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