Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 3 Ver. IV (May - June 2017), PP 09-21 www.iosrjournals.org

Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.

K. Gunasekaran1, S. Nandakumar2, D. Gunaseelan3 1, 3

Ramanujan Research Centre, PG and Research Department of Mathematics, Government Arts College (Autonomous), Kumbakonam-612002, Tamilnadu, India.

2 PG and Research Department of Mathematics, Government Arts College Ariyalur, Tamilnadu, India.

Abstract: The concept of a Bipolar intuitionistic M fuzzy group is a new algebraic structure of a bipolar intuitionistic M fuzzy subgroup of a M fuzzy group and anti M fuzzy group are defined and some related properties are investigated. The purpose of the study is to implement the fuzzy set theory and group theory of bipolar intuitionistic M fuzzy subgroup of a M fuzzy group and anti M fuzzy group. The relation between of a bipolar intuitionistic M fuzzy group and bipolar intuitionistic anti M fuzzy group are established.

Keywords: M fuzzy group, anti M fuzzy group, bipolar intuitionistic fuzzy set, bipolar intuitionistic M fuzzy group, bipolar intuitionistic anti M fuzzy group.

I. Introduction The concept of fuzzy sets was initiated by L.A. Zadeh [13] then it has become a vigorous area of research in engineering, medical science, graph theory. Rosenfeld [12] gave the idea of fuzzy subgroups. Bipolar valued fuzzy sets was introduced by K.M. Lee [5] are an extension of fuzzy sets whose membership degree range is enlarged from the interval [0,1] to [-1,1]. In a bipolar valued fuzzy set, the membership degree 0 means that the elements are irrelevant to the corresponding property, the membership degree (0,1] indicates that elements somewhat satisfy the property and the membership degree[-1,0) indicates that elements somewhat satisfy the implicit counter property. The author W. R. Zhang [15] commenced the concept of bipolar fuzzy sets as a generalization of fuzzy sets in 1994. The author Mourad Oqla [6] commenced the concept of an intuitionistic anti M fuzzy group. Chakrabarthy and R.Nanda [1] investicated note on union and intersection of intuitionistic fuzzy sets. P.S. Das, A. Rajeshkumar [2,3] were analyzed fuzzy groups and level subgroups. R. Muthuraj [8,9] introduced the concept of bipolar fuzzy subgroup of a M fuzzy group and bipolar anti M fuzzy group. He was introduced the notion of an image and pre-image of a bipolar fuzzy subset of a bipolar fuzzy subgroup of a group and also discuss some of its properties of bipolar M fuzzy subgroup under M homomorphism and M anti homomorphism. We discuss some of its properties with bipolar intuitionistic M fuzzy subgroup of M fuzzy group and anti M fuzzy group are established under M homomorphism and M anti homomorphism.

II. Preliminaries In this paper G  (G , ) is a finite groups, e is the identity element of G, and xy mean x*y the fundamental definitions that will be used in the sequel.

Definition.2.1

Let G be a non empty set, A bipolar intuitionistic fuzzy set (IFS) A in G is an 











object of the form A  { x ,  A ( x ),  A ( x ),  A ( x ),  A ( x ) / x  G} where  A : G  [0,1] and  A : G  [0,1] , 

 A : G  [ 1, 0]



and  A : G  [ 1, 0] is

called degree of positive membership, degree of negative

membership and the degree of positive non membership, degree respectively.

of negative non membership

Definition.2.2 [8] Let G be a group. A bipolar valued intuitionistic fuzzy set (IFS) A of G is called a bipolar intuitionistic fuzzy subgroup of G, if for all x, y  G       i)  A ( xy )  min (  A ( x ),  A ( y )) and  A ( xy )  max ( A ( x ),  A ( y ))

      ii)  A ( xy )  max (  A ( x ),  A ( y )) and  A ( xy )  min ( A ( x ), A ( y ))

 1   1   1   1  iii)  A ( x )   A ( x ),  A ( x )   A ( x ) and  A ( x )   A ( x ),  A ( x )   A (x ) .

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.

Example.2.3

 0.7 if



x 1

 0.2 if



x 1

 0.8 if



x 1

 0.1 if



x 1

 A ( x )   0.6 if x  1 ;  A ( x )   0.3 if x  1 and  A ( x )   0.5 if x  1 ;  A ( x )   0.4 if x  1

0.4 if x  i , i

0.5 if x  i , i

0.3 if x  i , i

0.6 if x  i , i

Definition.2.4 [7] Let G be a group. A bipolar valued IFS (or) bipolar IFS A of G is called a bipolar intuitionistic anti fuzzy subgroup of G, if for all x, y  G       i)  A ( xy )  max (  A ( x ),  A ( y )) and  A ( xy )  min ( A ( x ),  A ( y ))

      ii)  A ( xy )  min (  A ( x ),  A ( y )) and  A ( xy )  max ( A ( x ), A ( y ))

 1   1   1   1  iii)  A ( x )   A ( x),  A ( x )   A ( x) and  A ( x )   A ( x), A ( x )   A ( x) .

Example.2.5

 0.4 if



x 1



 A ( x )   0.6 if x  1 ;  A ( x ) 

0.7 if x  i , i

 0.5 if x  1  0.3 if x  1 and 0.2 if x  i , i

 0.3 if



x 1



 A ( x )   0.5 if x  1 ;  A ( x ) 

0.8 if x  i , i

 0.6 if x  1  0.4 if x  1 0.1 if x  i , i

Definition.2.6 Let G be an M group and A be a bipolar intuitionistic fuzzy subgroup of G, then A is called a bipolar intuitionistic M fuzzy group of G, if for all x  G and m  M then,         i)  A ( mx )   A ( x ) and  A ( mx )   A ( x ). ii)  A ( mx )   A ( x ) and  A ( mx )   A ( x).

Example.2.7 Consider 1  M   A ( x) 

 0.7 if x  1  0.6 if x  1 0.4 if x  i , i

 ;  A ( x) 

 0.2 if x  1  0.3 if x  1 and 0.5 if x  i , i

  A ( x) 

 0.8 if x  1  0.5 if x  1 0.3 if x  i , i

 ;  A ( x) 

 0.1 if x  1  0.4 if x  1 0.6 if x  i , i

Definition.2.8 Let G be an M group and A be a bipolar intuitionistic anti fuzzy subgroup of G, then A is called a bipolar intuitionistic anti M fuzzy group of G, if for all x  G and m  M then,         i)  A ( mx )   A ( x ) and  A ( mx )   A ( x ). ii)  A ( mx )   A ( x ) and  A ( mx )   A ( x ).

Example.2.9 Consider 1  M 

 0.4 if

x 1



 0.5 if

x 1

 A ( x )   0.6 if x  1 ;  A ( x )   0.3 if x  1

0.7 if x  i , i

0.2 if x  i , i



 0.3 if

x 1



 0.6 if

x 1

and  A ( x )   0.5 if x  1 ;  A ( x )   0.4 if x  1

0.8 if x  i , i

0.1 if x  i , i

Theorem.2.10 If A and B are bipolar intuitionistic M fuzzy group of G, then A  B is a bipolar intuitionistic M fuzzy group of G. Proof Consider m  M and x  A  B implies x  A, x  B

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.       ( mx )  min(  A ( mx ),  B ( mx ))  min(  A ( x ),  B ( x ))   ( x ).

A B A B   Therefore  ( mx )   ( x ).

A B A B Consider 

      ( mx )  max( A ( mx),  B ( mx))  max( A ( x),  B ( x))   ( x).

A B A B   Therefore  ( mx )   ( x ).

A B A B Consider 

      ( mx )  max(  A ( mx ),  B ( mx ))  max(  A ( x),  B ( x))   ( x).

A B A B   Therefore  ( mx )   ( x).

A B A B Consider 

      ( mx )  min( A ( mx ),  B ( mx ))  min( A ( x ),  B ( x ))   ( x ).

A B A B   Therefore  ( mx )   ( x ).

A B A B Consider 

Therefore A  B is a bipolar intuitionistic M fuzzy group of G Theorem.2.11 If A is a bipolar intuitionistic M fuzzy group of G, then A  A is also a bipolar intuitionistic M fuzzy group of G. Proof

Let m  M and x  A

    (mx)   (mx)   A( mx)   A( x).

A A     Consider  (mx)   (mx)   A(mx)   A( x).

A A     Consider  (mx)   (mx)   A(mx)   A( x).

A A     Consider  (mx)   (mx)   A(mx)   A( x).

A A

Consider 

  Therefore  ( mx)   A( x).

A   Therefore  ( mx)   A( x).

A   Therefore  ( mx)   A( x).

A   Therefore  ( mx)   A( x).

A

Therefore A  A is a bipolar intuitionistic M fuzzy group of G. Theorem.2.12 Union of any two bipolar intuitionistic M fuzzy group is also a bipolar intuitionistic M fuzzy group if either is contained in the other.

Proof

Let A and B be a bipolar intuitionistic M fuzzy group of G .

To prove that A  B is a bipolar intuitionistic M fuzzy group of G if A  B (or ) B  A

If

A  B  A  B  B (or) B  A  A  B  A

Let m  M & x  A  B       Consider  (mx )  max( A (mx ),  B (mx ))  max( A ( x ),  B ( x ))   ( x ).

A B A B   Therefore  (mx )   ( x ).

A B A B       Consider  ( mx )  min( A ( mx),  B ( mx))  min( A ( x),  B ( x))   ( x).

A B A B   Therefore  ( mx )   ( x ).

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.       ( mx )  min(  A ( mx ),  B ( mx ))  min(  A ( x),  B ( x))   ( x).

A B A B   Therefore  ( mx )   ( x).

A B A B       Consider  ( mx )  max( A ( mx),  B ( mx))  max( A ( x),  B ( x))   ( x).

A B A B   Therefore  ( mx )   ( x ).

A B A B Hence union of any two bipolar intuitionistic M fuzzy group is also a bipolar intuitionistic M fuzzy group if either is contained in the other.

Consider 

Theorem.2.13 If A is a bipolar intuitionistic anti M fuzzy group of G, then A  A is also a bipolar intuitionistic anti M fuzzy group of G. Proof

Consider m  M and x  A     Consider  (mx )   (mx )   A(mx )   A( x ).

A A     Consider  (mx )   (mx )   A (mx )   A ( x ).

A A     Consider  (mx)   (mx)   A (mx)   A ( x).

A A     Consider  (mx )   (mx )   A (mx )  A ( x ).

A A

  (mx )   A (x ).

A   Therefore  (mx )   A (x ).

A   Therefore  ( mx)   A ( x).

A   Therefore  (mx )   A (x ).

A

Therefore 

Therefore A  A is a bipolar intuitionistic anti M fuzzy group of G. Theorem.2.14 Union of any two bipolar intuitionistic anti M fuzzy group is also a bipolar intuitionistic anti M fuzzy group if either is contained in the other.

Proof Let A and B be a bipolar intuitionistic anti M fuzzy group of G. To prove that A  B is also a bipolar intuitionistic anti M fuzzy group of G if A  B (or ) B  A

If A  B  A  B  B (or ) B  A  A  B  A. Consider m  M and x  A  B.       Consider  ( mx )  max(  A( mx),  B ( mx))  max(  A( x),  B ( x))   ( x).

A B A B   Therefore  ( mx )   ( x).

A B A B       Consider  ( mx )  min( A( mx ),  B ( mx ))  min( A( x ),  B ( x ))   ( x ).

A B A B   Therefore  ( mx )   ( x ).

A B A B       Consider  ( mx )  min(  A( mx ),  B ( mx ))  min(  A( x ),  B ( x ))   ( x ).

A B A B   Therefore  ( mx )   ( x ).

A B A B       Consider  ( mx )  max( A( mx ),  B ( mx ))  max( A( x ),  B ( x ))   ( x ).

A B A B   Therefore  ( mx )   ( x ).

A B A B Therefore union of any two bipolar intuitionistic anti M fuzzy group is also a bipolar intuitionistic anti M fuzzy group if either is contained in the other.

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.

III. Some Result Based On Bipolar Intuitionistic M Fuzzy Group And Anti M Fuzzy Group Of G.

 and  be a bipolar intuitionistic fuzzy subset of an M fuzzy group then   (  ,  ) is a bipolar intuitionistic M fuzzy group of G if and only if   (  ,  ) is a bipolar

Theorem.3.1 

Let



intuitionistic anti M fuzzy group of G. Proof

Let

  (  ,  )





  (  ,  ) be a bipolar intuitionistic M fuzzy group of

G. To prove

is a bipolar intuitionistic anti M fuzzy group of G.

      i)  ( xy )  min{ ( x ),  ( y )}  1   ( xy )  min{1   ( x ), 1   ( y )}      ( xy )  1  min{1   ( x ), 1   ( y )}      ( xy )  max{ ( x ),  ( y )}.

      Therefore  ( xy )  min{ ( x),  ( y )}   ( xy )  max{ ( x),  ( y)}.       ii)  ( xy )  max{ ( x),  ( y )}  1   ( xy )  max{1   ( x), 1   ( y )}      ( xy )  1  max{1   ( x ), 1   ( y )}      ( xy )  min{ ( x ),  ( y )}.

      Therefore  ( xy )  max{ ( x),  ( y)}   ( xy)  min{ ( x),  ( y)}.  1   1   1  iii)  ( x )   ( x )  1   ( x )  1   ( x )   ( x )   ( x ). and 

 (x

1

  1   1  )   ( x )  1   ( x )  1   ( x )   ( x )   ( x ).

      iv)  (mx )   ( x )  1   (mx )  1   ( x )   (mx )   ( x ).

    Therefore  ( mx)   ( x) and  ( mx)   ( x).       v)  ( mx)   ( x)   1   ( mx)  1   ( x)   ( mx)   ( x).     Therefore  ( mx)   ( x) and  ( mx)   ( x).   Therefore   (  ,  ) is a bipolar intuitionistic M fuzzy group of G if and only if 



  ( ,  ) is a bipolar intuitionistic anti M fuzzy group of G.   Definition.3.2 [9] Let f and g be a mapping from a group G1 to a group G2 . Let   (  ,  )       and   ( ,  ) and   ( ,  ) ,   ( , ) are bipolar intuitionistic fuzzy subset in G1 and G2 respectively, then the image f ( ) and g ( ) is a bipolar intuitionistic fuzzy subset is defined by     f (  )  ( f (  ) , f (  ) ) and g ( )  ( g ( ) , g ( ) ) of G2 for all u, v  G2 .

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.   1 1   1 f (  )(u)  max{ ( x); x  f (u)} if f (u)   , 0 and g ( )(v)  min{ ( x); x  g (v)} if   1   1 1 g (v)   , 0 and f ( ) (u)  min{ ( x) ; x  f (u)}, if f (u)   , 0 g ( ) (v)  max{ ( x) ; 1 1 1 1 x  g (v)} if g (v)   , 0 . The preimage f ( ) is under f and g ( ) is under g is defined by the G1 for x  G1 , ( f 1( ) )( x)    ( f ( x)); bipolar intuitionistic fuzzy subset of all  1   1   1  ( f ( ) )( x)   ( f ( x)) and ( g ( ) )( x)   ( g ( x)); ( g ( ) )( x)   ( g ( x)).

Let G1 and G2 be any two bipolar intuitionistic M groups function f: G1  G2 and g: G1  G2 is said to be an intuitionistic M homomorphism if, Definition.3.3 [8]

i)

then the

f (xy) = f(x) f(y) for all x , y  G1

ii) f (mx) = m f(x) for all m  M and x  G1 iii) g (xy) = g(x) g(y) for all x , y  G1 iv) g (mx) = m g(x) for all m  M and x  G1 .

Let G1 and G2 be any two bipolar intuitionistic M groups (not necessarily commutative) then the function f : G1  G2 and g: G1  G2 is said to be an intuitionistic M anti Definition.3.4 [8]

homomorphism if , i)

f (xy) = f(x) f(y) for all x , y  G1

ii) f (mx) = m f(x) for all m  M and x  G1 iii) g (xy) = g(x) g(y) for all x , y  G1 iv) g (mx) = m g(x) for all m  M and x  G1 .

Let f and g be an intuitionistic M homomorphism from an M fuzzy group of G1   onto an M fuzzy group of G2 . If   (  ,  ) is a bipolar intuitionistic M fuzzy group of G1 then f ( ) the image of  under f is a bipolar intuitionistic M fuzzy group of G2 if and only if     ( ,  ) is a bipolar intuitionistic anti M fuzzy group of G1 then g ( ) is the image of  under g is a bipolar intuitionistic anti M fuzzy group of G2 .

Theorem.3.5

Proof

Let f : G1  G2 and g : G1  G2 be an intuitionistic M homomorphism.     Let   (  ,  ) and   ( ,  ) is a bipolar intuitionistic M fuzzy group and

bipolar intuitionistic anti M fuzzy group of G1 . To prove a bipolar intuitionistic fuzzy subset     f (  )  ( f (  ) , f (  ) ) and g ( )  ( g ( ) , g ( ) ) on G2 is a bipolar intuitionistic M fuzzy group and bipolar intuitionistic anti M fuzzy group.

Let u , v  G2 since f is a intuitionistic M homomorphism and so there exist x , y  G1 such that f(x)=u & f(y)=v it follows that xy  f

1

(uv) . We have to prove that g is an intuitionistic M 1 homomorphism so there exist x , y  G1 such that g(x)=u & g(y)=v it follows that xy  g (uv) .

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.   1 i) f (  ) (uv )  max{ ( z ) : z  xy  f (uv )}   1 1  max{min{ ( x ),  ( y )} : x  f (u ), y  f (v)}    min{ f (  ) (u ), f (  ) (v)}.    Therefore f (  ) (uv)  m in{ f (  ) (u ), f (  ) (v)}     1  g ( ) (uv)  min{(1  g ( ) )(u ), (1  g ( ) )( v)}     g ( ) (uv )  1  min{(1  g ( ) )(u ), (1  g ( ) )( v)}     g ( ) (uv )  max{ g ( ) (u ), g ( ) ( v)}.

   Hence f (  ) (uv)  min{ f (  ) (u ), f (  ) (v)}     g ( ) (uv)  max{g ( ) (u ), g ( ) (v )} .

  1 ii) f (  ) (uv )  max{ ( z ) : z  xy  f (uv )}   1 1  max{max{ ( x),  ( y )} : x  f (u ), y  f (v)}    max{ f (  ) (u ), f (  ) (v)}.    Therefore f (  ) (uv)  max{ f (  ) (u ), f (  ) (v)}     ( 1  g ( ) )(uv)  max{( 1  g ( ) )(u ), ( 1  g ( ) )( v)}     g ( ) (uv )  1  max{( 1  g ( ) )(u ), ( 1  g ( ) )( v)}     g ( ) (uv )  min{ g ( ) (u ), g ( ) (v)}.    Hence f (  ) (uv)  max{ f (  ) (u ), f ( ) (v )}     g ( ) (uv)  min{g ( ) (u ), g ( ) (v)}.

 1  1 1  1 1 1 iii) Now f ( ) (u )  max{ ( x ) : x  f (u )}  max{ (x ) : x  f (u ) }   f (  ) (u )  1   1  Therefore f (  ) (u )  f (  ) (u )  (1  g ( ) )(u )  (1  g ( ) )(u )  1   g ( ) (u )  g ( ) (u ).

 1   1  Hence f (  ) (u )  ( f (  ) (u)  g ( ) (u )  g ( ) (u) .  1  1 1  1 1 1 iv) f (  ) (u )  min{ ( x): x  f (u )}  min{ ( x ): x  f (u )}   f (  ) (u ).  1   1  Therefore f (  ) (u )  f (  ) (u )  ( 1  g ( ) )(u )  ( 1  g ( ) )(u )  1   g ( ) (u )  g ( ) (u ).

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.

Hence

 1   1  f ( ) (u )  f (  ) (u)  g ( ) (u )  g ( ) (u)

Therefore f ( ) and g ( ) is a bipolar fuzzy subgroup of G2 . v) Let m  M and u  G2 ,   1  f (  ) ( mu )  max{ ( x ) : x  f (u )}  f ( )(u ).

    Therefore f (  ) ( mu )  f (  ) (u )  (1  g ( ) )( mu )  (1  g ( ) )(u )    g ( ) ( mu )  g ( ) (u ).

Hence

    f (  ) ( mu )  f (  ) (u )  g ( ) ( mu )  g ( ) (u ).

  1  vi) f (  ) (mu)  min{ ( x) : x  f (u)}  f (  )(u).     Therefore f (  ) ( mu )  f (  ) (u )  ( 1  g ( ) )( mu )  ( 1  g ( ) )(u )    g ( ) ( mu )  g ( ) (u ).     Hence f (  ) ( mu )  f (  ) (u )  g ( ) ( mu )  g ( ) (u ).

 be a bipolar intuitionistic M fuzzy group of G1 then f ( ) is a bipolar intuitionistic M fuzzy group of G2 if and only if  be a bipolar intuitionistic anti M fuzzy group of Therefore if

G1 then g ( ) be a bipolar intuitionistic anti M fuzzy group of G2 .

The M homomorphic preimage of a bipolar intuitionistic M fuzzy group of G2 is a bipolar intuitionistic M fuzzy group of G1 if and only if M homomorphic preimage of a bipolar Theorem.3.6

intuitionistic anti M fuzzy group of G2 is a bipolar intuitionistic anti M fuzzy group of G1 .

  Let f : G1  G2 and g : G1  G2 be an intuitionistic M homomorphism. let   ( ,  ) is a   bipolar intuitionistic M fuzzy group of G2 and   ( , ) is a bipolar intuitionistic anti M fuzzy     group of G2 ,to prove a bipolar fuzzy subset   (  ,  ) and   ( ,  ) on G1 is a bipolar intuitionistic M fuzzy group and bipolar intuitionistic anti M fuzzy group where 1 1   f ( ) &   g ( ) Proof

i) Consider x,y  G1 (f

1

  ( )) ( xy )   ( f ( xy ))    min{ ( f ( x )),  ( f ( y ))}  min{( f

Therefore ( f

1

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1

 1  ( )) ( x ), ( f ( )) ( y )}.

 1  1  ( )) ( xy)  min{( f ( )) ( x), ( f ( )) ( y)}.

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.  (g

1

    ( )) ( xy )   ( g ( xy ))  max{ ( g ( x)), ( g ( y ))}  max{( g

Hence

1

(f

 1  ( )) ( x), ( g ( )) ( y )}.

 1  1  ( )) ( xy )  min{( f ( )) ( x ), ( f ( )) ( y )  (g

ii) Let x, y  G1

(f

1

1

 1  1  ( )) ( xy )  max{( g ( )) ( x ), ( g ( )) ( y )}.

    ( )) ( xy )   ( f ( xy ))  max{ ( f ( x)),  ( f ( y))} 1

 max{( f

Therefore ( f

1

1

 1  ( )) ( x), ( f ( )) ( y )}.

 1  1  ( )) ( xy)  max{( f ( )) ( x), ( f ( )) ( y)}  (g

1

  ( )) ( xy )   ( g ( xy ))    min{ ( g ( x)), ( g ( y ))}  min{( g

Hence

(f

1

1

 1  ( )) ( x), ( g ( )) ( y )}.

 1  1  ( )) ( xy )  max{( f ( )) ( x), ( f ( )) ( y )  (g

1

 1  1  ( )) ( xy )  min{( g ( )) ( x), ( g ( )) ( y )}

iii) Consider x  G1 (f

1

 1  1 ( )) ( x ))   ( f ( x ))    ( f ( x )) as  is a bipolar M fuzzygroup (f

Therefore ( f

1

1

 ( )) ( x ).

 1 1  ( )) ( x )  ( f ( )) ( x) (g

1

 1  1 ( )) ( x )   ( g ( x ))

 1   ( g ( x ) ) as g is an M homomorphism   ( g ( x )) as ψ is a bipolar anti M fuzzy group  (g

Hence ( f iv) ( f

1

1

1

 ( )) ( x ).

 1 1  1  1 1  ( )) ( x )  ( f ( )) ( x) ( g ( )) ( x )  ( g ( )) ( x).

 1  1 ( )) ( x ))   ( f ( x ))    ( f ( x )) as  is a bipolar M fuzzy group (f

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1

 ( )) ( x ).

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.

Therefore ( f

1

 1 1  ( )) ( x )  ( f ( )) ( x) (g

1

 1  1 ( )) ( x )   ( g ( x ))  1   ( g ( x ) ) as g is an M homomorphism   ( g ( x )) as  is a bipolar anti M fuzzy group  (g

Hence ( f v) ( f

1

1

  ( )) ( mx)   ( f ( mx))    ( f ( x )) as  is bipolar Mfuzzy group

1

Therefore ( f

1

 ( )) ( x ).

 1  ( )) ( mx )  ( f ( )) ( x )  (g

1

  ( )) ( mx )   ( g ( mx ))    ( m g ( x )) as g is an M homomorphism    ( g ( x )) as  is a bipolar anti Mfuzzygroup (g

1

Hence ( f 1

 ( )) ( x ).

 1 1  1  1 1  ( )) ( x )  ( f ( )) ( x) ( g ( )) ( x )  ( g ( )) ( x).

(f

vi) ( f

1

1

 ( )) ( x ).

 1  1  1  ( )) (mx)  ( f ( )) ( x)  ( g ( )) (mx) ( g ( )) ( x).

  ( )) (mx )   ( f (mx ))    ( f ( x )) as  is bipolar M fuzzy group (f

Therefore ( f

1

1

 ( )) ( x ).

 1  1  ( )) (mx )  ( f ( )) ( x )  ( g ( )) (mx )    ( g ( mx ))    ( m g ( x )) as g is an M homomorphism    ( g ( x )) as ψ is a bipolar anti M fuzzy group (g

Hence ( f

1

1

 ( )) ( x ).

 1  1  1  ( )) (mx)  ( f ( )) ( x)  ( g ( )) (mx) ( g ( )) ( x).

1 ( )   is a bipolar intuitionistic M fuzzy group of G1 and g ( )   is a bipolar intuitionistic anti M fuzzy group of G1 .

Hence f

1

Theorem.3.7

Let f and g be an intuitionistic M anti homomorphism from an M fuzzy group of   G1 onto an M fuzzy group of G2 . If   (  ,  ) is a bipolar intuitionistic M fuzzy group of G1 then f ( ) the image of  under f is a bipolar intuitionistic M fuzzy group of G2 if and only if DOI: 10.9790/5728-1303040921

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group. 



  ( ,  ) is a bipolar intuitionistic anti M fuzzy group of G1 then g ( ) the image of  under g is a bipolar intuitionistic anti M fuzzy group of G2 .

Let f : G1  G2 and g : G1  G2 be an intuitionistic M anti homomorphism and let     (  ,  ) and   ( ,  ) is a bipolar intuitionistic M fuzzy group and bipolar intuitionistic anti     M fuzzy group of G1 .  : G  [ 0, 1] &  G:1  [ 1, and 0]  G  & G1  :  [ are 1, 0] 1 1:  [ 0, 1]   mappings, to prove a bipolar intuitionistic fuzzy subset f (  )  ( f (  ) , f (  ) ) and   g ( )  ( g ( ) , g ( ) ) on G2 is a bipolar intuitionistic M fuzzy group and bipolar intuitionistic anti M fuzzy group.

Proof





Let u , v  G2 since f is an intuitionistic M anti homomorphism so there exist x , y  G1 such that f (x)= u and f (y)= v, it follows that x y  f

1

(uv) that g is intuitionistic M anti 1 homomorphism so there exist x , y  G1 such that g (x)=u, g (y)= v which implies x y  g (uv) i) Let

  1 1 f (  ) (uv)  max{ ( xy ) : x  f (u ), y  f (v)}   1 1  max{min{ ( x),  ( y )}: x  f (u ), y  f (v)}    min{ f (  ) (u ), f (  ) (v)}.

   Therefore f (  ) (uv)  min{ f (  ) (u ), f (  ) (v)}     (1  g ( ) )(uv )  min{(1  g ( ) )(u ), (1  g ( ) )( v)}     g ( ) (uv )  1  min{(1  g ( ) )(u ), (1  g ( ) )( v)}     g ( ) (uv )  max{ g ( ) (u ), g ( ) ( v)}.

   Hence ( f (  ) )(uv)  min{ f (  ) (u ), f (  ) ( v)}     g ( ) (uv)  max{g ( ) (u ), g ( ) (v)}. ii) Let

  1 1 f (  ) (uv)  max{ ( xy ) : x  f (u ), y  f (v)}   1 1  max{max{ ( x ),  ( y )}: x  f (u ), y  f (v )}    max{ f (  ) (u ), f (  ) (v )}.

   Therefore f (  ) (uv)  max{ f (  ) (u ), f (  ) (v)}     ( 1  g ( ) )(uv)  max{( 1  g ( ) )(u ), ( 1  g ( ) )( v)}     g ( ) (uv )  1  max{( 1  g ( ) )(u ), ( 1  g ( ) )( v)}     g ( ) (uv )  min{ g ( ) (u ), g ( ) (v)}.

   Hence f (  ) (uv)  max{ f (  ) (u ), f (  ) (v)}     g ( ) (uv)  min{g ( ) (u ), g ( ) (v)}.

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.  1  1 1 iii) Consider f ( ) (u )  max{ (x ); x  f (u )}  1 1 1  max{ ( x ); x  f (u )}   f (  ) (u ).

Therefore

 1   1  f (  ) (u )  f (  ) (u )  (1  g ( ) )(u )  (1  g ( ) )(u )  1   g ( ) (u )  g ( ) (u ).

Hence

 1   1  f (  ) (u )  f (  ) (u )  g ( ) (u )  g ( ) (u ).

- -1 -1 -1 - -1 -1 -1 iv) Consider f (  ) (u )  min{ ( x); x  f (u )}  min{ ( x ); x  f (u )}  ( f (  ) )(u ).  1   1  Therefore f (  ) (u )  f (  ) (u )  ( 1  g ( ) )(u )  ( 1  g ( ) )(u )  1   g ( ) (u )  g ( ) (u ).  1   1  Hence f (  ) (u )  f (  ) (u )  g ( ) (u )  g ( ) (u ).

Therefore f ( ) and g ( ) is a bipolar fuzzy subgroup of G2 . v) Consider m  M and u  G2

  1  1 f (  ) ( mu )  max{ ( mu ); x  f (u )}  max{ ( x); x  f (u )}   f (  ) (u ).

    Therefore f (  ) (mu )  f (  ) (u )  (1  g ( ) )( mu )  (1  g ( ) )(u )    g ( ) ( mu )  g ( ) (u ).     Hence f (  ) (mu )  f (  ) (u )  g ( ) (mu )  g ( ) (u ). vi ) Consider m  M and u  G2

  1  1 f (  ) ( mu )  min{ ( mu ); x  f (u )}  min{ ( x); x  f (u )}   f (  ) (u ).

Therefore

    f (  ) (mu )  f (  ) (u )  ( 1  g ( ) )( mu )  ( 1  g ( ) )(u )    g ( ) ( mu )  g ( ) (u ).

    Hence f (  ) ( mu )  f (  ) (u )  g ( ) ( mu )  g ( ) (u ).

 be a bipolar intuitionistic M fuzzy group of G1 then f ( ) is a bipolar M fuzzy group of G2 if and only if  be a bipolar anti M fuzzy group of G1 then g ( ) be a bipolar Hence if

intuitionistic anti M fuzzy group of G2 .

DOI: 10.9790/5728-1303040921

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Bipolar Intuitionistic M Fuzzy Group and Anti M Fuzzy Group.

IV. Conclusion The concept of a bipolar intuitionistic M fuzzy group is a new algebraic structure of a bipolar intuitionistic M fuzzy subgroup of a M fuzzy group and anti M fuzzy group are defined and some related properties are investigated. The purpose of the study is to implement the fuzzy set theory and group theory of bipolar intuitionistic M fuzzy subgroup of a M fuzzy group and anti M fuzzy group.

The relation between of a bipolar intuitionistic M fuzzy group and bipolar intuitionistic anti M fuzzy group are established. We hope that our results can also be extended to other algebraic system.

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DOI: 10.9790/5728-1303040921

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