The 3rd International Conference on Mathematics and Statistics (ICoMS-3)
Institut Pertanian Bogor, Indonesia, 5-6 August 2008
THE FUZZY VERSION OF THE FUNDAMENTAL THEOREM
OF SEMIGROUP HOMOMORPHISM
1,2
Karyati , 3Indah Emilia W, 4Sri Wahyuni, 5Budi Surodjo, 6Setiadji
1
Ph.D Student , Mathematics Department, Gadjah Mada University
Sekip Utara, Yogyakarta
2
Mathematics Education Department, Yogyakarta State University
Jl. Colombo No. 1, Karangmalang, Yogyakarta
3,4,5,6
Mathematics Department, Gadjah Mada University
Sekip Utara, Yogyakarta
e-mail: 1,2 yatiuny@yahoo.com, 3ind_wijayanti@yahoo.com, 4swahyuni@ugm.ac.id, ,5 surodjo_b@ugm.ac.id
Abstract . A semigroup is an algebra structure which is more general than a group. The fuzzy
version of the fundamental theorem of group homomorphism has been established by the
previous researchers. In this paper we are going to establish about the fuzzy version of the
fundamental theorem of semigroup homomorphism as a general form of group homomorphism.
Using the properties of level subset of a fuzzy subsemigroup, we have proved that the fuzzy
version of the fundamental theorem of semigroup homomorphism is held. If we have a surjective
semigroup homomorphism, then the fuzzy subsemigroup quotient is isomorphic with the
homomorphic image.
Keywords: Fuzzy subsemigroup, Level subset, Fuzzy subsemigroup quotient
1. Introduction
The basic concept of fuzzy sets was introduced by Zadeh since 1965 ( in Aktas; 2004). This concept
has developed in many areas including in algebra structure such as fuzzy subgroup, fuzzy subring, fuzzy
submodule and soon .
A fuzzy subgroup is constructed by generalized a group based on the concept of an ordinary set G
to a fuzzy set
on G , i.e. a mapping
:G
0,1 fulfills these axioms:
(i) for every x, y G ,
xy
min
x ,
y , (ii)for every x G ,
x
x 1
.
Asaad (1991) has
proved, if there is a surjective homomorphism group, then the quotient subgroup fuzzy is isomorphic to the
homomorphic image.
Every a group is a semigroup. So we can say that a semigroup is a generalization of a group. Based
on the set, we can generalize a group to a fuzzy subgroup which is defined as above. Kandasamy (2003)
introduced how to generalize a semigroup to a fuzzy subsemigroup. If S is a semigroup, so a fuzzy
subsemigroup is a mapping : S [0,1] such that fulfill this axiom: xy min
x , y for every
In this paper we are going to establish the fuzzy version of the fundamental theorem of semigroup
homomrphism.
x, y
S.
2. Theoritical Review
Let S be an ordinary non empty set,
is called a fuzzy subset of S if
is a mapping from S
into interval 0,1 . Asaad (1991), Kandasamy (2003) and Mordeson & Malik (1998) have introduced how to
construct a fuzzy subgroup and a fuzzy normal subgroup. They can prove that the level subset
and strong
t
level subset t , t 0, e of a fuzzy subset
is an ordinary subgroup of G , with e is an identity element
of a group G .
Ajmal (1994) defined a homomorphism of a fuzzy subgroup, establishes their properties related to
the homomorphism, constructs the fuzzy quotient subgroup and proves the fundamental theorem of group
homomorphism in fuzzy version.
Semigroup
A semigroup is a non empty set with an associative binary operation is defined into this set. A
semigruop S has an identity element if there is e S such that ex x and xe x . The set A S is
called subsemigroup if A is a semigroup of S relative to the same binary operation which is defined in S .
Lemma 2.1.( Howie :p.7) The non empty of the intersection of all subsemigroup S is a semigroup.
Proof:
Let S i be a subsemigroup of S , for i 1,2,3,... Take x, y  S i , so x, y Si for every i 1,2,3,... Since S i is a
i
subsemigroup of S for
 Si
1,2,3,..., then xy Si for every i 1,2,3,... The consequence is xy
i
 Si , hence
i
is a semigroup.
i

( xy)
Let S , S ' be semigroups. The mapping : S S ' is called a semigroup homomorphism if
is defined as : ker
( x) ( y) for every x, y S . The kernel of
( x, y) S S
( x)
( y) . It is a
congruency relation, so it divides S into congruency classes. The set of congruency classes denoted by
( xy) ker
S / ker . This set is a semigroup under the following operation: x ker . y ker
See the following diagram:
S
:S
S'
S'
I
(ker ) : S
(ker ) I
: S / ker
: semigroup homomorphism
S / ker
S'
: canonic mapping
: monomorphism with Im
This diagram commutes i.e.
S / ker
Im
 (ker ) I
If
is surjective, then Im S ' . The consequency is Im
S' , in other word is surjective. On the other
hand
is monomorphism, so
is isomorphism. If
and
are congruency on S with
, so the
mapping from S / onto S / is surjective and the following diagram commutes (Howie:p.23).
I
S
S/
I
S/
Lemma 2.2. If
:S
S ' is a semigroup homomorphism, then
(S ) is a subsemigroup of
S'
Proof: Let y, y' (S ) , so there is x, x' S such that y
( x' ) . The next we must prove that
(x) and y'
yy'
( S ) . Construct z xx' , so we have z S , such that:
( z)
( xx' )
( x) ( x' ) yy' . Since there is
z S such that ( z ) yy' , therefore yy' (S ) . Hence
(S ) is a subsemigroup of S ' .

2.2. Subsemigroup Fuzzy
A semigroup is the a simple algebra structure. The first we will give a definition of a fuzzy
subsemigroup and the other basic definitions which refer to Asaad (1991), Kandasamy (2003), Mordeson &
Malik (1998), Ajmal (1994), Shabir (2005).
Definition 2.1. Let S be a set, a fuzzy subset
be a fuzzy subset of S and t
Definition 2.2. Let
t
(i)
(ii)
of the fuzzy subset
x
t
t
S
x
t
x
S
t
of S is a mapping from S into 0,1 ,i.e.
0,1 , then the level subset
t
:S
0,1
dan the strong level subset
is defined as follow:
t
Definition 2.3. Let S be a semigroup, then the mapping
xy min x , y , for every x, y S
0,1 is called a fuzzy semigroup if
:S
3.Results
As the initial result, we give the property of a fuzzy subsemigroup as follow:
Proposition 3.1. Let S be a semigroup and
be a fuzzy subset of S . The fuzzy subset
subsemigroup if and only if for every non empty level subset is a subsemigroup of S
Proof:
(⇒)
Let x, y
( xy)
t
min
, so
( x) t and
( x), ( y )
( y) t . We know that xy
min t, t
t . Hence
( xy)
t
if and only if
t or xy
is a fuzzy
( xy) t . On the other hand,
t
(⇐)
It is known that t is a subsemigroup of S , so for every
consequence is: ( xy) t min ( x), ( y )
( x)
t and
( y)
t , so
( xy)
t . The

Proposition 3.2. Let S be a semigroup and
be a fuzzy subset of S . The fuzzy subset
subsemigroup if and only if for every non empty strong level subset
t
is a fuzzy
is a subsemigroup of S .
Proof:
(⇒)
Take x, y
( xy)
min
t
, so
( x), ( y )
( x)
t and
min t, t
( y)
t . We know that xy
t . Hence, we have
( xy)
t
t or xy
if and only if
t
( xy)
t . We have:
(⇐)
The strong level subset t is a subsemigroup of S , so for every
The consequence is: ( xy) t min ( x), ( y)
( x)
t and
( y)
t , we have
( xy)
t.

Definition 3.1. Let f be a mapping from a semigroup S to semigroup S ' , : S [0,1] and : S ' [0,1] be
fuzzy subsets of S and S ' , respectively. The Image homomorphic, denoted by f ( ) , is defined as follow:
1
1
f ( x)
f ( )( y )
sup
( x) . The pre-image f ( ) is defined as f ( )( x)
x f 1( y)
If f is an isomorphism, so f ( ) is called isomorphic to . A fuzzy subsemigroup
is called isomorphic
to if there is an isomorphism from S onto S ' , or vice versa, such that
f ( ) and
f( )
Proposition 3.3. If f is a mapping from a semigroup S to a semigroup S ' and
of S , then f ( ) t
 f ( t ) , for t 0,1
t
Proof:
We remember : f ( ) t
Let y
is a fuzzy subsemigroup
0
y S'
f ( )( y ) t and f
f (x) be in S ' which fulfill y
f x
t
S
( x) t
, so t
.
t
f ( ) t . Then the consequence is f ( ) f ( x)
( x)
sup
t . So,
x f 1 f ( x)
for
0 , we have y f ( x0 ) f ( t
Finally, we have f ( ) t
 f( t )
t
Take y
t
y

f(
t
) . Hence, y
f( t
).
(1)
0
) , so y
f( t
0 . The consequence, there is x0
) for every
such that
t
0
f ( x0 ) . So, for every
0 there is x0
f 1 ( y ) with
f ( )( y )
( xi )
sup
xi f 1 ( y )
Hence f ( )( y)
t or y
( x0 )
sup {t
t
t . Based on this fact, we have :
t
From equation (1) and (2), we get f ( ) t
t
 f(
t
t
such that :
} t
0
 f(
t
f ( )t
)
(2)
0
)
0

Proposition 3.4. Let f be a semigroup homomorphism from S into S ' and
then f ( ) is a fuzzy subsemigroup of S '
Proof:
Methode I: We must prove that f ( )t is a subsemigroup of S ' . For t
f ( )t = f ( )0 y S ' f ( )( y) 0 = S ' , so f ( ) t is a semigroup. For t
be a fuzzy subsemigroup of S ,
0 , then
0,1 , f ( )t
t
f(
t
).
We know
0
is a subsemigroup of S , since
is a fuzzy subsemigroup of S . And we know
t
f is a homomorphism, so it’s image homomorphic, denoted by f ( t ) , is a subsemigrup of S ' . Hence, the
intersection of all f ( t ) is a subsemigroup. The consquence f ( ) t is a subsemigroup of S' .
that for
0,
Methode II:
is a fuzzy subsemigrup fuzzy of S , so t is a subsemigroup S . The next we must
prove that f ( ) t is a subsemigroup of S ' :
Take a, b f ( ) t , so f ( )(a) t and f ( )(b) t . We get sup
( x) t dan sup
( x' ) t . The next step,
It is known that
x f 1 (a)
we must prove ab
f ( ) t , i.e.
( x' ' )
sup
x ' f 1 (b )
t:
x '' f 1 ( ab)
If x f 1(a) , then f ( x) a and if x' f 1(b) , then f ( x' ) b . The next we get ab
Construct x' ' xx' , so x' ' f 1(ab) and we get :
( x' ' )
sup
x '' f 1 ( ab)
( xx' )
sup
xx ' f 1 ( ab)
sup
min
( x), ( x' )
f ( x) f ( x' )
f ( xx' ) .
t
xx ' f 1 ( ab)
Hence ab f ( ) t , or in the other word f ( ) t is a
subsemigroup of S '
subsemigroup. It means that f ( ) is a fuzzy

f be a homomorphism from semigroup S into semigroup S ' . If
Corrolary 3.1. Let
subsemigroup of S' , then f
1
Proposition 3.5. Let f : S
1
then f ( t )
Proof :
We have
x
f
1
1
f
( ) is a fuzzy subsemigroup of S
S ' be a semigroup homomorphism and
: S'
[0,1] be a fuzzy subsemigroup,
( )t
1
y
( t) = f
f 1( y ) , with
is a fuzzy
(f
1
S'
)( y )
( y)
t = f
t , so we get f
1
( y)
1
( f 1 )( y )
S
( t) = x
S f
1
t . For any y S ' there is
( )( x)
t = f
1
x S such that
( )t

Proposition 3.6. If f is a semigroup homomorphism from S to S ' and
then f is a homomorphism from t to f ( t )
Proof:
We know that
f
t
: t
t is
a subsemigroup of S and f (
t)
is a fuzzy subsemigroup of S ,
is a subsemigroup of S ' . So we get:
f ( ) t is a homomorphism.

Proposition 3.7. If f : S
S ' is an epimorphism semigroup and
the mapping
: t
f ( ) t is an epimorphism.
is a fuzzy subsemigroup of S , then
Proof:
Based on Proposition 3.6, so
is a homomorphism which is induced by f . The next, we must prove that
: t
f ( ) t is surjective. Take y f ( )t , so we have f ( )( y) t or
sup
( x ) t . We know that
x f 1( y)
( x)
t , with y
f (x) . Since f is surjective, so for every y S ' has an element in pre image S . Since
every
( x) t such that
y f ( ) t , there is x S and
f ( ) t = f ( t ) S ' , so for
(x)
sup
( x)
f ( )( y )
x f 1( y)
3.1. Fuzzy Quotient Subsemigroup
If we discuss about semigroup, we always talk about the quotient semigroup i.e. if we have a
homomorphism : S
S ' with kernel K , then we have a set S / K , S / K {xK x S} . This set is a
semigroup with respect to a binary operation which is defined by xKyK xyK . This semigroup is called a
quotient semigroup. The following proposition talks about fuzzy quotient subsemigroup.
Proposition 3.8. Let f : S S ' be a semigroup homomorphism with kernel K and
be a fuzzy
subsemigroup of S . A mapping K : S / K [0,1] is defined as K ( xK ) sup { ( xk )} , is a fuzzy subsemigrup
k K
Proof:
a.
K is a mapping:
S / K with xK yK . If the consequence is
Let xK , yK
sup { ( xk )} sup { ( yk )} . So, there is k0
k K
k
t
( xK )
K
xk 0
b.
( xk 0 )
such that
( xK )
K
( yK ) , so we have :
( yk ) for every k K . Finally, for every
if t
. But it is known that xk 0
( xk 0 ) . The consequence is t yK
t and
. So we have
xk 0 yK . It contradicts with the fact that t yK
xK . It means that
K , yk
K
K,
k K
K
K
( yK )
is a fuzzy subsemigroup
Take xK , yK
K (xKyK )
S / K , so we have :
K
(xyK )
sup {xyk}
sup { ( xkyk )}
sup min ( xk ), ( yk )
k K
k
k K
min sup { ( xk )}, sup { ( yk )}
k K
min
k K
K
K ( xK ),
K ( yK )

This semigroup
K
is called fuzzy quotient subsemigroup.
Proposition 3.9. Let f : S
subsemigroup of S ,then
Proof:
Take xK
K t,
so
a semigroup homomorphism with kernel K and
S ' be
K t
K ( xK )
be a fuzzy
t /K
t or sup
k K
( xk ) t . As the consequence, there is k0
K , the result is
xk0
t . The next, if we multiply it by
.
xK
/
K
t
By reverses this proof, we get the proof of this proposition completely.
we have
xk0 K
t
K
such that ( xk0 ) t , so
/ K . Since k 0 K
K , then

S ' be a semigroup homomorphism with kernel K and
Proposition 3.10. Let f : S
subsemigroup of S and
:
Proof:
See the following diagrams:
In the following diagram, t
t
f ( ) t . Then
K denoted by K '
t
K
t
t
K
be a fuzzy
f ( )t
t
'
t
1'
1
`
t
'
g
/K
t
K'
By the fundamental theorem of homomorphism semigroup, we obtain
and ' are monomorphism. We
K
'
K
know that K , K ' are congruencies , with
. Therefore g is surjective. The next, we prove that g is
g
 g ( x' K ' ) . Since
injective. It is easy to verify that
get
 g ( xK ' )
' . The first, take g ( xK ' )
g
' , so we obtain
' is monomorphism. Hence g is injective and
xK ' x' K ' since
t
is a mapping, so we
g ( x' K ' ) . Since
' ( xK ' )
' ( x' K ' ) . Finally, we have
t
K
t
K

Theorem 3.1. Let f : S S ' be a semigroup homomorphism with kernel K and
be a fuzzy subsemigroup
of S , then for every t [0,1] the following diagram commutes and the mapping
: Kt
f ( )t is
monomorphism with Im
Im
f ( )t
t
I
K t
Proof:
See the following diagram:
f ( )t
t
I
'
t
t
The kernel of
is
t
monomorphism with Im
consequence ,
K
K . Based on the fundamental theorem of semigroup homomorphism , so
Im ' . Refer to the Proposition 3.10,
is monomorphism with Im
Im
we obtain
t
K
t
t
K
.
' is
The
.

Corollary 3.2. Let f : S S ' be a semigroup epimorphism with kernel
of S , then for every t [0,1] the set K t is isomorphic with f ( )t
Proof:
Using Theorem 3.1 and Proposition 3.7, we get a mapping
and
the consequence,
Im
Im
:
K
K t
, and
f ( )t
be a fuzzy subsemigroup
is a monomorphism with
 I
. Since f epimorphism, then based on Proposition 3.7
is an isomorphism. Finally,
K t is isomorphic with f ( )t .
is ephimorphism. As

Theorem 3.2. Let f : S S ' be a semigroup epimorphism with kernel
S , then f or every t [0,1] the set K is isomorphic with f ( ) .
Proof:
Using Corollary 3.2 we obtain an isomorphism
f ( )( y )
sup
:
K t
f ( )t .
K
and
be fuzzy subsemigroup of
The next we prove that f ( )
( x) .
K
:
(3.1)
x f 1( y)
For x
f
1
( y ) , then y
other word, xK
f ( )( y )
sup
x f 1( y)
So, we get f ( )
f (x) for any x
1
( y ) . Substitute xK
sup
( xK ) =
( x)
xK
1
S . We can see that y
1
K
(xK ) for any xK
K t
. In the
( y ) to the equation (3.1), we obtain:
( y)
( y)
K

Conclusion
Based on our discussion , we can prove that the fundamental theorem of the fuzzy subsemigroup
homorphism is hold. The side result is : if f : S S ' is an epimorphism semigroup with kernel K and
is a
fuzzy subsemigroup of S , then for every t [0,1] , the set
K is isomorphic with f ( ) .
References
Ajmal, Naseem. 1994.Homomorphism of Fuzzy groups, Corrrespondence Theorrm and Fuzzy Quotient
Groups. Fuzzy Sets and Systems 61, p:329-339. North-Holland
Aktaş, Haci. 2004. On Fuzzy Relation and Fuzzy Quotient Groups. International Journal of Computational
Cognition Vol 2 ,No 2, p: 71-79
Asaad, Mohamed.1991. Group and Fuzzy Subgroup. Fuzzy Sets and Systems 39, p:323-328. North-Holland
Howie, J.M. 1976. An Introduction to Semigroup Theory. Academic Press. London
Kandasamy, W.B.V. 2003. Smarandache Fuzzy Algebra. American Research Press and W.B. Vasantha
Kandasamy Rehoboth. USA
Klir, G.J, Clair, U.S, Yuan, B. 1997. Fuzzy Set Theory: Foundation and Applications. Prentice-Hall, Inc. USA
Mordeson, J.N, Malik, D.S. 1998. Fuzzy Commutative Algebra. World Scientifics Publishing Co. Pte. Ltd.
Singapore
Shabir, M. 2005. Fully Fuzzy Prime Semigroups. International Journal of Mathematics and Mathematical
Sciences:1 p:163-168