� CI X� it follows

that x � I�� so that

��x � supf� � �jx � I�g � m � minf���x � �y � x � z� ��zg�

For the case �ii� we have �x � �y � x � z �� I� and z � I�� It follows that either

x � I� or x �� I�� If x � I�� then

��x � n � minf���x � �y � x � z� ��zg�

If x �� I� then x � I� � I� for some � �� and so

��x � m � minf���x � �y � x � z� ��zg�

Finally case �iii implies �x� �y �x�z �� I� and z �� I�� Then we also have either

x � I� or x �� I�� If x � I�� then obviously

��x � minf���x � �y � x � z� ��zg�

ON FUZZY IMPLICATIVE IDEALS OF BCK�ALGEBRAS �

If x �� I� then x � I� � I� for some � �� and thus

��x � m � minf���x � �y � x � z� ��zg�

This completes the proof�

Lemma ����� If � CFI X� then two level implicative ideals ��� and ���

�with �� � ��� of � are equal if and only if there is no x � X such that �� �

��x � ���

Proof� Assume that ��� � ��� for �� � �� and that there exists x � X

such that �� � ��x � ��� Then ��� � ��� � which is impossible�

Conversely suppose that there is no x � X such that �� � ��x � ��� Clearly

��� � ��� � If x � ��� � then ��x � �� and so ��x � ��� because ��x � ���

Hence x � ��� � This shows that ��� � ��� � This completes the proof�

Remark ����� Let � CFI X� As a consequence of Lemma ����� the level

implicative ideals of � form a chain� But ��� � ��x for all x � X� Therefore ��� �

where �� � ���� is the smallest level implicative ideal but not always ��� � f�g

as shown in the following example� and hence we have the chain�

��� ��� � � � ��r � X�

where �� � �� � � � � � �r�

Example ����� Let I be a nonzero implicative ideal of X and let � be the

fuzzy implicative ideal of X as in Example ���� Then Im�� � f�� �g� Further�

the two level implicative ideals of � are �� � X and �� � I� Thus we have that

��� � � but �� � I �� f�g�

Theorem ����� Let � CFI X with Im�� � f��� ��� � � � � �ng� where �� �

�� � � � � � �n� Then the family of implicative ideals ��i�i � �� �� � � � � n consti�

tutes all the level implicative ideals of ��

Proof� Let � � ��� �� and � �� Im��� If � � ��� then ��� � ��� Since

��� � X� therefore �� � X and �� � ��� � If �i � � � �i�� �� � i � n� �� then

there is no x � X such that � � ��x � �i��� It follows from Lemma ���� that

� YOUNG BAE JUN� EUN HWAN ROH AND SAMY M� MOSTAFA

�� � ��i�� � This shows that for any � � ��� ��� the level implicative ideal �� is in

f��i � i � �� �� � � � � ng�

The following example shows that two fuzzy implicative ideals of X may have

an identical family of level implicative ideals but the fuzzy implicative ideals may

not be equal�

Example ����� Let X and � be as in Example ���� Then the level implica�

tive ideals of � are ��� � f�g� f�� � f�� ag and ��� � X� Now let ��� ��� �� � ��� ��

be such that �� � �� � �� and �i �� �j for i � �� �� �� j � �� �� �� We note that

a fuzzy set � � X � ��� �� de�ned by ��� � ��� ��a � �� and ��b � ��c � ��

is a fuzzy implicative ideal of X� The level implicative ideals of � are ��� � f�g�

��� � f�� ag and ��� � X� Thus the two fuzzy implicative ideals � and � have

the same family of level implicative ideals� However it is clear that � is not equal

to ��

Lemma ����� Let � CFI X� If � and � belong to Im�� such that �� � ���

then � � ��

Proof� Assume that � �� �� say � � �� Then there is x � X such that

��x � � � �� and so x � �� and x �� ��� Thus �� �� ��� a contradiction� The

proof is complete�

Theorem ����� Let � CFI X and � CFI X with identical family of level

implicative ideals� If

Im�� � f��� ��� � � � � �mg and Im�� � f��� ��� � � � � �ng�

where �� � �� � � � � � �m and �� � �� � � � � � �n� then

�i m � n�

�ii ��i � ��i � i � �� � � � �m�

�iii if x � X such that ��x � �i then ��x � �i� i � �� � � � �m�

Proof� Using Theorem ����� we know that the only level implicative ideals

of � and � are ��i and ��i � respectively� Since � and � have the identical family

of level implicative ideals� it follows that m � n� Thus �i holds� Using Theorem

ON FUZZY IMPLICATIVE IDEALS OF BCK�ALGEBRAS

���� again� we get that f��� � � � � � ��mg � f��� � � � � � ��ng� and by Lemma ���� we

have

��� ��� � � � ��m � X and ��� ��� � � � ��n � X�

Hence ��i � ��i � i � �� � � � �m and �ii holds�

Let x � X be such that ��x � �i and let ��x � �j � Noticing that x � ��i �

that is� ��x � �i� we obtain �j � �i� Thus ��j � ��i � Since x � ��j and

��j � ��j � therefore x � ��j and so �i � ��x � �j � It follows that ��i � ��j �

By �ii� ��i � ��i � ��j � ��j � Consequently ��i � ��j � and by Lemma ��� we

conclude that �i � �j � Thus ��x � �i� The proof is complete�

Theorem ���� Let � and � be as in Theorem ����� Then � � � if and only

if Im���Im���

Proof� � It is clear�

�� Assume that Im�� � Im�� � f��� � � � � �ng where �� � � � � � �n� Let

x�� x�� � � � � xn be distinct elements of X such that ��xi � �i�� � i � n� By

Theorem �����iii� ��xi � �i�� � i � n� Since for any x � X there exists some

�i such that ��x � �i� therefore x � ��i � ��i � Hence ��x � �i � ��x� By the

same argument� we obtain ��x � ��x� Consequently ��x � ��x for all x � X�

This completes the proof�

Theorem ���� Let � be a fuzzy set in X with Im�� � f��� ��� � � � � �kg

where �� � �� � � � � � �k� If there exists a chain of implicative ideals of X �

I� I� � � � Ik � X

such that ��I�n � �n� where I�n � In � In��� I�� � �� n � �� �� � � � � k� then

� CFI X�

Proof� Since � � I�� we have ��� � �� � ��x for all x � X� We divide

into the following cases to prove that � satis�es �F�� If �x � �y � x � z � I�n and

z � I�n� then x � In because In CI X� Thus

��x � �n � minf���x � �y � x � z� ��zg�

If �x � �y � x � z �� I�n and z �� I�n� then the following four cases arise�

YOUNG BAE JUN� EUN HWAN ROH AND SAMY M� MOSTAFA

�� �x � �y � x � z � X � In and z � X � In�

�� �x � �y � x � z � In�� and z � In���

�� �x � �y � x � z � X � In and z � In���

�� �x � �y � x � z � In�� and z � X � In�

But� in either case� we know that

��x � minf���x � �y � x � z� ��zg�

If �x � �y � x � z � I�n and z �� I�n� then either z � In�� or z � X � In� It follows

that either x � In or x � X � In� Thus

��x � �n � minf���x � �y � x � z� ��zg�

If �x � �y � x � z �� I�n and z � I�n� then by similar process we have

��x � minf���x � �y � x � z� ��zg�

Summarizing the above results� we obtain

��x � minf���x � �y � x � z� ��zg

for all x� y� z � X� Consequently � satis�es the condition �F�� This completes

the proof�

Theorem ����� Let � CFI X with Im�� � f��� ��� � � � � �kg where �� �

�� � � � � � �k� Then

�i In � ��n CI X for n � �� �� � � � � k�

�ii ��I�n � �n for n � �� �� � � � � k� where I�n � In � In�� and I�� � ��

Proof� �i follows from Theorem �����

�ii Obviously ��I� � ��� Since ��I� � f��� ��g� for x � I�� we have ��x �

��� Hence ��I�� � ��� Repeating the above argument� we have ��I�n � �n for

n � �� �� � � � � k� This completes the proof�

Let F denote the class of fuzzy implicative ideals of a �nite BCK�algebra X�

De�ne a relation � on F as follows� for any �� � � F � � � � if and only if �

and � have the identical family of level implicative ideals� We note that by the

Example ����� two elements � and � of F may be such that � � � but � and �

need not be equal� The following lemma is easy to prove�

ON FUZZY IMPLICATIVE IDEALS OF BCK�ALGEBRAS �

Lemma ����� The relation � is an equivalence relation�

If � � F � let ��� denote the equivalence class determined by �� Since X is

�nite� the number of distinct level implicative ideals of X is �nite because each

level implicative ideal is an implicative ideal of X� From Remark ���� it follows

that the number of possible chains of level implicative ideals is also �nite� Since

each equivalence class is characterized completely by its chain of level implicative

ideals� we have the following theorem�

Theorem ����� If X is a �nite BCK�algebra� then the number of distinct

equivalence classes in F is �nite�

It follows from Theorem ���� that F can be written as a disjoint union of

�nite distinct equivalence classes�

F � ���� � � � � � ��k�� ��i� � ��j� � � �i �� j�

Let us denote fIi�j j� � j � �i� � � i � kg to be the chain of level implicative

ideals associated with the equivalence class ��i�� Let �� denote the collection

of all level implicative ideals of the members of F � It follows from Remark ���

that each implicative ideal of X is of the form Ii�j � De�ne a relation � on ��

as follows� Ii�j � Ir�k if and only if Ii�j � Ir�k � This relation � is clearly an

equivalence relation� If we denote the equivalence class containing Ii�j as �Ii�j��

then we have that the correspondence Ii�j � �Ii�j � is a ��� correspondence between

the implicative ideals of X and certain equivalence classes of ��� Hence we have

the following theorem�

Theorem ����� Let F be the collection of all fuzzy implicative ideals of a

�nite BCK�algebra X and let �� be the collection of all level implicative ideals

of the member of F � Then there is a ��� correspondence between the implicative

ideals of X and the equivalence classes of level implicative ideals under a suitable

equivalence relation on ���

�� Fuzzy Characteristic Implicative Ideals

De�nition ���� If � CFI X and � is a mapping from X into itself� we

de�ne a map ���� � X � ��� �� by �����x �� ����x for all x � X�

� YOUNG BAE JUN� EUN HWAN ROH AND SAMY M� MOSTAFA

Theorem ���� If � CFI X and � is an endomorphism of X� then

���� CFI X�

Proof� For any x� y� z � X� we have

�����x � ����x � ��� � ����� � ������

and

minf������x � �y � x � z� �����zg

�minf�����x � ���y ���x ���z� ����zg

� ����x �by �F��

� �����x�

This completes the proof�

De�nition ���� An implicative ideal I of X is said to be characteristic if

��I � I for all � � Aut�X�

De�nition ���� A fuzzy implicative ideal � of X is said to be fuzzy charac�

teristic if ���� � � for all � � Aut�X�

Theorem ���� If � is a fuzzy characteristic implicative ideal of X� then

each level implicative ideal of � is a characteristic implicative ideal of X�

Proof� Assume that � is a fuzzy characteristic implicative ideal of X� Let

� � Im��� � � Aut�X and x � ��� Then ����x � �����x � ��x � ��

and so ��x � ��� This implies ���� � ��� To prove the reverse inclusion�

let x � �� and let y � X be such that ��y � x� Then ��y � �����y �

����y � ��x � �� whence y � ��� which implies that x � ��y � �����

Thus �� � ����� Consequently ��� � � Im��� is a characteristic implicative

ideal of X� completing the proof�

Lemma ���� Let � CFI X and let x � X� Then ��x � � if and only if

x � �� and x �� �� for all � � ��

Proof� Straightforward�

ON FUZZY IMPLICATIVE IDEALS OF BCK�ALGEBRAS �

Now we provides the converse of Theorem ����

Theorem ���� If � CFI X in which every level implicative ideal is charac�

teristic� then � is a fuzzy characteristic implicative ideal of X�

Proof� Let x � X� � � Aut�X and ��x � �� It follows from Lemma ��

that x � �� and x �� �� for all � � �� Since ���� � �� by hypothesis� therefore

��x � ���� � �� and hence �����x � ����x � �� Let �����x � ��

Then we must have � � �� For if � � �� then ��x � �� � ����� Since � is one�

one� it follows that x � ��� which is a contradiction� Hence �����x � � � ��x

or ���� � �� showing that � is a fuzzy characteristic implicative ideal of X�

For given fuzzy implicative ideal� we will construct a new fuzzy implicative

ideal� Let � � � be a real number� We denote m�� m � ��� ��� shall mean the non�

negative ��th root in case � � �� We de�ne �� � X � ��� �� by ���x �� ���x�

for all x � X�

Theorem ��� If � CFI X� then �� CFI X and X�� � X� for every real

number � � �� where X� �� fx � Xj��x � ���g�

Proof� For any x� y� z � X� we have

���� � ����� � ���x� � ���x

and

���x � ���x�

� �minf���x � �y � x � z� ��zg�

�minf����x � �y � x � z�� ���z�g

�minf����x � �y � x � z� ���zg�

Thus �� CFI X� Also clearly we have X�� � X��

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��� K� Is�eki� An algebra related with a propositional calculus� Proc� Japan� Acad�� ��������

��������

�� YOUNG BAE JUN� EUN HWAN ROH AND SAMY M� MOSTAFA

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�������� �����

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��� J� Meng and Y� B� Jun� BCK�algebras� Kyung Moon Sa Co�� Seoul� Korea� ��� �

���� J� Meng� Y� B� Jun and H� S� Kim� Fuzzy implicative ideals of BCK�algebras� Fuzzy Sets

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���� S� M� Mostafa� Fuzzy implicative ideals in BCK�algebras� Fuzzy Sets and Systems� ��������

�������

���� O� G� Xi� Fuzzy BCK�algebras� Math� Japon�� ����������� ����� ��

���� L� A� Zadeh� Fuzzy sets� Inform� Control�� ������� �������

�Department of Mathematics Education� Gyeongsang National University� Chinju �������� Ko�

rea�

E�mail� ybjun�nongae�gsnu�ac�kr

�Department of Mathematics Education� Chinju National University of Education� Chinju ����

���� Korea�

E�mail� ehroh�ns�chinju�e�ac�kr

�Department of Mathematics� Faculty of Education� Ain Shams University� Roxy� Cairo� Egypt�