THE COMMUTATOR OF TWO FUZZY SUBSETS

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The Commutator of Two Fuzzy Subsets L. N. M. Tawfiq , R. H. Shihab Department of Mathematics, College of Education Ibn Al-Haitham, University of Baghdad.

Received in:2 March 2011 Accepted in: 12 April 2011

Abstract

In this paper we introduce the idea of the commutator of two fuzzy subsets of a group and study the concept of the commutator of two fuzzy subsets of a group .We introduce and study some of its properties .

Key ward: fuzzy set , fuzzy group, normal fuzzy subgroup.

1.Introduction

Applying the concept of fuzzy sets of Zadeh to the group theory, 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] .

In this paper, we use our notion of commutator of two fuzzy subsets of a group.

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 µ , v be fuzzy subsets of G, if µ ( x ) ≤ v( x ) for every x ∈ G , then we say that µ is contained in v (or v contains µ ) and we write µ ⊆ v (or ν ⊇ µ ).

If µ ⊆ v and µ ≠ v , then µ is said to be properly contained in v (or v properly contains

µ ) and we write µ ⊂ v ( or ν ⊃ µ ).[3] Note that: µ = v if and only if µ ( x ) = v( x ) for all x ∈ G .[4] Definition 1.3 [3] Let µ , v be two fuzzy subsets of G. Then µ ∪ v and µ ∩ v are fuzzy subsets as follows: (i) (µ ∪ v )( x) = max{µ ( x), v( x)} (i) (µ ∩ v )( x) = min{µ ( x), v( x)} , for all x ∈ G

‫ﻣﺠﻠﺔ ﺇﺑﻦ ﺍﻟﻬﻴﺜﻢ ﻟﻠﻌﻠﻮﻡ ﺍﻟﺼﺮﻓﺔ ﻭ ﺍﻟﺘﻄﺒﻴﻘﻴﺔ‬

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Then µ ∪ v and µ ∩ v are called the union and intersection of µ and v , respectively.

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

}

= = ∈ G and x a* b For all x ∈ G . ( µ  v )( x ) sup {min {µ( a ),v( b )} a,b We call µ  v the product of µ 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 µ of a group G is a fuzzy subgroup of G if: (i) (ii)

min {µ ( a ) , µ ( b )

} ≤ µ ( a* b )

µ (a −1 ) = µ (a ) , for all a, b ∈ G .

Theorem 1.6 [3] If µ is a fuzzy subset of G, then µ is a fuzzy subgroup of G, if and only if, µ satisfies the following conditions: (i) µµ⊆µ (ii) µ −1 = µ where µ−1(x) = µ(x), ∀ x ∈G. Proposition 1.7 [6] Let µ be a fuzzy group. Then µ (a ) ≤ µ (e )

∀a ∈ G .

Definition 1.8 [7] If µ is a fuzzy subgroup of G, then µ is said to be abelian if ∀ x, y ∈ G , µ (x ) > 0, µ ( y ) > 0 , then µ (xy ) = µ ( yx ) .

Definition 1.9 [8] , [9] A fuzzy subgroup µ of G is said to be normal fuzzy subgroup if

µ (= x* y ) µ ( y* x ) , ∀x, y ∈ G .

2. The Commutator of Two Fuzzy Subsets of a Group In this section we introduce the idea of the commutator of two fuzzy subsets of a group and prove some of its properties.

Definition 2.1 Let λ and µ be two fuzzy subsets of G. The commutator of λ and µ is the fuzzy subgroup [λ , µ ]of G generated by the fuzzy subset (λ , µ ) of G which is defined as follows for any x ∈ G:

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sup{λ ( a ) ∧ µ (b)}

(λ , µ )( x) =

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if x is a commutator

x= [a, b]

0

otherwise,

Next, we will introduce some theorems about the commutator of two fuzzy subsets of a group which is useful in fuzzy mathematics .

Theorem 2. 2 If A, B are subsets of G, then [ χ A , χ B ] = χ  , where for all x ∈ G: A, B  

1, if a ∈ A 0, if a ∉ A

χ A (a) =  Proof:

and  χ , χ  = < (χ , χ ) >

 A

B

A

B

{χ (a ) ^ χ B (b)}, if x is acommutator   xSup A [ , ] a b = (χ , χ B ) =  A otherwise  0 , Then:

1 [χ A, χ B ] =

if x = [a, b]∈ [A, B ]

0 0

otherwise

(1)

On the other hand ,

χ

A, B  

( x) =

1

if x ∈ [A, B ]

0 0

otherwise

(2)

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‫ﻣﺠﻠﺔ ﺇﺑﻦ ﺍﻟﻬﻴﺜﻢ ﻟﻠﻌﻠﻮﻡ ﺍﻟﺼﺮﻓﺔ ﻭ ﺍﻟﺘﻄﺒﻴﻘﻴﺔ‬

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From (1) and (2), we get [ χ A , χ B ] = χ  A, B  .   Theorem 2. 3 For any two fuzzy subsets λ , µ of G , [λ , µ ] = [µ , λ ] Proof : The result follows from definition (2.1) and definition(1.5 ).

For more explanation we give the following example: Example 2. 4 Let λandµ be two fuzzy subsets of S 3 ( the group of all permutations on the set {1,2,3,} ) defined as follows, for any x ∈ S3 :

λ (x) =

µ (x) =

1

if x = {e}

½

if x ∈ Α 3 − {e}

¼

if x ∈ S 3 − Α 3

1

if x = {e}

⅓

if

0

otherwise

x ∈ {(12 ), (13), (23)}

By definition ( 2.1 ) :

(λ , µ )( x) =

=

sup{λ (a ) ∧ µ (b)}

if x is a commutator

x = [ a , b]

0

otherwise

1 ⅓

if x = {e} if x ∈ Α 3 − {e}

0

otherwise

‫ﻣﺠﻠﺔ ﺇﺑﻦ ﺍﻟﻬﻴﺜﻢ ﻟﻠﻌﻠﻮﻡ ﺍﻟﺼﺮﻓﺔ ﻭ ﺍﻟﺘﻄﺒﻴﻘﻴﺔ‬

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Hence [λ , µ ] = 〈(λ , µ )〉 =

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1

if x = {e}

⅓

if x ∈ Α 3 − {e}

0

otherwise

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On the other hand :

(µ , λ )( x) =

1

if x = {e}

⅓

if x ∈ Α 3 − {e}

0

otherwise

Hence [µ , λ ] = 〈(µ , λ )〉 =

1 ⅓

if x = {e} if x ∈ Α 3 − {e}

0

otherwise

Thus [λ , µ ] = [µ , λ ] .

Theorem 2. 5 If λ , µ , β and δ are fuzzy subsets of G such that λ ⊆ µ and β ⊆ δ , then:

[λ , β ] ⊆ [µ , δ ] .

Proof:

[λ , β ] = 〈(λ , β )〉

,

by definition ( 2.1 )

for all x ∈ G ,

sup{λ (a) ∧ β (b)}

(λ , β )( x) =

≤

if x is a commutator

x = [a, b]

sup{µ (a) ∧ δ (b)}

if x is a commutator

x = [a, b]

0

otherwise

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= (µ , δ )( x) Thus,

[λ , β ] = 〈(λ , β )〉 ≤ 〈(µ , δ )〉 = [µ , δ ] Hence, [λ , β ] ⊆ [µ , δ ] . .

Corollary 2. 6 If λ , µ are fuzzy subsets of G such that λ ⊆ µ , then [λ , δ ] ⊆ [µ , δ ] for every fuzzy subset δ of G. Proof: The result follows from theorem (2. 5) by taking β = δ .

Now, we introduce an important concept about the fuzzy subset.

Definition 2. 7 Let λ be a fuzzy subset of G. Then the tip of λ is the supremum of the set {λ ( x) x ∈ G}.

Theorem 2. 8 Let λandµ be fuzzy subsets of G. Then the tip of [λ , µ ] is the minimum of tip of λ and tip of µ .

Proof: We want to prove that the tip of= [λ , µ ] tip of λ ∧ tip of µ Let tip of λ = sup{λ ( x) / x ∈ G} = L And, let tip of µ = sup{µ ( x) / x ∈ G} = M Such that L, M ∈ [0,1] Now,

Tip of

(λ , µ )

= sup{(λ , µ )( x) / x ∈ G} = sup{sup{λ ( a) ∧ µ (b)}/ x = [a, b], x ∈ G} = sup sup{λ ( a ) ∧ µ (b) / x = [a, b], x ∈ G} = sup{λ (a ) ∧ µ (b) / x = [a, b] and a, b ∈ G = {sup λ (a ) / a ∈ G} ∧ {sup µ (b) / b ∈ G} = L∧M

Since, [λ , µ ] = 〈 (λ , µ )〉 Therefore, tip of [λ , µ ] = L ∧ M

That is tip of= [λ , µ ] tip of λ ∧ tip of µ .

Theorem 2. 9

}

‫ﻣﺠﻠﺔ ﺇﺑﻦ ﺍﻟﻬﻴﺜﻢ ﻟﻠﻌﻠﻮﻡ ﺍﻟﺼﺮﻓﺔ ﻭ ﺍﻟﺘﻄﺒﻴﻘﻴﺔ‬

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Let

λ, µ

be

S ([λ , µ ]) = [Η , K ] .

Proof:

Year fuzzy

2012

2012 subsets

of

G.

If

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S (λ ) = Η and S (µ ) = K ,

2

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then

:

First, we have S ( λ ) = { x ∈ G | λ ( x)〉0} = H and S ( µ ) = { x ∈ G | µ ( x)〉0} = K . Then :

[H , K ] =〈 { [ a, b]〉 / a ∈ H , b ∈ K } = {〈[ a, b]〉 / λ (a) > 0 µ (b)〉0}

= {x x

is a commutaltor }

(1)

and,

S ([λ , µ ]) = S (〈 (λ , µ )〉 )

sup{λ (a) ∧ µ (b)}

(<

>)

if x is a commutator

x = [a, b]

=S

0

otherwise

= {x / x : is a commutator }

(2)

From (1) and (2), we get S ([λ , µ ]) = [H , K ] .

The following example illustrates theorems (2. 8) and (2. 9).

Example 2. 10 Let λandµ be a fuzzy subsets of S 3 which are defined as follows:

λ (x) =

and,

1

if x = {e}

⅓

if x ∈ Α 3 − {e}

¼

if x ∈ S 3 − Α 3 ½

µ (x) = 0

if x ∈ Α 3

otherwise

,x∈S

‫ﻣﺠﻠﺔ ﺇﺑﻦ ﺍﻟﻬﻴﺜﻢ ﻟﻠﻌﻠﻮﻡ ﺍﻟﺼﺮﻓﺔ ﻭ ﺍﻟﺘﻄﺒﻴﻘﻴﺔ‬

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Then, tip of λ = 1 and tip of µ = ½

sup{λ (a ) ∧ µ (b)}

if x is a commutator

x = [a, b]

(λ , µ )( x) =

0

=

otherwise

½

if x = {e}

⅓

if x ∈ Α 3 − {e}

0

otherwise , x ∈ S3

Since, [λ , µ ] =

(λ , µ )

[λ , µ ]( x) =

. Then :

½

if x = {e}

⅓

if x ∈ Α 3 − {e}

0

otherwise

Then tip of [λ , µ ] = ½

= tip of λ ∧ tip of µ Also, S (λ ) = S

3

and S (µ ) = Α

[S (λ ), S (µ )] = { [a, b] Also, S ([λ , µ ]) = A3

3

a ∈ S (λ ), b ∈ S (µ )} = A3

2

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‫ﻣﺠﻠﺔ ﺇﺑﻦ ﺍﻟﻬﻴﺜﻢ ﻟﻠﻌﻠﻮﻡ ﺍﻟﺼﺮﻓﺔ ﻭ ﺍﻟﺘﻄﺒﻴﻘﻴﺔ‬

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Then, we get : S ([λ , µ ]) = [S (λ ), S (µ )] .

Next, we will give and prove the following propositions, which we will be needed later.

Proposition 2. 11 If λ is a fuzzy subgroup of G, then : [λ , λ ] ⊆ λ .

Proof: For all x ∈ G ,

(λ , λ )( x) =

sup{λ (a) ∧ λ (b)}

if x is a commutator

x = [a, b]

0

otherwise

{

}

sup λ (a ) ∧ λ (b) ∧ λ (a −1 ) ∧ λ (b −1 ) =

x = [a, b]

0

otherwise

{(

sup λ aba −1b −1

≤

if x is a commutator

)}

if x is a commutator

x = [a, b]

0

otherwise

= λ (x) That is (λ , λ ) ⊆ λ .

Hence [λ, λ] ⊆ λ.

From theorem (2. 5) and proposition (2. 11) we obtain the following corollary : Corollary 2. 12 Let λ , µ , β , δ ,ν and α be fuzzy subsets of G. If [β , δ ] ⊆ [λ , µ ] and [ν , α ] ⊆ [λ , µ ] .

Then [[β , δ ], [ν ,α ]] ⊆ [λ , µ ] .

Proof: Since [β , δ ] ⊆ [λ , µ ] and [v, α ] ⊆ [λ , µ ]

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Then : [[β , δ ], [v,α ]] ⊆ [[λ , µ ], [λ , µ ]] ⊆ [λ , µ ] Hence : [[β , δ ], [ν ,α ]] ⊆ [λ , µ ] .

Proposition 2. 13 Let λ , µ be fuzzy subsets of G. Then: [λ , µ ]  [λ , µ ] = [λ , µ ] .

Proof: From definition (2. 1), [λ , µ ] is fuzzy subgroup of G and by theorem (1.6) : [λ , µ ]  [λ , µ ] ⊆ [λ , µ ] (1) Now, let x ∈ G

([λ , µ ] [λ , µ ]) ( x)= sup {[λ , µ ] (a) ∧ [λ , µ ] (b), x= ≥ {[λ , µ ]( x) ∧ [λ , µ ](e), x = xe}

a * b}

= [λ , µ ]( x )

That is [λ , µ ] ⊆ [λ , µ ]  [λ , µ ] (2) From (1) and (2), we get : [λ , µ ]  [λ , µ ] = [λ , µ ] .

Proposition 2. 14 Let λ , µ , β , δ ,ν and α be fuzzy subsets of G, such that [β , δ ] ⊆ [λ , µ ] and [ν , α ] ⊆ [λ , µ ] . Then [β , δ ]  [v, α ] ⊆ [λ , µ ] .

Proof: For all x ∈ G ,

([ β ,δ ] [ν ,α ]) ( x=)

sup {[ β , δ ] (a) ∧ [ν ,α ] (b), x= a * b}

a * b} ≤ sup {[ λ , µ ] (a ) ∧ [ λ , µ ] (b), x =

= ([λ , µ ]  [λ , µ ])( x) = [λ , µ ]( x) ( by proposition (2. 13) ) Hence, [β , δ ]  [v,α ] ⊆ [λ , µ ] .

Now, we can give the following corollary: Corollary 2. 15 If λ , µ , β , δ ,ν and α are fuzzy subsets of G, such that [λ , β ] ⊆ [µ , δ ] , then : (i) [λ , β ]  [ν , α ] ⊆ [µ , δ ]  [ν , α ] (ii) [λ , β ]  [µ , δ ] ⊆ [µ , δ ] (iii) If [λ , β ] ⊆ [ν , α ] , then [λ , β ] ⊆ [µ , δ ]  [ν , α ] .

Proof: The result follows from proposition (2. 14).

Next, we will give and prove the following proposition : Proposition 2. 16 Let [λ , β ], [µ , δ ] be two fuzzy subgroups of G. Then [λ , β ](e) = [µ , δ ](e) if and only if, [λ , β ] ⊆ [λ , β ]  [µ , δ ] and [µ , δ ] ⊆ [λ , β ]  [µ , δ ] .

Proof: First, if [λ, β](e) = [µ, δ](e), we prove : [λ , β ] ⊆ [λ , β ]  [µ , δ ] and [µ , δ ] ⊆ [λ , β ]  [µ , δ ] .

Let x ∈ G ,

([λ , β ] [ µ , δ ]) ( x)=

sup {[ λ , β ] (a ) ∧ [ µ , δ ] (b), x= a * b}

‫ﻣﺠﻠﺔ ﺇﺑﻦ ﺍﻟﻬﻴﺜﻢ ﻟﻠﻌﻠﻮﻡ ﺍﻟﺼﺮﻓﺔ ﻭ ﺍﻟﺘﻄﺒﻴﻘﻴﺔ‬

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≥ {[ λ , β ] ( x) ∧ [ µ , δ ] (e), x = x * e}

={[ λ , β ] ( x) ∧ [ λ , β ] (e), x =x * e}

= [λ , β ]( x) That is, [λ , β ]( x ) ≤ ([λ , β ]  [µ , δ ])( x ) for all x ∈ G .

Hence, [λ , β ] ⊆ [λ , β ]  [µ , δ ] Also,

([λ , β ] [ µ , δ ]) ( x)= sup {[λ , β ] (a) ∧ [ µ , δ ] (b), x= a * b} ≥ {[ λ , β ] (e) ∧ [ µ , δ ] ( x ), x = e * x} ={[ µ , δ ] (e) ∧ [ µ , δ ] ( x ), x =e * x}

That is [µ , δ ]( x ) ≤

= [µ , δ ]( x)

([λ , β ]  [µ , δ ])( x)

for all x ∈ G

Hence [µ , δ ] ⊆ [λ , β ]  [µ , δ ] .

Conversely, we prove [λ , β ](e) = [µ , δ ](e) .

[λ , β ](e) ≠ [µ , δ ](e) , then if [λ , β ](e) ≥ [µ , δ ](e) . [λ , β ](e) ≤ ([λ , β ] [µ , δ ])(e) = sup {[ λ , β ] ( a ) ∧ [ µ , δ ] (b), e= a * b} ≤ {[ λ , β ] (e) ∧ [ µ , δ ] (b), e = e * e}

Suppose

= [µ , δ ](e) That is [λ , β ](e) ≤ [µ , δ ](e) , which is a contradiction .

Now, if [µ , δ ](e) ≥ [λ , β ](e) ,then in the same way we get a contradiction.

Therefore [λ , β ](e) = [µ , δ ](e) .

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,.50, 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 . l.8. 133-139,.

6.Abou-Zaid. , (1988)” On normal fuzzy subgroups ” , J.Facu.Edu.,.13, .

7. Seselja .B and Tepavcevic A. , (1997) “Anote on fuzzy groups” , J.Yugoslav. Oper.

Rese ,.7, No.1, pp.49-54, .

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

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

10.Bandler. W. and Kohout. L. , (2000) Semantics of implication operators and fuzzy relational products, Internat. J. Man- Machine studies 12 89 -116 .

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‫‪Ibn Al-Haitham Journal for Pure and Applied Science‬‬

‫‪2012‬‬

‫‪2012‬‬

‫‪Year‬‬

‫‪25‬‬

‫‪Vol.‬‬

‫‪2‬‬

‫‪No.‬‬

‫اﻟﻤﺒﺎدﻝ ﻟﻤﺠﻤوﻋﺘﺎن ﺠزﺌﻴﺘﺎن ﻀﺒﺎﺒﻴﺘﺎن‬ ‫ﻟﻤﻰ ﻨﺎﺠﻲ ﻤﺤﻤد ﺘوﻓﻴق ‪،‬رﺠﺎء ﺤﺎﻤد ﺸﻬﺎب‬

‫ﻗﺴم اﻟرﻴﺎﻀﻴﺎت – ﻛﻠﻴﺔ اﻟﺘرﺒﻴﺔ أﺒن اﻟﻬﻴﺜم – ﺠﺎﻤﻌﺔ ﺒﻐداد‪.‬‬

‫اﺴﺘﻠم اﻟﺒﺤث‪ 2:‬اذار ‪ 2011‬ﻗﺒﻝ اﻟﺒﺤث ﻓﻲ ‪ 12‬ﻨﻴﺴﺎن ‪.2011‬‬

‫اﻟﺨﻼﺼﺔ‬

‫ﻴﺘﻀﻤن اﻟﺒﺤث ﺘﻘدﻴم ﻓﻛرة اﻟﻤﺒﺎدﻝ ﻟﻤﺠﻤوﻋﺘﺎن ﺠزﺌﻴﺘﺎن ﻀﺒﺎﺒﻴﺘﺎن ودراﺴﺔ ﻤﻔﻬوم اﻟﻤﺒﺎدﻝ ﻟﻤﺠﻤوﻋﺘﺎن ﺠزﺌﻴﺘﺎن ﻀﺒﺎﺒﻴﺘﺎن ﻤن‬

‫زﻤرة و دراﺴﺔ ﺨواﺼﻬﺎ وﺘﻘدﻴم اﻟﺒراﻫﻴن اﻟﻤﻬﻤﺔ ﺤوﻝ اﻟﻤﻔﻬوم ‪.‬‬ ‫ﺍﻟﻜﻠﻤﺎﺕ ﺍﻟﻤﻔﺘﺎﺣﻴﺔ ‪ :‬ﺍﻟﻤﺠﻤﻮﻋﺎﺕ ﺍﻟﻀﺒﺎﺑﻴﺔ ‪ ،‬ﺍﻟﺰﻣﺮ ﺍﻟﻀﺒﺎﺑﻴﺔ‬