FUZZY NEUTROSOPHIC SUBGROUPS

BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH

Vol.4.Issue.1.2016 (January-March)

A Peer Reviewed International Research Journal http://www.bomsr.com Email:editorbomsr@gmail.com RESEARCH ARTICLE

FUZZY NEUTROSOPHIC SUBGROUPS J. MARTINA JENCY, I.AROCKIARANI Nirmala College for women, Coimbatore, Tamilnadu, India. martinajency@gmail.com ABSTRACT In this paper we introduce the notion of fuzzy neutrosophic subgroups. Also we obtain the fuzzy neutrosophic subgroups generated by fuzzy neutrosophic set and investigate some of their properties.

MSC 2010:03B99,03E99,20K27 Keywords:fuzzy neutrosophic ŠKY PUBLICATIONS 1.INTRODUCTION Smarandache [13 ] initiated the concept of neutrosophic set which overcomes the inherent difficulties that existed in fuzzy sets[14] and intuitionistic fuzzy sets [5,6].Following this, the neutrosophic sets are explored to different heights in all fields of science and engineering.I.Arockiarani et al. defined the notion of fuzzy neutrosophic sets [1].In 1989, R.Biswas [8] introduced the concept of intuitionistic fuzzy subgroups and studied some of their properties . In this paper we define fuzzy neutrosophic subgroups and discuss their properties.

2.PRELIMINARIES: Definition 2.1:[1] A Fuzzy neutrosophic set A on the universe of discourse X is defined as A= đ?‘Ľ, đ?‘‡đ??´ đ?‘Ľ , đ??źđ??´ đ?‘Ľ , đ??šđ??´ đ?‘Ľ , đ?‘Ľ ∈ đ?‘‹ where đ?‘‡, đ??ź, đ??š: đ?‘‹ →[0,1] and 0 ď‚Ł TA ( x)  I A ( x)  FA ( x) ď‚Ł 3 Definition 2.2: [1] Let X be a non empty set, and

A  x, TA ( x), I A ( x), FA ( x) , B  x, TB ( x), I B ( x), FB ( x) (i)

A ďƒ?B for all x if TA ( x) ď‚Ł TB ( x) , I A ( x) ď‚Ł I B ( x) , FA ( x) ď‚ł FB ( x)

(ii)

A ďƒˆ B  x, max (TA ( x), TB ( x)) , max( I A ( x), I B ( x)) , min( FA ( x), FB ( x))

(iii)

A ďƒ‡ B  x, min (TA ( x), TB ( x)) , min( I A ( x), I B ( x)) , max( FA ( x), FB ( x))

(iv)

A\B (x)= x, min (TA ( x), FB ( x)) , min( I A ( x),1  I B ( x)) , max( FA ( x), TB ( x))

Definition 2.3:[1] A Fuzzy neutrosophic set A over the universe X is said to be null or empty Fuzzy neutrosophic set if TA(x) = 0 ,IA(x) = 0 , FA(x) = 1 for all x ďƒŽ X. It is denoted by 0 N

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Definition 2.4:[1] A Fuzzy neutrosophic set A over the universe X is said to be absolute fuzzy neutrosophic set if TA(x) = 1 , IA(x) = 1 , FA(x) = 0 for all x  X. It is denoted by 1 N Definition 2.5: [1]The complement of a Fuzzy neutrosophic set A is denoted by Ac and is defined as Ac = x, TAc ( x) , I Ac ( x) , FAc ( x) where TAc ( x)  FA ( x) , I Ac ( x)  1  I A ( x) , FAc ( x)  TA ( x) The complement of a Fuzzy neutrosophic set A can also be defined as Ac = 1N  A .

Definition 2.6:[2]Let X and Y be two non empty sets and f : X Y be a function . (i)





If B  y, TB ( y), I B ( y), FB ( y) : y  Y is a fuzzy neutrosophic set in Y then the pre image of B under f ,denoted by f

f 1 ( B) 

 x, f

1

( B) , is the fuzzy neutrosophic set in X defined by

(TB ( x)), f 1 ( I B ( x)), f 1 ( FB ( x)) : x  X

Where f 1 (TB ( x))  TB ( f ( x)) (ii)

1







If A  x, TB ( x), I B ( x), FB ( x) : x  X is a fuzzy neutrosophic set in X then the image of A under f ,denoted by f (A) ,is the fuzzy neutrosophic set in Y defined by

f ( A)   y, f (TA ( y)), f ( I A ( y)), f ~ ( FA ( y)) : y  Y 

 Sup T A ( x)  f (T A ( y ))   x f 1 ( y ) 0

if f

1

( y)  0 N otherwise

 Sup I A ( x) if f 1 ( y )  0 N  where f ( I A ( y ))   x f 1 ( y ) 0 otherwise  inf F A ( x) if f 1 ( y )  0 N f ~ ( F A ( y ))   x f 1 ( y ) 1 otherwise And f ~ ( FA ( y))  (1  f (1  FA )) y

.

Definition 2.7:[3] Let ( X , ) be a group and let A be fuzzy neutrosophic set in

X . Then A is called a fuzzy neutrosophic group (in short, FNG) in X if it satisfies the following conditions: (i) TA ( xy )  TA ( x)  TA ( y), I A ( xy )  I A ( x)  I A ( y) and

FA ( xy )  FA ( x)  FA ( y) (ii) TA ( x 1 )  TA ( x), I A ( x 1 )  I A ( x), FA ( x 1 )  FA ( x)

.

Definition 2.8:[3]Let ( X , ) be a groupoid and let A and B be two fuzzy neutrosophic sets in X .Then the fuzzy neutrosophic product of A and B , A  B ,is defined as follows: for any x X ,

   TA ( y )  TB ( z ) TA B ( x)   yz  x  0    I A ( y )  I B ( z ) I A B ( x)   yz  x  0    FA ( y )  FB ( z ) FA B ( x)   yz  x  1

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for each ( y, z )  X  X with yz  x, otherwise

for each ( y, z )  X  X with yz  x, otherwise for each ( y, z )  X  X with yz  x, otherwise

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Definiion 2.9:[4] Let G be a groupoid and let A FNS (G) .Then A is called a : (1) fuzzy neutrosophic left ideal (in short FNLI ) of G if for any x, y  G , A( xy )  A( y) .(i.e.,)

TA ( xy )  TA ( y), I A ( xy )  I A ( y) and FA ( xy )  FA ( y) (2) fuzzy neutrosophic right ideal (in short FNRI ) of G if for any x, y  G , A( xy )  A( x) .(i.e.,)

TA ( xy )  TA ( x), I A ( xy )  I A ( x) and FA ( xy )  FA ( x) (3) fuzzy neutrosophic ideal (in short FNI ) of G if it is both a FNLI and FNRI It is clear that A is a FNI of G if and only if for any

x, y  G ,

TA ( xy )  TA ( x)  TA ( y), I A ( xy )  I A ( x)  I A ( y) and FA ( xy )  FA ( x)  FA ( y) .Moreover ,a FNI (respectively FNLI , FNRI ) is a FNSGP of G .Note that for any FNSGP A of G we have TA ( x n )  TA ( x), I A ( x n )  I A ( x) and FA ( x n )  FA ( x) for each x  G , where x

n

is any composite of x ’s.

We will denote the set of all FNSGP s of G as FNSGP(G) .

Definition 2.10:[4]Let (G, ) be a groupoid and let 0 N  A  FNS (G) .Then A is called a fuzzy

.

neutrosophic subgroupoid in G (in short , FNSGP in G ) if A  A  A .

Definition 2.11:[4]Let (G, ) be a groupoid and let A FNS (X ) .Then A is called a fuzzy

.

neutrosophic

subgroupoid

in

G (in short , FNSGP

in

G ) if for any

x, y  G ,

TA ( xy )  TA ( x)  TA ( y), I A ( xy )  I A ( x)  I A ( y) and FA ( xy )  FA ( x)  FA ( y) .It is clear that 0 N and 1N are both FNSGP s of G .

Definition 2.12:[4]Let A FNS (G) .Then A is said to have the sup property if for any T  P(G) ,there exists a t 0  T such that A(t 0 )   A(t ) .i.e., tT

TA (t0 )   TA (t ), I A (t0 )   I A (t ), FA (t0 )   FA (t ) ,where P(G) denotes the power set of G . tT

tT

tT

Definition 2.13:[4]Let A be a fuzzy neutrosophic set in X and let  ,  ,  I with       3





.Then the set X A(  , , )  x  X : A( x)  C(  , , ) ( x)  x  X : TA ( x)   , I A   , FA    is called a ( ,  , )  level subset of A .

3. FUZZY NEUTROSOPHIC SUBGROUPS Definition 3.1: Let G be a group and let A FNSGP(G) .Then A is called a fuzzy neutrosophic subgroup (in short , FNSG ) of G if A( x 1 )  A( x) .(i.e.,) T A ( x 1 )  T A ( x), I A ( x 1 )  I A ( x) and FA ( x 1 )  FA ( x) for each x  G .

Proposition 3.2: Let

A 

 FNSG (G). Then  A  FNSG (G).  

Proposition 3.3: Let A and B be any two FNSG s of a group G .Then the following conditions are equivalent: (1) A  B  FNSG (G).

(2) A  B  B  A Proof: Proof is immediate. 1 Proposition 3.4: Let A FNSG (G) .Then A( x )  A( x) ,

(i.e.,) TA ( x 1 )  TA ( x), I A ( x 1 )  I A ( x), FA ( x 1 )  FA ( x) and A( x)  A(e) ,

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(i.e.,) TA ( x)  TA (e) , I A ( x)  I A (e) , FA ( x)  FA (e) for each x  G ,where e is the identity element of G .

Proof: Let x  G . Then TA ( x)  TA (( x 1 ) 1 )  TA ( x 1 ) , for each x  G .

I A ( x)  I A (( x 1 ) 1 )  I A ( x 1 ) , for each x  G .

FA ( x)  FA (( x 1 ) 1 )  FA ( x 1 ) , for each x  G .

Since A FNSG (G) , T A ( x 1 )  T A ( x), I A ( x 1 )  I A ( x) and FA ( x 1 )  FA ( x) for each x  G .

Hence TA ( x 1 )  TA ( x), I A ( x 1 )  I A ( x), FA ( x 1 )  FA ( x) . (i.e.,) A( x 1 )  A( x) .

On the other hand ,

TA (e)  TA ( xx 1 )  TA ( x)  TA ( x 1 )  TA ( x) , I A (e)  I A ( xx 1 )  I A ( x)  I A ( x 1 )  I A ( x) , FA (e)  FA ( xx 1 )  FA ( x)  FA ( x 1 )  FA ( x) Hence TA ( x)  TA (e) , I A ( x)  I A (e) , FA ( x)  FA (e) for each x  G .(i.e.,) A( x)  A(e) .

This completes the proof.

Proposition 3.5:If A FNSG (G) ,then

G A  {x  G : A( x)  A(e), i.e., TA ( x)  TA (e), I A ( x)  I A (e), FA ( x)  FA (e)} is a subgroup of G. Proof: Let x, y  G A .Then TA ( x)  TA (e), I A ( x)  I A (e), FA ( x)  FA (e) and

TA ( y)  TA (e), I A ( y)  I A (e), FA ( y)  FA (e) .

Thus TA ( xy 1 )  TA ( x)  TA ( y 1 )

 TA ( x)  TA ( y) (by Proposition 3.4)  TA (e)  TA (e)  TA (e) .

Similarly , I A ( xy 1 )  I A (e) .

FA ( xy 1 )  FA ( x)  FA ( y 1 )

 FA ( x)  FA ( y) (by Proposition 3.4)  FA (e)  FA (e)  FA (e) .

On the other hand , by proposition 3.4 TA ( xy 1 )  TA (e) , I A ( xy 1 )  I A (e) , FA ( xy 1 )  FA (e) 1 1 1 1 .So TA ( xy )  TA (e) , I A ( xy )  I A (e) , FA ( xy )  FA (e) .(i.e.,) A( xy )  A(e) .Thus

xy 1  G A .Hence G A is a subgroup of G .

Proposition 3.6: 1 1 1 Let A FNSG (G) . If A( xy )  A(e) . (i.e.,) TA ( xy )  TA (e) , I A ( xy )  I A (e) ,

FA ( xy 1 )  FA (e) for any x, y  G ,then A( x)  A( y) ,(i.e.,)

TA ( x)  TA ( y), I A ( x)  I A ( y), FA ( x)  FA ( y) .

Proof: 1 1 Let x, y  G . Then TA ( x)  TA (( xy ) y)  TA ( xy )  TA ( y)  TA (e)  TA ( y)  TA ( y) .

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On the other hand , by Proposition 3.4 TA ( x 1 )  TA ( x) ,then we have

TA ( xy 1 )  TA (( yx 1 ) 1 )  TA ( yx 1 ) and thus TA ( y)  TA (( yx 1 ) x)  TA ( yx 1 )  TA ( x)  TA ( xy 1 )  TA ( x)  TA (e)  TA ( x)  TA ( x).

So TA ( x)  TA ( y) .By the similar arguments, we have I A ( x)  I A ( y), FA ( x)  FA ( y).

This completes the proof.

Corollary 3.7 : Let A FNSG (G) .If G A is a normal subgroup of G , then A is constant on each coset of G A .

Proof: Let a  G and let x  aG A .Then there exists x' G A such that x  ax' .Since G A is normal and

x' G A , xa 1  ax' a 1  G A .Thus TA ( xa 1 )  TA (e) , I A ( xa 1 )  I A (e) and FA ( xa 1 )  FA (e) .By Proposition 3.6 , TA ( x)  TA (a) , I A ( x)  I A (a) and FA ( x)  FA (a) .So A is constant on aG A for each a  G . By the similar arguments , we can see that A is constant on G A a for each a  G .

Hence A is constant on each coset of G A .

Note: Let H be a subgroup of G .Then the number of right [respectively left] cosets of H in G is called index of H in G and denoted by G : H  . If G is a finite group ,then there can be only a finite

number of distinct right [respectively left] cosets of H .Hence the index G : H  is finite .If G is an infinite group ,then G : H  may be either finite or infinite.

Corollary 3.8 : Let A FNSG (G) and let G A be normal .If G A has a finite index, then A has the sup -property.

Proof: Let T  G .Since G A has finite index ,let the index [G : GA ]  n ,say

Α  a1GA ,.....anGA ,where ai  G(i  1,....n) and ai GA  a j GA   for any i  j .Let t  T .Since n

G  Α   ai Gi ,there exists i {1,2,...n} such that t  ai G A .Since G A is normal ,by corollary i 1

3.7, TA (t )  TA (ai ), I A (t )  I A (ai ), FA (t )  FA (ai ) on ai G A ,say TA (t )   i , I A (t )   i , FA (t )   i ,where  i ,  i ,  i  I and  i   i   i  3 .Thus there exists a t 0  T such that n

n

n

TA (t 0 )   TA (t )    i , I A (t 0 )   I A (t )    i , FA (t 0 )   FA (t )    i . Hence A has the tT

i 1

tT

i 1

tT

i 1

sup- property.

Proposition 3.9 :

A FNSG (G) if and only if TA ( xy 1 )  TA ( x)  TA ( y) , I A ( xy 1 )  I A ( x)  I A ( y) , FA ( xy 1 )  FA ( x)  FA ( y) for any x, y  G .

Proof: Proof follows from Definition 3.1 and Proposition 3.4.

Proposition 3.10 : A group G cannot be the union of two proper FNSGs .

Proof: Let A and B be proper FNSG s of a group G such that A  B  1N , A  1N and B  1N .

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A  B  1N  TA  TB  1, I A  I B  1, FA  FB  0 .Then TA  1 or TB  1 , I A  1 or I B  1 ,

FA  0 or FB  0 .Since A  1N and B  1N , TA  1or I A  1or FA  0 and TB  1or I B  1or FB  0 .In either cases,this is a contradiction.This completes the proof.

Proposition 3.11: If A is a FNSGP of a group G ,then A is a FNSG of G .

Proof: Let x  G .Since G is finite , x has the finite order say n .Then x  e ,where e is the identity of G n

. Thus x

1

 x n1 .Since A is a FNSGP of a group G , TA ( x 1 )  TA ( x n1 )  TA ( x n2 x)  TA ( x)

I A ( x 1 )  I A ( x n1 )  I A ( x n2 x)  I A ( x) FA ( x 1 )  FA ( x n1 )  FA ( x n2 x)  FA ( x) Hence A is a FNSG of G .

Proposition 3.12: Let

A be

a FNSG

of

a

group

G

and

let

x  G .Then

A( xy )  A( y) ,(i.e.,)

TA ( xy )  TA ( x), I A ( xy )  I A ( x), FA ( xy )  FA ( x) for each y  G if and only if A( x)  A(e) .(i.e.,) TA ( x)  TA (e), I A ( x)  I A (e), FA ( x)  FA (e) ,where e is the identity of G .

Proof: Suppose A( xy )  A( y) for each y  G .Then clearly A( x)  A(e) .

Conversely, suppose A( x)  A(e) .Then by Proposition 3.4,

TA ( y)  TA ( x), I A ( y)  I A ( x), FA ( y)  FA ( x) for each y  G .Since A is a FNSG of G ,then TA ( xy )  TA ( x)  TA ( y), I A ( xy )  I A ( x)  I A ( y), FA ( xy )  FA ( x)  F A( y) .Thus TA ( xy )  TA ( y), I A ( xy )  I A ( y), FA ( xy )  F A( y) for each y  G .

On the other hand ,by Proposition 3.4,

TA ( y)  TA ( x 1 xy )  TA ( x)  TA ( xy ), I A ( y)  I A ( x)  I A ( xy ), FA ( y)  FA ( x)  F A( xy ) .

Since TA ( x)  TA ( y), I A ( x)  I A ( y), FA ( x)  F A( y) for each y  G .

TA ( x)  TA ( xy )  TA ( xy ), I A ( x)  I A ( xy )  I A ( xy ), FA ( x)  FA ( xy )  FA ( xy ) .So TA ( y)  TA ( xy ), I A ( y)  I A ( xy ), FA ( y)  F A( xy ) for each y  G .Hence TA ( xy )  TA ( y), I A ( xy )  I A ( y), FA ( xy )  F A( y) for each y  G .

Proposition 3.13: Let f : G  G' be a group homomorphism, let A FNSG (G) and let B  FNSG (G' ) .Then the following hold: (i) If A has the sup- property , then f ( A)  FNG (G' ) . (ii) f 1 ( B)  FNSG (G) .

Proof: (i)

By Proposition 5.4 in [4],(i.e.,)Let f : G  G" be a groupoid homomorphism and let

A FNS (G) have the sup property .

(1) If A FNSGP(G) , then f ( A)  FNSGP(G") .

(2) If A is a FNI ( FNLI , FNRI ) of G , then f (A) is a FNI ( FNLI , FNRI ) of G" .

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Since f ( A)  FNSGP(G) , it is enough to show that

T f ( A) ( y 1 )  T f ( A) ( y), I f ( A) ( y 1 )  I f ( A) ( y), F f ( A) ( y 1 )  F f ( A) ( y) for each y  f (G) .

Let y  f (G) .Then   f 1 ( y)  G. Since A has the sup- property , there exists x0  f 1 ( y) such that TA ( x0 ) 

 TA (t ), I A ( x0 ) 

t f 1 ( y )

Thus T f ( A) ( y 1 )  f (TA )( y 1 ) 

I f ( A) ( y 1 )  f ( I A )( y 1 )  1

1



t f 1 ( y 1 ) t f

Ff ( A) ( y )  f ( FA )( y ) 

 FA (t ) .

t f 1 ( y )

TA (t )  TA ( x01 )  TA ( x 0 )  T f ( A) ( y) .



1

 I A (t ), FA ( x0 ) 

t f 1 ( y )

1

(y )



I A (t )  I A ( x01 )  I A ( x 0 )  I f ( A) ( y) .

t f 1 ( y 1 )

FA (t )  FA ( x01 )  FA ( x 0 )  Ff ( A) ( y) .

Hence f ( A)  FNSG (G) . (ii)

By Proposition 5.1 in [ ],(i.e.,) Let f : G  G" be a groupoid homomorphism and let

B  FNS (G" ) (1) If B  FNSGP(G" ) , then f 1 ( B)  FNSGP(G) .

(2) If B is a FNI ( FNLI , FNRI ) of G" then f 1 ( B) is a FNI ( FNLI , FNRI ) of G . Since

f 1 ( B)  FNSGP(G) , it is enough to show that f 1 ( B)( x 1 )  f 1 ( B)( x) for each x G .

Let x  G .Then

T f 1 ( B ) ( x 1 )  f

1

(TB )( x 1 )  TB ( f ( x 1 ))  TB ((( f ( x)) 1 )  TB ( f ( x))  T f 1 ( B ) ( x) ,

I f 1 ( B ) ( x 1 )  f

1

( I B )( x 1 )  I B ( f ( x 1 ))  I B ((( f ( x)) 1 )  I B ( f ( x))  I f 1 ( B ) ( x) ,and

F f 1 ( B ) ( x 1 )  f

1

( FB )( x 1 )  FB ( f ( x 1 ))  FB ((( f ( x)) 1 )  FB ( f ( x))  F f 1 ( B ) ( x) .

Hence f 1 ( B)  FNSG (G) .

Proposition 3.14: Let G p be the cyclic group of prime order

p .Then A FNSG (G p ) if and only if

A( x)  A(1)  A(0) , (i.e.,) TA ( x)  TA (1)  TA (0) , I A ( x)  I A (1)  I A (0) , FA ( x)  FA (1)  FA (0) for each 0  x  G p .

Proof: Suppose A FNSG (G p ) and let 0  x  G p .Then

TA ( xy )  TA ( x)  TA ( xy ), I A ( xy )  I A ( x)  I A ( xy ), FA ( xy )  FA ( x)  FA ( xy ) for any x, y  G p .Since G p is the cyclic group of prime order p , Gp  {0,1,2,.... p  1} .Since x is the sum of 1' s and 1 is the sum of x' s , TA ( x)  TA (1)  TA ( x), I A ( x)  I A (1)  I A ( x) and FA ( x)  I A (1)  I A ( x) .Thus TA ( x)  TA (1), I A ( x)  I A (1), FA ( x)  FA (1). Since 0 is the identity element of G p ,

TA ( x)  TA (0), I A ( x)  I A (0), FA ( x)  FA (0) .Hence the necessary conditions hold.

Conversely, suppose the necessary conditions hold, and let x, y  G p .Then we have four cases : (i)

x  0 , y  0 and x  y (ii) x  0 , y  0 (iii) x  0 , y  0 (iv) x  0 , y  0 and x  y .

Case (i) Suppose x  0, y  0 and x  y .Then by the hypothesis, TA ( x)  TA ( y)  TA (1)  TA (0) ,

I A ( x)  I A ( y)  I A (1)  I A (0) and FA ( x)  FA ( y)  FA (1)  FA (0) .So

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TA ( x  y)  TA (0)  TA ( x)  TA ( y), I A ( x  y)  I A (0)  I A ( x)  I A ( y) , FA ( x  y)  FA (0)  FA ( x)  FA ( y).

Case (ii) Suppose x  0 and y  0. Since x  y  0 ,by the hypothesis,

TA ( x  y)  TA ( x)  TA (1)  TA (0)  TA ( y) , I A ( x  y)  I A ( x)  I A (1)  I A (0)  I A ( y) and FA ( x  y)  FA ( x)  FA (1)  FA (0)  FA ( y) .So TA ( x  y)  TA ( x)  TA ( y), I A ( x  y)  I A ( x)  I A ( y) , FA ( x  y)  FA ( x)  FA ( y) .

Case (iii) is the same as Case (ii).

Case (iv) Suppose x  0, y  0 and x  y .Since x  y  0, by the hypothesis,

TA ( x  y)  TA ( x)  TA ( y)  TA (1)  TA (0) , I A ( x  y)  I A ( x)  I A ( y)  I A (1)  I A (0) , FA ( x  y)  FA ( x)  FA ( y)  FA (1)  FA (0) .So TA ( x  y)  TA ( x)  TA ( y), I A ( x  y)  I A ( x)  I A ( y) , FA ( x  y)  FA ( x)  FA ( y) .In all, TA ( x  y)  TA ( x)  TA ( y), I A ( x  y)  I A ( x)  I A ( y) , FA ( x  y)  FA ( x)  FA ( y) .Hence by Proposition 3.5, A FNSG (G p ).

Proposition 3.15: The FNI ( FNLI , FNRI ) in a group G are just the constant mappings.

Proof: Suppose A is a constant mapping and let x, y  G .Then TA ( xy )  TA ( x)  TA ( y) ,

I A ( xy )  I A ( x)  I A ( y) , FA ( xy )  FA ( x)  FA ( y) .So A is a FNI of G .

Now suppose A is a FNLI of G . Then

TA ( xy )  TA ( y), I A ( xy )  I A ( y), FA ( xy )  FA ( y)

for any x, y  G .In particular , TA ( x)  TA (e), I A ( x)  I A (e), FA ( x)  FA (e) for each x  G .

Moreover , TA (e)  TA ( x 1 x)  TA ( x), I A (e)  I A ( x 1 x)  I A ( x), FA (e)  FA ( x 1 x)  FA ( x) For each x  G .So TA ( x)  TA (e), I A ( x)  I A (e) , FA ( x)  FA (e) for each x  G . Hence A is a constant mapping.

Proposition 3.16: Let A be a FNSG of a group G .Then for each ( ,  , )  I 3 with ( ,  , )  A(e) , (i.e.,)   TA (e),   I A (e),  F (e) , G A(  ,  , ) is a subgroup of G ,where e is the identity of G .

Proof: Clearly, G A(  ,  , )  . Let x, y  G A(  ,  , ) .Then A( x)  ( ,  , ) and A( y)  ( ,  , ) .(i.e.,)

TA ( x)  , I A ( x)   , FA ( x)   and TA ( y)   , I A ( y)   , FA ( y)   .Since A  FNSG (G) , TA ( xy )  TA ( x)  TA ( y)   , I A ( xy )  I A ( x)  I A ( y)   , FA ( xy )  FA ( x)  FA ( y)   .Thus A( xy )  ( ,  , ) .So xy  G A(  ,  , ) .On the other hand ,

TA ( x 1 )  TA ( x)   , I A ( x 1 )  I A ( x)   , FA ( x 1 )  FA ( x)   .Thus A( x 1 )  ( ,  , ) .So

x 1  G A(  ,  , ) .Hence G A(  ,  , ) is a subgroup of G .

Proposition 3.17: (  ,  , ) Let A be a fuzzy neutrosophic set in a group G such that G A is a subgroup of G for each

( ,  , )  I 3 with ( ,  , )  A(e) .Then A is a FNSG of a group G .

Proof:

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For any x, y  G ,let A( x)  (t1 , s1 , r1 ) and let A( y)  (t 2 , s 2 , r2 ) .Then clearly, x  G A 1

( t , s1 , r1 )

y G

( t 2 , s2 , r2 ) .Suppose A

Since G

( t1 , s1 , r1 ) A

t1  t 2 , s1  s 2 and r1  r2 .Then G

is a subgroup of G , xy  G

( t1 , s1 , r1 ) A

( t 2 , s2 , r2 ) A

G

( t1 , s1 , r1 ) .Thus A

y G

and

( t1 , s1 , r1 ) A

.

.Then A( xy )  (t1 , s1 , r1 ) .

(i.e.,) TA ( xy )  t1 , I A ( xy )  s1 , FA ( xy )  r1 .

So TA ( xy )  TA ( x)  TA ( y), I A ( xy )  I A ( x)  I A ( y) , FA ( xy )  FA ( x)  FA ( y) .For each x  G , let A( xy )  ( ,  , ) .Then x  G A(  ,  , ) .Since G A(  ,  , ) is a subgroup of G , x 1  G A(  ,  , ) .So

A( x 1 )  ( ,  , ) .(i.e.,) TA ( x 1 )  TA ( x), I A ( x 1 )  I A ( x), FA ( x 1 )  FA ( x) .Hence A is a FNSG of a group G .

Proposition 3.18: Let A be a fuzzy neutrosophic set in X and let (1 , 1 , 1 ), (2 ,  2 , 2 ) Im( A) .If

1  2 , 1   2 , 1   2 then A( , , )  A( , 1

1

1

2

2 , 2 )

.

Result 3.19: Let A be a fuzzy neutrosophic set in a group G .Then A is a FNSG of G if and only if

A(  ,  , ) is a subgroup of G for each ( ,  , )  Im( A) .

Definition 3.20: Let A be a FNSG of group G and let ( ,  , )  Im( A) .Then the subgroup A(  ,  , ) is called a

( ,  , )  level subgroup of A .

Lemma 3.21:





Let A be any fuzzy neutrosophic set in X.Then TA ( x)    : x  A(  ,  , ) ,









I A ( x)    : x  A( ,  , ) , FA ( x)    : x  A( ,  , ) where x  X and ( ,  , )  I 3 with      3 .





    ,      : x  A ,      : x  A

Proof: Let     : x  A(  ,  , ) ,     : x  A(  ,  , ) ,     : x  A(  ,  , ) and let   0 be



arbitrary. Then       : x  A(  ,  , )

(  ,  , )

(  ,  , )



.Thus there exist  ,  ,  I with       3 such that x  A(  , , ) ,

    ,      ,      .Since x  A( , , ) , TA ( x)  , I A ( x)   , FA ( x)   .Thus TA ( x)     , I A ( x)     , FA ( x)     .Since   0 is arbitrary , TA ( x)   , I A ( x)   , FA ( x)   .

We now show that TA ( x)   , I A ( x)   , FA ( x)   .Suppose TA ( x)  t1 , I A ( x)  t 2 , FA ( x)  t 3









.Then t1  t 2  t 3  3 .Thus x  A(t1 ,t2 ,t3 ) .So t1   : x  A(  ,  , ) , t 2   : x  A(  ,  , ) ,















t 3   : x  A( , , ) .Thus, t1    : x  A( ,  , ) t 2    : x  A( ,  , ) , t 3    : x  A(  ,  , )



.(i.e.,) TA ( x)   , I A ( x)   , FA ( x)   .

This completes the proof.

We shall denote by  A the FNSG generated by the fuzzy neutrosophic set A in G .We shall use





the same notation A (  ,  , ) for the ordinary subgroup of the group generated by the level subset (  ,  , )

.

Theorem 3.22:

A

* Let G be a group and let A  FNS (G) .Let A  FNS (G) be defined as follows :for each

x  G , TA* ( x)   : x  A( ,  , )  I A* ( x)   : x  A( ,  , ) , FA* ( x)   : x  A( ,  , ) ,where

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 ,  ,  I with       3 .Then A* is a FNSG of G such that A*  B  FNSG (G) : A  B.In this case , A* is called the fuzzy neutrosophic subgroup

generated by A in G and denoted by  A .

Proof:

1 1 1 ,   t 2  ,   t 3  where n is any sufficiently n n n *(t1 .t 2 ,t3 ) large positive number .Let x  G .Suppose x  A .Then TA* ( x)  t1 , I A* ( x)  t 2 , FA* ( x)  t 3 Let (t1 , t 2 , t 3 )  Im( A* ) and let   t1 

.Thus there exist  ,  ,  I with       3 such that    ,    ,   and x  A(  ,  , ) .



( ,  ,  )  ( ,  , ) and       3 , A( , , )  A( , , ) .So x  A( ,  , ) .(i.e.,) x  A ( ,  , )



 ,    : x  A ,    : x  A ,    : x  A .So

.Now suppose x  A( , , ) .Then

   : x  A( , , )

(  ,  , )

   : x  A( , , )

(  ,  , )

.Thus    : x  A



(  ,  , )

,

(  ,  , )

1 1 1  TA* ( x), t 2   I A* ( x), t 3   FA* ( x) . n n n (i.e.,) t1  TA* ( x), t 2  I A* ( x), t 3  FA* ( x) . t1 

*(t1 .t 2 ,t3 )

Hence x A

*(t1 .t 2 ,t3 )

Hence A



  .Since A

. (i.e.,) A( ,  , )  A

 A( ,  , )

*(t1 .t 2 ,t3 )

( ,  , )

.

 is a subgroup of G , A

*(t1 .t 2 ,t3 )

Result 3.19, A* is a FNSG of G .



is a subgroup of G. By





Now , we show that A  A* .Let x  G .Then by Lemma 3.21, TA ( x)    : x  A(  , , ) ,

  T ( x)   : x  A

(  , , )

I A ( x)    : x  A

(  , , )

A

, F ( x)   : x  A , I ( x)   : x  A

(  , , )

A

(  , , )

A

.Thus , F ( x)   : x  A

(  , , )

A

.So A  A . *

Finally , let B be any FNSG of G such that A  B .We show that A*  B. Let x  G and





1 1 1 A* ( x)  (t1 , t2 , t3 ) .Then A*(t1.t2 ,t3 )  A( , , ) ,where   t1  ,   t2  ,   t3  , and n is n n n ( , , ) any sufficiently large positive integer. Thus x  A .So x  a1a2 ......am ,where ai or ai1 belongs





to A( , , ) (i  1,....m).

On the other hand, TB ( x)  TB (a1a2 ......am )

 TB (a1 )  TB (a2 )  TB (a3 )......  TB (am )

1  TA (a1 )  TA (a 2 )  .....  TA (am )    t1  . n Similarly I B ( x)  I B (a1a2 ......am )  I B (a1 )  I B (a2 )  I B (a3 )......  I B (am )

 I A (a1 )  I A (a 2 )  .....  I A (am )    t2 

1 and n

FB ( x)  FB (a1a2 ......am )  FB (a1 )  FB (a 2 )  FB (a3 )......  FB (a m )

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1  FA (a1 )  FA (a 2 )  .....  FA (am )    t3  . n Since n is sufficiently large positive integer , TB ( x)  t1 , I B ( x)  t2 , FB ( x)  t3 .So A*  B. Hence A*  B  FNSG (G) : A  B.This completes the proof.

Lemma 3.23: Let G be a finite group.Suppose there exists a FNSG A of G satisfying the following conditions:for any x, y  G ,

(i) A( x)  A( y)  ( x)  ( y) (ii ) TA ( x)  TA ( y) , I A ( x)  I A ( y) , FA ( x)  FA ( y)  ( x)  ( y). Then G is cyclic.

Proof: Suppose A is constant on G .Then A( x)  A( y) for any x, y  G .By the condition (i) , ( x)  ( y) .So G  (x) .Now suppose A is not constant on G .Let

Im( A)  (t0 , s0 , r0 ), (t1 , s1 , r1 ),.......( tn , sn , rn ),where t0  t1  .....  tn , s0  s1  .....  sn , r0  r1  .....  rn .Then by Proposition 3.18 and Result 3.19, we obtain the chain of level subgroups of A : A(t0 ,s0 ,r0 )  A(t1 ,s1 ,r1 )  .....  A(tn ,sn ,rn )  G .

Let x G  A

( tn1 ,sn1 ,rn 1 )

.We show that G  (x) .Let g G  A

( tn1 ,sn1 ,rn 1 )

. Since

t0  t1  .....  tn , s0  s1  .....  sn , r0  r1  .....  rn , A( g )  A( x)  A(tn1 ,sn1 ,rn1 ) . By the condition (i) , ( g )  ( x) .Thus G  A

( tn 1 ,sn 1 ,rn 1 )

 ( x) .Now let g  A (t n 1 , sn 1 , rn 1 ) .Then

TA ( g )  tn 1  tn  TA ( x), I A ( g )  sn 1  sn  I A ( x), FA ( g )  rn 1  rn  FA ( x) .By the condition (ii) , ( g )  ( x) .Thus A(t n1 , sn1 , rn1 )  ( x) .So G  (x) .Hence in either case , G is cyclic.

Lemma 3.24: Let G be a cyclic group of order p n ,where p is prime .Then there exists a FNSG A of G satisfying the following conditions: for any x, y  G ,

(i) A( x)  A( y)  ( x)  ( y) (ii ) TA ( x)  TA ( y) , I A ( x)  I A ( y) , FA ( x)  FA ( y)  ( x)  ( y).

Proof: Consider the following chain of subgroups of G :

(e)  G0  G1  ....  Gn 1  Gn  G ,where Gi is the subgroup of G generated by an element of order pi , i  0,1,...n and e is the identity of G . We define a complex mapping

A  (TA , I A , FA ) : G  I 3 as follows: for each x  G, A(e)  (t0 , s0 , r0 ) and A( x)  (ti , si , ri ) if x  Gi  Gi 1 for any i  1,2,...n, where ti , si , ri  I such that ti  si  ri  3,

t0  t1  .....  tn , s0  s1  .....  sn , r0  r1  .....  rn .Then we can easily check that A is a FNSG of G satisfying the conditions (i) and (ii).

From Lemma 3.23 and Lemma 3.24 we obtain the following : Theorem 3.25: Let G be a group of order p n . Then G is cyclic if and only if there exists a FNSG A of G such that for any x, y  G ,

(i) A( x)  A( y)  ( x)  ( y)

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(ii ) TA ( x)  TA ( y) , I A ( x)  I A ( y) , FA ( x)  FA ( y)  ( x)  ( y).

4.REFERENCES [1] I.Arockiarani, J.Martina Jency,More on Fuzzy Neutrosophic sets and Fuzzy Neutrosophic Topological spaces,International journal of innovative research and studies .May (2014) ,vol 3, Issue 5,643-652.

[2] I.Arockiarani, J.Martina Jency,On FN continuity in Fuzzy Neutrosophic Topological spaces,Asian academic research journal of multidisciplinary,Vol . 1,Issue 24,Aug 2014,330340.

[3] I.Arockiarani,I.R.Sumathi, Fuzzy neutrosophic groups ,Advances in Fuzzy mathematics,10(2)(2015),117-122.

[4] I.Arockiarani, J.Martina Jency ,Fuzzy neutrosophic subgroupoids (Communicated) [5] K.Atanassov ,Intuitionistic fuzzy sets,Fuzzy sets and systems ,1986,20:87-96 [6] K.Atanassov,More on intuitionistic fuzzy sets, Fuzzy sets and systems,1989,33(1):37- 46.

[7] BaldevBanerjee and DhirenKr.Basnet ,Intuitionistic fuzzy subrings and ideals,J.Fuzzy mathematics,11(1)(2003)139-155.

[8] R.Biswas ,Intuitionistic fuzzy subgroups,Mathematical forum X(1989) 37-46.

[9] Kul Hur,SuYounJang,Hee Won Kang, Intuitionistic fuzzy subgroupoids , International journal of Fuzzy logic and intelligent systems,Vol3,No.1,June 2003,72-77.

[10] Kul Hur,SuYounJang,Hee Won Kang, Intuitionistic fuzzy subgroups and subrings, Honam mathematical journal,25,No.2,19-41.

[11] Kul Hur,SuYounJang,Hee Won Kang, Intuitionistic fuzzy subgroups and cosets, Honam mathematical journal,26(2004).17-41.

[12] A.Rosenfeld, Fuzzy groups,J.Math.Anal.Appl.,35(1971),512-517.

[13] F.Smarandache, Neutrosophic set, a generialization of the intuituionistics fuzzy sets, Inter.J.Pure Appl.Math.,24 (2005),287 – 297.

[14] L.A.Zadeh,Fuzzy sets ,Information and control,8(1965)338-353.

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