SOME RESULTS ON SOLVABLE FUZZY SUBGROUP OF A GROUP

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 SOME RESULTS ON SOLVABLE FUZZY SUBGROUP OF A GROUP Luma. N. M. Tawfiq

and

Ragaa. H. Shihab

Department of Mathematics, College of Education Ibn Al-Haitham, Baghdad University .

Abstract In this paper we introduce an alternative definition of a solvable fuzzy group and study some of its properties . Many of new results are also proved, which is useful and important in fuzzy mathematics .

1. Introduction The concept of fuzzy sets is introduced by Zadeh in 1965, Rosenfeld introduced the notion of a fuzzy group as early as 1971.

The technique of generating a fuzzy group (the smallest fuzzy group) containing an arbitrarily chosen fuzzy set was developed only in 1992 by Malik ,Mordeson and Nair, [1] .

Then many research study properties of fuzzy group , fuzzy subgroup of group ,fuzzy coset , and fuzzy normal sub group of group . In this paper we introduce the definition of solvable fuzzy group and study some of its properties .

Now we introduce the following definitions which is necessary and needed in the next section : Definition 1.1 [1], [2] A mapping from a nonempty set X to the interval [0, 1] is called a fuzzy subset of X .

Next, we shall give some definitions and concepts related to fuzzy subsets of G. Definition 1.2 Let  , v be fuzzy subsets of G, if  x   vx  for every x  G , then we say that  is contained in v (or v contains  ) and we write   v (or    ).

If   v and   v , then  is said to be properly contained in v (or v properly contains

 ) and we write   v ( or    ).[3] Note that:   v if and only if  x   vx  for all x  G .[4] Definition 1.3 [3] Let  , v be two fuzzy subsets of G. Then   v and   v are fuzzy subsets as follows: (i)

  v( x)  max ( x), v( x)  1 

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011   v( x)  min ( x), v( x), for all

(ii)

x G

Then   v and   v are called the union and intersection of  and v , respectively.

Definition 1.4[5] For  , v are two fuzzy subsets of G, we define the operation   v as follows:



  v  x   sup min( a ),v( b ) a,b  G

and x  a* b For all x  G .

We call   v the product of  and v .

Now, we are ready to give the definition of a fuzzy subgroup of a group : Definition 1.5[1], [6] A fuzzy subset  of a group G is a fuzzy subgroup of G if: (i)

min  a  ,   b 

(ii)

 a 1    a  , for all a, b  G .

    a* b 

Theorem 1.6 [3]

If  is a fuzzy subset of G, then  is a fuzzy subgroup of G, if and only if,  satisfies the following conditions:

(i)



(ii)  1   where 1(x)  (x),  x G. Proposition 1.7 [6] Let  be a fuzzy group. Then  a    e 

a  G .

Definition 1.8 [7] If  is a fuzzy subgroup of G, then  is said to be abelian ,if  x, y  G ,

 x   0,   y   0 , then  xy     yx  .

Definition 1.9 [8] , [9] A

fuzzy

subgroup



of

G

is

said

  x* y     y* x  , x, y  G .

 2 

to

be

normal

fuzzy

subgroup

if

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 Definition 1.10 [ 10 ] Let  and  be two fuzzy subsets of G. The commutator of

 and  is the fuzzy

subgroup  ,  of G generated by the fuzzy subset  ,   of G which is defined as follows, for any x  G:

,  ( x) 

sup (a)   (b)

if x is a commutator

0

otherwise,

x= a, b

Now, we introduce the following theorems about the commutator of two fuzzy subsets of a group which are needed in the next section :

Theorem 1. 11[ 10 ] If A, B are subsets of G, then [  A ,  B ]    

A, B 

1, if a  A 0, if a  A

where for all x  G:  A (a)  

Theorem 1.12[ 10 ] If  ,  ,  and  are fuzzy subsets of G such that    and    , then

,    ,   .

Definition 1.13 [11] A fuzzy subgroup  of G is said to be normal fuzzy subgroup if   x* y     y* x  , x, y  G .

Corollary 1.14[ 10 ] If  ,  are normal fuzzy subgroups of  and  , respectively. Then  ,   is a normal fuzzy subgroup of  ,  .

Proposition 1.15 [2] Let λ be a fuzzy subgroup of a fuzzy group  , then  is a normal fuzzy subgroup in  if and only if  is normal subgroup in  t , t  0,1 , where  (e)   (e) . t

 3 

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 Now, we introduce an important concept about the fuzzy subset.

Definition 1.16[ 10 ] Let  be a fuzzy subset of G. Then the tip of  is the supremum of the set

 ( x) x  G.

Theorem 1. 17[ 10 ] Let and be fuzzy subsets of G. Then the tip of  ,   is the minimum of

tip of 

and tip of  .

Now, we are ready to define the concept of derived chain, which is of great importance in the next section :

Definition 1. 18[ 10 ] Let  be a fuzzy subgroup of G. We call the chain

  0   1  ......  n   .... of fuzzy subgroups of G the derived chain of  .

2. Solvable Fuzzy Subgroups of A Group In this section, we propose an alternative definition of a solvable fuzzy group and study some of its properties :

Definition 2.1 Let



be a fuzzy subgroup of G with tip

  0   1  ............  n   ...... of 

 . If the derived chain

terminates finitely to (e) , then we call  

a solvable fuzzy subgroup of G. If k is the least nonnegative integer such that

( k )  (e) , then we call the series   0   1 .........  (k )  (e) the derived series of   .

Now, we introduce another definition of solvable fuzzy groups.

 4 

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 Definition 2.2 Let



be

a

fuzzy

subgroup

of

G

with

tip

 . We call

a

series

  0  1  ............  n  (e) of fuzzy subgroups of G a solvable series for  if for 

0  i  n , [i , i ]  i1 . This is equivalent to saying that : min

i (a), i (b) i  1a, b,

for 0  i  n.

To prove the equivalence between definitions (2.1) and (2.2), we need firstly the following proposition .

Proposition 2.3 If   0   1  ............  n   (e)

  0  1  .......    (e) be n 

a

be a solvable

derived series

series ,

and

then

:

(i )   , i  0,1........., n. i

Proof We show by induction that (i )  i , i  0,1,....n .we have 0     0 , and we have

1  0  , 0    0 , 0   1 .

Therefore, the result holds for n  1 .

Now,

let

i   i

for

some

i  0,1,....n  1

then

by

theorem

(1.12),

i 1  i  , i    i , i   i 1 i  Hence,   i for i  0,1,.....n .

Now, we are ready to prove the equivalence between definitions (2.1) and (2.2)

Proposition 2.4 Definition ( 2.1 )

 Definition ( 2.2 ).

Proof Let 

be a fuzzy subgroup of G with tip  . First, suppose definition

(2.1) is hold then the derived series of  is a solvable series for  .That is definition (2.2) hold .

 5 

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 Conversely, suppose definition (2.2) is hold that is,



have a solvable series

  0  1  ............  n  (e) such that [i , i ]   for some 0  i  n . i1  Form proposition (2.3),

 i   i , for 0  i  n .

Therefore, we get (e)

 ( n )    e . Consequently,  is solvable. n



That is, definition (2.1) is hold.

Now ,we introduce a nontrivial example of a solvable fuzzy subgroup of the group

s 4 ( the group of all permutations on the set 1,2,3,4) .

Example 2.5 Let

D4  1 , 12  34  , 13  24  , 14  23  ,  24  , 1234  , 1432  ,(13) .Which dihedral subgroup of

is

S 4 with center C  1, 1324.

Let  be the fuzzy subset of S 4 defined by :

 (x) 

Clearly,

1 ½ ¼ 0

if if if if

x C  (1), (13)(24) x   (1234) \ C x  D4 \ (1234)

x  S 4 \ D4

 is a fuzzy subgroup of S 4 .The fuzzy subgroup 1 has the following definition:

(1) ( x) 

if x  (1) if x  (13)(24) otherwise , x  S4

1 ¼ 0

And ,

( 2) ( x) 

if x  (1) otherwise

1 0

Thus, we have   0   1  2   e1 Hence,  is a solvable fuzzy subgroup of

S4 .

We will give the following remark :

 6 

, x  S4

a

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 Remark 2.6 1) If G is a solvable group, then 1

G is a solvable fuzzy subgroup of G.

Proof Let G be a solvable group, then

G  G 0   G 1  ...........  G n   e . Now :

1G  1 ( 0)  1 (1)  ..........  1 ( n )  1e   e1 G G G Hence, from definition (2.1), 1

G is a solvable fuzzy subgroup of G.

2) If H is a subgroup of G, then for all n  1, ( 

H

) n   

H n  . [1]

Now, we can give the following example :

Example 2.7

S 3 ,

is solvable group with S 3  A3  e. Now ,let  ( x)  1S that is  ( x)  1 for 3

all x  S 3 :

(1) ( x)  1 A

3

1

if x  A3

0

otherwise

1

if x  {e}

0

otherwise

(1) that is  ( x)  

and

( 2) ( x)  1

{e}

( 2) that is  ( x)  

  ( 0)  (1`)  ( 2)  (e)1

then That is

1S  1 A  1  (e)1 {e} 3 3

Thus , 1S is a solvable fuzzy subgroup of S 3 .

3

Now we introduce the following theorem :

Theorem 2.8 A subgroup A of G is solvable if and only if ,χA is a solvable fuzzy subgroup of G Proof Let

A

be any subgroup of G. Consider the derived chain of

A  A0   A1  ............  An   ...........  7 

A

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 From Remark (2.6) , (  )(n)   for each n  0 and therefore, (  A ) A A(n) only if, A

( n)

( n)

 e1 if and

 e  . The result now is clear .

Now, we have the following result :

Theorem 2.9 If and are two solvable fuzzy subgroups of G. Then  ,   is a solvable fuzzy subgroup of G.

Proof Let  is solvable fuzzy subgroup of G with tip  , then :

1

  0   1  ..........  n   e1 Also,  is solvable fuzzy subgroup of G with tip  , then :

2

   0    1  .........   n   e2 From theorem (1.17 ), the tip of

,    1   2 . Therefore ,

,    0 ,  0   1 ,  1   .............  n  ,  n    e

1



2

Then  ,   is solvable fuzzy subgroup of G.

Next, we shall state the following theorem :

Theorem 2.10 Let



be a solvable fuzzy subgroup of G. Then



t is a solvable, t  (0, 1] .

Proof Suppose  is a solvable fuzzy subgroup of G with tip  . Then 0  1 n  series       ..........    e

Now,

  0  t  1 t  ..............  n  t  e t  e t

 8 

 has a solvable

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 Hence, t is a solvable subgroup of G.

Now, we will give the following interesting theorem :

Theorem 2.11 Every fuzzy subgroup of a solvable group is solvable.

Proof Let G be a solvable group. Then :

G  G 0   G 1  ..............  G n   e Let  be any fuzzy subgroup of G with tip  for 0  i  n . Define i by  (x)  i

 (x)

if x  G (i ) otherwise

0

  0    ..........  n

Then

1

 

, x G

is a finite chain of fuzzy subgroups of G,

i  such that S i  G for each i , clearly n  (e)

i  Let a, b  G and o  i  n  1 ,if a  G i  or b  G then :





a, b . on the other hand, if min  a ,  b  0   i i i 1

a, b G i 1 . And therefore,



a, b  G i  , then



min  a ,  b   min (a),  (b) i i

  a, b a, b  i 1

Thus ,

  0  1  ..........  n  (e) 

is a solvable series

for  , and  is

solvable.

Now, we can obtain the following result : Corollary 2.12 If  ,  are fuzzy subgroups of a solvable group G, then  ,   is solvable Proof  9 

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011  ,  are solvable fuzzy subgroups of G from theorem (2.11).

Then  ,   is solvable fuzzy subgroups of G from theorem (2.9).

Theorem 2.13 If

 ,  are fuzzy subsets of a solvable group G. Then  ,  is solvable.

Proof Let G be a solvable group, then :

G  G 0   G 1  ..............  G n   (e) .

Now, if

 ,  are the tips of  ,  , respectively, then from theorem (1.17 )      is

the tip of  ,  .

Define  i ,  i  as follows:

i , i  ( x) 

[ ,  ] (x)

if x  G (i ) otherwise

0

, x G

Then  ,    0 , 0   1 , 1   ...........  n ,  n  is a finite chain of fuzzy i  subgroups of G, such that S i ,  i   G for each i .

Clearly n ,  n   (e)



a, b  G

Let

and

0  i  n 1

,

If

a  G i 

min i , i (a), i , i (b)  0  i 1 , i 1 a, b a, b  G i  , then a, b G i 1 , and therefore,

mini , i (a), i , i (b)  min ,  (a),  ,  (b)

  ,  ([a, b])

 i 1 ,  i 1 ([a, b])

Thus ,

,    0 ,  0   1 , 1   ...........  n ,  n   (e) is a solvable series for  ,  , and  ,  is solvable.

Now, we shall state and prove the following theorem :

 11 

or

b  G i  ,

then

on the other hand, if

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 Theorem 2.14 Every fuzzy subgroup of solvable fuzzy group is solvable.

Proof Let  be a solvable fuzzy subgroup of a group G with tip  and let  be a fuzzy subgroup of G, such that    in view. By theorem (1.12 ), we get

  n    n 

for all

n  0 . Also,  is a solvable then n   e . n  n  Thus we have e      e , then

 n   e

Hence,  is a solvable fuzzy subgroup.

Now, we introduce the following theorem :

Theorem 2.15  Let  be a fuzzy subgroup of G, and t  inf  ( x) x  G, suppose that t   0 . Then the following are equivalent: (i)

G is solvable

(ii)



(iii)

 is solvable t  0,1 . t

is solvable

Proof

  1 then from theorem G is solvable, but G

Now, G is solvable if and only if, 1 (2.11) .  is solvable then (i)  (ii) by theorem (2.10), (ii)  (iii).

Now, suppose (iii) holds, G  

t

Thus G is solvable, that is (iii)  (i).

In the following proposition we obtain a sufficient condition for truth of the converse of theorem (2.14).

Proposition 2.16 Let  ,  be fuzzy subgroups of G, such that S    S  ,    and  is solvable.

Then  is solvable.

Proof  11 

AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 From theorem (2.10),  is solvable,  t  (0, 1]

t

Since S     , then by theorem (2.14), S   is solvable and S   is solvable.

t

And from theorem (2.15), G is solvable .

Consequently,  is solvable.

Theorem 2.17 Let and be fuzzy subgroups of G, such that  is a normal fuzzy in  . If  is

(t ) is solvable fuzzy group, t  0,1 .

a solvable fuzzy group, then    

Proof Since  is solvable fuzzy group, then from theorem (2.10),  is solvable for all

t

t  0,1 .

Also, by proposition (1.15 ),

 is normal in  t  0,1 . t t

 t t  is solvable . But ,  t

Then 

Then    

(t )

is solvable for all

t     (t )



t  0,1 .

Also, we have the following theorem :

Theorem 2.18 Let

and be fuzzy subgroups of G, such that  is a normal fuzzy in  . If  is a

solvable fuzzy group, then    is a solvable fuzzy semi-group.

Proof Since  is solvable fuzzy group. Then from theorem (2.10),  is solvable for all

t

t  0,1 .

Also, by proposition (1.15 ),



 t is normal fuzzy subgroup in t for all t  0,1 .

 t t  is solvable, But t t     t .

Then 

Hence    is solvable fuzzy semi-group.

Next , we introduce some important propositions :

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AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 Proposition 2.19 Let  , and be fuzzy subgroups of G such that If and are solvable fuzzy then





is normal fuzzy in and .

 ,    is solvable.

Proof Since  ,  are solvable fuzzy subgroups of G and theorem (2.18),

  and  

Also, by theorem (2.9).



normal in and , then from

are solvable fuzzy semi-group.

  ,    . Is solvable.

Proposition 2.20 Let



be a normal fuzzy subgroup of and be a normal fuzzy subgroup of  . If

 ,  are solvable fuzzy subgroups of G, then  ,  

 ,  is solvable fuzzy semi-group.

Proof Let

 ,  two solvable fuzzy subgroups of G, then from theorem (2.9),  ,   is

solvable fuzzy subgroup of G. And since  ,  are normal fuzzy in  ,  respectively then by corollary (1.14),  ,   is normal fuzzy in  ,  



Thus from theorem (2.18),  ,  

 ,   is solvable fuzzy semi-group.

References 1. Malik . D. s. , Mordeson . J. N. and Nair. P. S., (1992), ” Fuzzy Generators and Fuzzy Direct Sums of Abelian Groups”, Fuzzy sets and systems, vol.50, pp.193-199.

2. Majeed. S. N., (1999) ,” On fuzzy subgroups of abelian groups ”, M.Sc.Thesis , University of Baghdad .

3. Mordesn J.N., (1996 ) , ”L-subspaces and L-subfields” .

4. Hussein. R. W., (1999), ”Some results of fuzzy rings” , M.Sc.

Thesis , University of

Baghdad .

5. Liu. W.J., (1982), ”Fuzzy invariant subgroups and fuzzy ideals”, fuzzy sets and systems .vol.8. , pp.133-139 .

6. Abou-Zaid. , (1988) , ” On normal fuzzy subgroups ” , J.Facu.Edu., No.13.

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AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011 7. Seselja .B and Tepavcevic A. ,(1997), “Anote on fuzzy groups” , J.Yugoslav. Oper. Rese , vol.7, No.1, pp.49-54.

8. Gupta K.c and Sarma B.K., (1999), “nilpotent fuzzy groups” ,fuzzy set and systems , vol.101, pp.167-176 .

9. Seselja . B . and Tepavcevic A. , (1996) ,“ Fuzzy groups and collections of subgroups ” , fuzzy sets and systems , vol.83 , pp.85-91.

10. Luma. N. M. T., and Ragaa. H. S., THE COMMUTATOR OF TWO FUZZY SUBSETS , M.Sc. Thesis , University of Baghdad . , 11. Seselja .B. and Tepavcevic A. ,(1996) , “Fuzzy groups and collections of subgroups” , fuzzy sets and systems, vol.83, pp. 85 – 91.

12. Luma Naji Mohammed Tawfiq, On Training of Artificial Neural Networks, Al-Fatih journal, Vol. 1, No. 23, 130-139.

13. Luma Naji Mohammed Tawfiq,2004, Design and training artificial neural networks for solving differential equations, Ph.D. Thesis, University of Baghdad, College of Education-Ibn-Al-Haitham, Baghdad, Iraq.

13. Luma Naji Mohammed Tawfiq, (2013), Design feed forward neural network to solve singular boundary value problems, ISRN Applied Mathematics, 14. Luma Naji Mohammed Tawfiq and QH Eqhaar , (2009), On Multilayer Neural Networks and its Application For Approximation Problem, 3 rd scientific conference of the College of Science, University of Baghdad, Vol. 24.

15. Luma Naji Mohammed Tawfiq, Kareem A. Jasim and E. O. Abdulhmeed, (2015), Mathematical Model for Estimation the Concentration of Heavy Metals in Soil for Any Depth and Time and its Application in Iraq, International Journal of Advanced Scientific and Technical Research, Vol.4, No.5, 718-726.

16. Luma Naji Mohammed Tawfiq, Kareem A. Jasim, and E. O. Abdulhmeed, (2015), Pollution of Soils by Heavy Metals in East Baghdad in Iraq, IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 6, 17. Luma Naji Mohammed Tawfiq and Heba Waleed , (2013), On Solving Singular Boundary Value Problems and Its Applications, LAP LAMBERT Academic Publishing 18. E. O. Abdulhmeed, Luma Naji Mohammed Tawfiq, Kareem A. Jasim, (2016), NUMERICAL MODEL FOR ESTIMATION THE CONCENTRATION OF HEAVY METALS IN SOIL AND ITS APPLICATION IN IRAQ, Global Journal Of Engineering Science And Researches, Vol.3, No.3, 75-81.

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‫‪AL-Qadisiya Journal for Science Vol.16 No. 4 Year 2011‬‬

‫بعض النتائج حول الزمر الضبابية القابلة للحل‬ ‫ا‪.‬د‪.‬لمى ناجي محمد توفيق‬

‫& رجاء حامد شهاب‬

‫قسم الرياضيات – كلية التربية أبن الهيثم – جامعة بغداد‬ ‫املستـخلـص‬ ‫يتضمن البحث تعريف الزمرة الضبابية القابلة للحل بأكثر من صيغة ثم أثبات تكافئ‬ ‫الصيغ المختلفة للتعريف و دراسة خواصها وتقديم البراهين المهمة حول المفهوم ‪.‬‬

‫‪ 15 ‬‬