The Kernel Of Fuzzy and Anti-Fuzzy Groups

Katy D. Ahmad1,*, Mikail Bal2 , Malath Aswad3

1Islamic University Of Gaza, Palestine

2 Gaziantep University, Turkey

3Malath Aswad, Albaath university, Syria

Emails : katyon765@gmail.com1,*, mikailbal46@hotmail.com2, malaz.aswad@yahoo.com3

 

                                                                                                       

Abstract: The aim of this paper is to define  the concept of kernel subgroup of a fuzzy group and anti-fuzzy group respectively. Also, we prove that these kernels are groups in the ordinary algebraic meaning, as well as presenting many results about fuzzy groups and anti-fuzzy groups.

Keywords: Fuzzy group, anti-fuzzy group, fuzzy kernel, anti-fuzzy kernel

1.Introduction

Fuzzy set theory began with the work of Zadeh [1], were he has defined fuzzy subsets and relations.

These ideas have been used by many authors to study the algebra of fuzzy sets such as fuzzy groups [2,3], anti-fuzzy groups [20], intuitionistic fuzzy algebras [11] and some other interesting generalizations such as neutrosophic structures [8-9, 12-14].

The concept of neutrosophic group was firstly defined in [2], and studied on a wide range in [4-7,15-19], as well as anti-fuzzy group theory [20], where we find concepts such as fuzzy abelian subgroups, fuzzy nilpotency, anti fuzzy normality, and many other algebraic concepts applied to fuzzy set theory.

In this work, we use the definition of fuzzy and anti fuzzy groups to derive a new subgroup of fuzzy and anti fuzzy group which we have called the fuzzy\anti fuzzy kernel. Also, we define and study the closed normal factors in these groups.

Main discussion          

Definition 1

Let  be a group, , then  is called a fuzzy group if:

1.     .

2.      for all

Definition 2  

Let  be a group, , then  is called anti-fuzzy group if:

1.     .

2.      for all

Definition 3  

Let  be a group, , then  is called an intuitionstic-fuzzy group if:

.

.

For all

Remark 4:

The intuitionstic-fuzzy group is fuzzy and anti fuzzy ogether.

Example 5:

Let  Be the group of integers modulo 5 with multiplication modulo 5, we define:

.

.

For all .

 is an intuitionstic fuzzy group.

Theorem 6:

1.     Let  be a fuzzy group with  , then:

.

2.     Let  be an anti-fuzzy group with  , then:

.

Theorem 7:

Let  be a fuzzy group with  ,  be a normal subgroup of  with the property , then there exists a function  , such that  is a fuzzy group.

Proof.

By the normality of , we get that  is a group.

Define

is well define mapping.

Assume that , then .

On the other hand, we have , this implies that , thus .

Now, we check the conditions of a fuzzy group:

.

Also, .

If , we have:

.

Thus,  is a fuzzy group.

Definition 8:

Let  be a fuzzy group,  be normal subgroup with  for all .

 is called a fuzzy closed normal factor of  with respect to .

Example 9:

Consider the group .

Define  such that .

We have  is a normal subgroup of , and  for all , thus is closed normal factor.

Definition 10:

Let  be a fuzzy group, we define the fuzzy kernel of with respect to  as follows:

.

Theorem 11:

 is a subgroup of .

Proof.

 is not empty, that is because .

Let  be two arbitrary elements of , we have.

, thus .

.

So that  and  is a subgroup of .

Remark 12:

The fuzzy kernel of containts any closed normal factor.

Example 13:

Let  be the group of untegers modulo with multiplication.

Define

 which is a subgroup of .

Theorem 14:

Let  be a fuzzy group,  be its fuzzy kernel, then.

1.     .

2.       If  is normal, then   for all .

Proof.

1.     .

2.       Assume that  is normal, hence  for all  and .

This implies , thus .

Theorem 15:

Let  be an anti fuzzy group with  ,  be a normal subgroup of  with the property , then there exists a function  , such that  is anti fuzzy group.

Proof.

By the normality of , we get that  is a group.

Define

is well define mapping.

Assume that , then .

On the other hand, we have , this implies that , thus .

Now, we check the conditions of anti fuzzy group:

.

Also, .

If , we have:

.

Thus,  is anti fuzzy group.

Definition 16:

Let  be anti fuzzy group,  be normal subgroup with  for all .

 is called anti fuzzy closed normal factor of  with respect to .

Example 17:

Consider the group .

Define  such that .

We have  is a normal subgroup of , and  for all , thus is closed normal factor.

Definition 18:

Let  be anti fuzzy group, we define the anti fuzzy kernel of  with respect to  as follows:

.

Theorem 19:

 is a subgroup of .

Proof.

 is not empty, that is because .

Let  be two arbitrary elements of , we have.

, thus .

.

So that  and  is a subgroup of .

Remark 20:

The anti fuzzy kernel of containts any closed normal factor.

Example 21:

Let  be the group of untegers modulo with multiplication.

Define

 which is a subgroup of .

Theorem 22:

Let  be anti fuzzy group,  be its anti fuzzy kernel, then.

.

2. If  is normal, then   for all .

Proof.

.

2. Assume that  is normal, hence  for all  and .

This implies , thus

 

 

 

 

Conclusion

In this paper, we have introduced the concept of fuzzy kernel of a fuzzy group and anti-fuzzy kernel of an anti-fuzzy group. Also, we have proved that these kernels are subgroups by classical algebraic meaning, as well as, we have presented many other properties of fuzzy and anti-fuzzy groups with many examples to clarify the validity of our work.

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