N-Generated Fuzzy Groups and Its Level Subgroups

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N-Generated Fuzzy Groups and Its Level

Subgroups

Dr. M. Mary JansiRani1, B. Bakkiyalakshmi2, P. Sudhalakshmi3 1 .Head & Assistant professor, PG& Research Department of Mathematics,

Thanthai Hans Roever college of Arts & Science, Perambalur.

2&3. Research scholar, PG& Research Department of Mathematics,

Thanthai Hans Roever college of Arts & Science, Perambalur.

Abstract- In this paper, we define the algebraic structures of ngenerated fuzzy subgroups and some related properties are investigated. The purpose of this study is to implement the fuzzy set theory and group theory in n generated fuzzy subgroups. Characterizations of ngenerated level subsets of a n generated fuzzy subgroups of a group are given

Keywords- Fuzzy set, multi-fuzzy set, fuzzy subgroup, multi-fuzzy subgroup, anti-fuzzy subgroup, multi-anti fuzzy subgroup,ngenerated fuzzy subset, ngenerated fuzzy subgroups, ngenerated fuzzy level subsets,n-generated fuzzy level subgroups

1.INTRODUCTION

After the introduction of the concept of fuzzy sets by L.A.Zadeh [1], researchers were conducted the generalizations of the notion of fuzzy sets A. Rosenfeld [2] Introduced the concept of fuzzy group and the idea of “Intuitionistic Fuzzy set” was first published by K.T. Atanassov [3]. W.D.Blizard [4] Introduced the concept of fuzzy multi-set theory. Also Shinoj .T.K and Sunil Jacob [6] produced some results in Intuitionistic Fuzzy Multi-sets. In this chapter we define ngenerated fuzzy sets and n

generated fuzzy subgroups and some of their properties.

2. PRELIMINARIES

2.1Definition

Let X be a non-empty set. A fuzzy set A on X is a mappingA X: 

 

0,1 and is defined as







/ , ( )



A xX x  x

2.2. Definition

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1( )

x f  y 2.3. Definition

Let X and Y be any two sets. Let f X: Y be a function. If S is a fuzzy set on Y then the preimage of S under f is a fuzzy set onX and is defined by



_f1_{(S) ( )}



_x _{S f x}



_{( )}



2.4. Definition

Let A be a fuzzy subset of asetX. For t[0, 1], A_t 



xX A x/ ( )t



is called a levelfuzzy subset of A

2.5. Definition

Let X be a non empty set. An Intuitionistic Fuzzy set A on X is an object having

the formA



x, ( ),x ( ) /x x X



A A

 

  , where :X

 

0,1 & :X

 

0,1

A A

    are the

degree of membership and non- membership functions respectively with 0_A( )x _A( ) 1x 

2.6. Definition

Let X be a non-empty set. A Fuzzy Multi set (FMS) A drawn from X is characterized by a function „ Count membership‟ of A denoted by CM_A such that CM :X Q

A  where Q is the set of all crisp finite set drawn from the unit interval

 

0,1 .Then for any xX , the

value CM ( )x

A is a crisp multi set drawn from

 

0,1 . For each xX , the membership sequence is defined as the decreasingly ordered sequence of elements in CM_A( )x . It is

denoted by ( ), ( ),... ( )

1 2

x x x

A A A

k

  













where 1( ) 2( ) ... ( )

x x x

A A A

k

   

: ( ), ( )... ( ) :

1 2

A x x x x x X

A A A

k

  

 _ _ 

 

 _ _  

 

 

 

Example 2.7

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2.8. Definition

Let X be a non- empty universal set and let A be an Fuzzy Multi set on X. The n-generated Fuzzy set on X is constructed from the Fuzzy Multi set and is defined as







1



, ( ) ( ) ... ( ) :

1 2

n n n

x x x x x X

A A A

k k

     

where ( ) ( ) ... ( )

1 2

n n n

x x x

A A _Ak

    and n is the dimension of the Fuzzy Multi setA

2.9.Definition Multi-level subset

Let A be a multi-fuzzy subset of X. For

 

0,1 , 1, 2,..., ,



/ ( )



i

i t i

t  i k A  x X A x t is called multi-level subset of A

2.10. Definition Multi-Fuzzy Mapping

Let 



 ₁, ₂,...,_k



and  



 ₁, ₂,...,_k



be two multi-fuzzy sets in X of dimenstion k and n respectively. A multi-fuzzy mapping is a mapping

: k ( ) n ( )

F M FS X M FS X which maps each M FS Xk ( ) into a unique multi-fuzzy set ( )

n

M FS X



2.11. Definition Atanassov Intuitionistic Fuzzy Sets Generating Maps (AIFSGM)

A mapping 2

: k ( ) ( )

F M FS X M FS X is said to be an Atanassov Intuitionistic Fuzzy Sets Generating Maps(AIFSGM) if F( ) is an Intuitionistic fuzzy set in

2

( ) M FS X

2.12. Definition Multi-Fuzzy extensions of functions

Let f X: Y and :h M_i  L_j be functions . The Multi-fuzzy extension and the inverse of the extension are f :M_iX  L_jY, f1:L_jY  M_iX defined by





1

( )( ) ( ) , ,

( )

X i

f A y Sup h A x A M y Y

x f y

  

 and





1 1

( )( ) ( ( ) , _jY,

f B x h B f x BL xX where 1

h is the upper adjoint of h .

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2.13. Definition

Let X and Ybe any two sets. Let f X: Y be a function. If  is a ngenerated fuzzy set on X then the image of  under f is a ngenerated fuzzy set onY and is defined by  

1

( ) ( ) ( ),

( )

f y v y Sup x y Y

x f y

 



   

 is called image of  under f

Let X and Y be any two sets. Let f X: Y b a function. If  is an ngenerated fuzzy set on Y then the pre image of  under f is a ngenerated fuzzy set on X and is

defined _f1( ) ( ) _ x 



f x( )



2.15. Definition:

Let  be an ngenerated fuzzy set onX. For t[0, 1], a level ngenerated fuzzy subset of _t is defined by



_t

 



x

X

/ ( )



x



t



2.16. Properties of n - generated fuzzy set

Let k be a positive integer and A and B be two fuzzy multi- sets of dimension kand if

 







, ;



G

A  x  x xX & BG 





x, ( ) ; x



xX



 

1 1

, where ( ), ( ) ( )

1 1

k _n k _n

n N x _i x x _i x

k_i k_i

   

    

  Then

 

(1). AG  BG   x ( )x

 

(2). AG  BG   x ( )x

 

(3).AG BG 



x 



( )x

_



x, max_

 

x , ( ) x _



;xX_

 

(4).AGBG  x ( )x





_



x

, min







 

x

, ( )



x







;

x



X



_





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Let G be a group. A fuzzy subset A of G is said to be a fuzzy subgroup of G if





( ). (

i A xy

)



min

A x A y

( ), ( )

(

1

)

( ).

ii A x

-

_³

A x

( )

_"

x y

,

_Î

G

2.18 . Definition

Let G be a group. A fuzzy subset A of G is said to be an anti-fuzzy subgroup of G if (i). 𝐴 𝑥𝑦 ≤ 𝑚𝑎𝑥 𝐴 𝑥 , 𝐴(𝑦) , (ii). 𝐴 𝑥−1 = 𝐴 𝑥 ∀ 𝑥, 𝑦 ∈ G

2.19. Definition

Let G be a group. A multi-fuzzy subset A of G is said to be an multi-fuzzy

subgroup of G if ( ). (i A xy)min



A x A y( ), ( )



( ).ii A x

(

- 1

)

³ A x( ) " x y, Î G

2.20. Definition

Let G be a group. A multi-fuzzy subset A of G is said to bean multi-anti-fuzzy

subgroup of G if ( ). (i A xy) max



A x A y( ), ( )



( ). (ii A x-1)

=

A x( ) " x y, Î G

2.21. Definition

Let G be a group. A ngenerated fuzzy subset  of a group G is called a ngenerated fuzzy subgroup of G if





( ). (i  xy)min ( ), ( )x  y

 

1

( ).ii  x ( )x x y, G where

 

1 ( ), ( ) 1 ( )

1 1

k _n k _n

x _i x y _i y

k_i k_i

       

 

1

& ( ) ( )

1 k _n

xy _i xy

k i

   

 2.22. Definition

Let G be a group. An ngenerated fuzzy subset  of a group G is called an n generated anti- fuzzy subgroup of G if





( ). (i  xy)max ( ), ( )x  y

 

1

( ).ii  x ( )x x y, G

3. Properties of n-generated- Level Subsets of an n- generated Fuzzy subgroups

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Let  be a ngenerated fuzzy subgroup of a group G . For any



, ,.... ,..



1 2

t t t t

k 

where t_i[0,1] for all i , we define the n generated level subset of  as

  

; / ( )



L  t  xG  x t

Theorem.3.2:

Let  be an ngenerated fuzzy subgroup of a group G . For any



, ,.... ,..



1 2

t t t t

k 

 

where _ti 0,1 for all i suchthat t( )e where ' 'e is the identity element of G ,

 

; is a subgroup of .

L  t G

Proof:

 

Let x y,  L ;t ( )x t and ( ) y t

 

1





Now,  xy Min ( ), ( )x  y

Min ,

 

t t

 

1

xy t

 

 

 

1

; xy L  t

 

 

; is a subgroup of .

L  t G



Theorem3.3:

Let G be a group and let  be an ngenerated fuzzy subset of a group G such that

 

;

L  t is a subgroup of G. Then for any t



t t_{1 2}, ,.... ,..t



k

 where _ti

 

0,1 for all isuch

that t( )e where ' 'e is the identity element of G, is an ngenerated fuzzy subgroup of G

Proof:

Let ,x yG and ( ) x r & ( ) y s









where , ,.... ,... , , ,... ,.... ,

1 2 1 2

r r r r s s s s

k k

  for ,r s_{i i}

 

0,1 for all i

Suppose rs

 

Now ( ) x   r x L ;r

 

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

1







That is,  xy min ( ), ( )x  y

.

Hence  is a ngenerated fuzzy subgroup of G

Theorem3.4:

Let  be an ngenerated fuzzy subgroup of a group Gand ' 'e is the identity element of GIf two ngenerated level fuzzy subgroups L

   

;r , L ;s for



, ,.... ,..



1 2 3

r r r r ,



, ,... ,...



1 2

s s s s

k

 where r s_{i i}, 

 

0,1 for all and ,i r s( )e with rs of  are equal,

then There is no x in G such that r( )x s Proof:

Let L

   

;r L ;s

Suppose there exists xG such that r( )x s Then L

 

;s L

 

;r

 

; , but

 

;

x L  r x L  s

  

This contradicts our assumption that L

   

;r L ;s Hence there is no xG such that r( )x s Conversely, suppose that there is no xG such that

( )

r x s, then by definition L

 

;s L

 

;r

Let xL

 

;r and there is no xGsuchthat r( )x s Hence xL

 

;s and therefore L

 

;r L

 

;s

Hence L

   

;r L ;s

CONCLUSION

In this chapter we have propounded the concept of n-generated fuzzy sets. It is directly proportional to Multi-fuzzy set theory

REFERENCES

[1] L.A.Zadeh, Fuzzy sets, Information and control, 8(1965), 338-353. [2] A.Rosenfeld, Fuz groups,J.Math.Anal.Appl.,35(1971 512- 517

[3] K.T.Atanassov “ Intuitionistic fuzzy sets”,Fuzzy sets and Systems 20(1986) no.1, 87-96 [4] W.D.Blizard, “Multiset Theory”, Notre Dame Journal of Formal Logic,Vo.30, No.1, (1989) 36-66

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[7] Choudhury.F.P. and Chakraborty.A.B. and Khare.S.S., A note on fuzzy subgroups and

homomorphism .Journal of mathematical analysis and applications 131, 537-553(1988)

[8] N.Palaniappan and R.Muthuraj, Anti-fuzzy group and Its lower level subgroups,