naganathan t

INTRODUCTION

1. Fuzzy subsets :

Fuzzy subsets were introduced by Zadeh in 1965 to represent /

manipulate data and information possessing non-statistical uncertainties.

It was specifically designed to mathematically represent uncertainty and

vagueness and to provide formalized tools for dealing with the

imprecision intrinsic to many problems.

The first publication in fuzzy subset theory by Zadeh (1965 ) and

then by Goguen (1967, 1969 ) show the intention of the authors to

generalize the classical set. In classical set theory, a subset A of a set X

can be defined by its characteristic function A : X → {0, 1} is defined

by A( x ) = 0, if xA and A( x ) = 1, if xA.

The mapping may be represented as a set of ordered pairs

{ ( x, A( x ) ) } with exactly one ordered pair present for each element

of X. The first element of the ordered pair is an element of the set X and

the second is its value in { 0, 1 }. The value ‘0’ is used to represent

non-membership and the value ‘1’ is used to represent membership of

the element A. The truth or falsity of the statement “x is in A” is

determined by the ordered pair. The statement is true, if the second

element of the ordered pair is ‘1’, and the statement is false, if it is ‘0’

Similarly, a fuzzy subset A of a set X can be defined as a set of

ordered pairs { ( x, A( x ) ) : xX }, each with the first element from X

and the second element from the interval [ 0, 1 ] with exactly one

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ordered pair present for each element of X. This defines a mapping, A

between elements of the set X and values in the interval [ 0, 1].

That is, A : X [ 0, 1 ].

The value ‘0’ is used to represent complete non-membership, the

value ‘1’ is used to represent complete membership and values in

between are used to represent intermediate degrees of membership.

The set X is referred to as the Universe of discourse for the fuzzy

subset A. Frequently, the mapping A is described as a function, the

membership function of A, the degree to which the statement ‘‘x is in

A’’ is true, is determined by finding the ordered pair ( x, A(x) ). The

degree of truth of the statement is the second element of the ordered

pair.

2. Intuitionistic fuzzy subsets :

Prof. K.T. Atanassov, a Bulgarian Engineer, introduced a new

component which determines the degree of non-membership also in

defining Intuitionistic Fuzzy Subset( IFS ) theory. In 1983, he came

across A.Kauffmann’s book ‘‘Introduction to the theory of Fuzzy

subsets’’ Academic Press, New York, 1975, then he tried to introduce

IFS to study the properties of the new objects so defined. He defined

ordinary operations as , , + and – over the new sets, then defined

operators similar to the operators ‘necessity’ and ‘possibility’.

George Gargov named new sets as the ‘‘Intuitionistic Fuzzy

Subsets’’, as their fuzzification denies the law of the excluded middle,

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A Ac = X. This has encouraged Prof. K.T.Atanassov to continue his

work on intuitionistic fuzzy subsets.

CHAPTER - I

PRELIMINARIES

1. Introduction :

This chapter contains the basic concepts required to devolop the

thesis.

1.1 Definition :

A non-empty set G together with a binary operation ‘’ that maps

G x G into G is called a group if the following conditions are satisfied:

(i) is associative,

(ii) there exists an element eG such that a e = e a = a, for

all aG, e is called the identity element of G,

(iii) for any element a in G there exists an element a'G such

that a a' = a' a = e, a' is called the inverse of a.

1.1 Example :

If Z is the set of integers, then ( Z, + ) is a group under the usual

addition.

1.2 Definition :

A subset H of a group G is called a subgroup of G if H forms a

group with respect to the induced binary operation of G.

1.2 Example :

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The additive group of even integers is a subgroup of the additive

group of all integers.

1.3 Definition :

A subgroup ( H, . ) of a group (G, . ) is called a normal

subgroup of G if aH = Ha, for all aG.

1.3 Example :

The multiplicative group { 1, –1 } is a normal subgroup of the

multiplicative group { 1, –1, i, –i }.

1.4 Definition :

If ( G, . ) and ( G׀, . ) are any two groups, then the function

f : G → G׀ is called a group homomorphism if f(xy) = f(x)f(y) , for all

x and yG.

1.4 Example :

Let Z be the set of integers and let G be the group ( Z, + ) and H

be the multiplicative group { 1, –1 }. Define f : G → H by f(x) = 1 , if x

is even and f(x) = –1 , if x is odd . Then f is a group homomorphism.

1.5 Definition :

If ( G, . ) and ( G׀, . ) are any two groups, then the function

f : G → G׀ is called a group anti-homomorphism if f(xy) = f(y)f(x),

for all x and yG.

1.5 Example :

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If R and C are additive groups of real and complex numbers

respectively and if the mapping f : C → R is defined by f ( x + iy ) = x ,

then f is a group anti-homomorphism.

1.6 Definition :

Let ( G, . ) and ( G׀, . ) be two groups. A map f : G → G׀ is

called a group isomorphism if the following conditions are satisfied:

(i) f is a bijection,

(ii) f(xy) = f(x)f(y) , for all x and yG.

1. 6 Example :

Let G = { 1, –1, i, –i } be a group under multiplication and

Z4 = { 0, 1, 2, 3 }, ( Z4 , ) is a group. Then f : G → ( Z4 , ) defined by

f (1) = 0 , f (–1 ) = 2 , f ( i ) = 1 , f (–i ) = 3, is an isomorphism.

1.7 Definition :

Let ( G, . ) and ( G׀, . ) be two groups. A map f : G → G׀ is

called a group anti-isomorphism if the following conditions are

satisfied:

(i) f is a bijection,

(ii) f(xy) = f(y)f(x), for all x and yG.

1.7 Example :

Let R be the set of all real numbers and R+ be the set of all

positive real numbers, ( R, + ) and ( R+, . ) are groups. Then

f : ( R, + ) → ( R+, . ). Defined by f(x) = ex. Then f is an anti-

isomorphism.

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1. 8 Definition :

An isomorphism of a group ( G, . ) to itself is called a group

automorphism of G. It is denoted by Aut G.

1. 8 Example :

Let ( G, . ) be any group. Let a G. Then a : G → G defined by

a (x) = axa-1 is an automorphism.

1. 9 Definition :

An anti-isomorphism of a group ( G, . ) to itself is called a group

anti-automorphism of G. It is denoted by anti-Aut G.

1.10 Definition :

Let ( G, . ) be any group. If aG, then the conjugate class of a is

defined as C (a) = { x G / a x },where a x means x = c-1ac , c

G.

1.11 Definition :

We call a group G, Hamiltonian if G is non-abelian and every

subgroup of G is normal .

1.12 Definition :

A Dedekind group is one which is abelian or Hamiltonian .

1.13 Definition :

A subgroup ( H, . ) of ( G, . ) is said to be a characteristic

subgroup of G if f( H ) H , for all automorphisms f of G.

1.14 Definition:

A poset ( L, ≤ ) is called a lattice if sup{a, b} and inf{a, b} exist,

for all a, b L.

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1.15 Definition:

An algebra ( L, , ) is called lattice if L is a nonvoid set, and

are binary operations on L, both and are idempotent, commutative

and associative and they satisfy the two absorption identities.

The interconnectedness of the two definitions above is given with the

following relations:

a ≤ b iff a b = a ,

a ≤ b iff a b = b.

1.9 Example:

The poset N with the usual ≤ is a lattice. If a, b N, then

a b = max {a, b} and a b = min {a, b}, where N is the set of all

natural numbers.

1.16 Definition:

A non-empty subset S of a lattice L is called a sublattice if

a, b S implies a b, a b in S.

1.10 Example:

Let ( N, ≤ ) be a lattice. Any non-empty subset of N is a

sub-lattice of N, where N is the set of all natural numbers.

1.17 Definition:

Lattice ( L, ≤ ) in which every subset X has GLB, inf( X ) and

LUB, sup (X) are called complete lattices.

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1.11 Example:

The unit interval, i.e. the closed interval [0, l] of real numbers is a

lattice with respect to the operations min and max for meet and join

respectively. The lattice ( [0, 1] , min, max ) is complete.

1.18 Definition :

Let X be a non-empty set. A fuzzy subset A of X is a function

A : X → [ 0, 1 ].

1.12 Example :

Let X = { a, b, c } be set. Then A = { a, 0.4 , b, 0.1 ,

c, 0.3 } is a fuzzy subset of X.

1.19 Definition :

The characteristic function A : X → { 0, 1 } is defined by

A(x) = 0, if xA and A(x) = 1, if xA.

1.20 Definition:

Let X be a non-empty set and L = (L, ≤) be a lattice with least

element 0 and greatest element 1. A L-fuzzy subset A of X is a function

A : X L.

1.13 Example:

Let Z be the ring of integers and A : Z L be a L-fuzzy subset

defined by

1, if x = 0

A(x) = 1/3, if x (2) – 0

0, if x Z – (2).

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1.21 Definition :

The union of two L-fuzzy subsets A and B of a set X is defined

by (AB)(x) = A(x) B(x), for all xX.

1.14 Example :

Let A = { a, 0.4 , b, 0.7 , c, 0.3 } and B = { a, 0.5 ,

b, 0.3 , c, 0.43 } be two L-fuzzy subsets of X= { a, b, c }. The

union of the L-fuzzy subsets A and B is AB = { a, 0.5 , b, 0.7 ,

c, 0.43 }.

1.22 Definition :

The intersection of two L-fuzzy subsets A and B of a set X is

defined by (AB)(x) = A(x) B(x), for all xX.

1.15 Example :

Let A = { a, 0.54 , b, 0.57 , c, 0.93 } and

B = { a, 0.75 , b, 0.63 , c, 0.43 } be two L-fuzzy subsets of

X = { a, b, c}. The intersection of the L-fuzzy subsets A and B is

AB = { a, 0.54 , b, 0.57 , c, 0.43 }.

1.23 Definition :

If A is a L-fuzzy subset of a set X , then the complement of A,

denoted Ac is the L-fuzzy subset of X, given by Ac(x) = 1– A(x), for

all x and yX.

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1.16 Example :

Let A = { a, 0.4 , b, 0.7 , c, 0.8 } be a L-fuzzy subset of

X = { a, b, c}. The complement of A is Ac ={ a, 0.6 , b, 0.3 ,

c, 0.2 }.

1.24 Definition :

Let ( G, . ) be a group. A L-fuzzy subset A of G is said to be a

L-fuzzy subgroup(FSG) of G if the following conditions are satisfied:

(i) A( xy ) A( x ) A( y ),

(ii) A( x -1) A( x ), for all x and yG.

1.17 Example :

Let Z be the additive group of all integers. For any integer n, nZ

denotes the set of all integer multiples of n.

That is nZ = { 0, n, 2n, 3n, ………….. }.

We have Z 2Z 4Z 8Z 16Z. Define A : Z → L by

A(x) = 1, if x16Z

= 0.7, if x8Z–16Z

= 0.5, if x4Z – 8Z

= 0.2, if x2Z – 4Z

= 0, if xZ – 2Z.

It can be easily verified that A is a fuzzy subgroup of Z.

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1.25 Definition :

Let A be a L-fuzzy subgroup of a group ( G, . ). Then for t L

and t ≤ A(e), the subgroup At = { x G : A(x) ≥ t } is called a level

subgroup of the L-fuzzy subgroup A.

1.26 Definition :

Let ( G, . ) be a group. A L-fuzzy subgroup A of G is said to be a

L-fuzzy normal subgroup(LFNSG) of G if A( xy ) = A( yx ), for all x

and yG.

1.27 Definition :

Let ( G, . ) be a group. A L-fuzzy subset A of G is said to be an

anti-L-fuzzy subgroup(ALFSG) of G if the following conditions are

satisfied:

(i) A( xy ) A( x ) A( y ),

(ii) A( x-1 ) A( x ), for all x and yG.

1.18 Example :

Let G be the Klein’s four group, G = { e, a, b, ab }, where

a2 = b2 = e and ab = ba. Then A = { e, 0.2 , a, 0.4 , b, 0.6 ,

ab, 0.6 } is an anti-L-fuzzy subgroup of G.

1.28 Definition :

Let ( G, . ) be a group. An anti-L-fuzzy subgroup A of G is said

to be an anti-L-fuzzy normal subgroup(ALFNSG) of G if

A( xy ) = A( yx ), for all x and yG.

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1.29 Definition [1]:

An intuitionistic fuzzy subset( IFS ) A in a set X is defined as

an object of the form A = { x, A(x), A(x) / xX }, where

A : X [0, 1] and A : X [0, 1] define the degree of membership

and the degree of non-membership of the element xX respectively

and for every xX satisfying 0 A(x) + A(x) 1.

1.19 Example :

Let X = { a, b, c } be a set. Then A={ a, 0.5, 0.3 , b, 0.1, 0.7 ,

c, 0.5, 0.4 } is an intuitionistic fuzzy subset of X.

1.30 Definition[2] :

Let A = { x, A(x), A(x) / xX } be an intuitionistic fuzzy

subset of a set X. Let X*= X x [ 0,1 ] x [ 0,1 ]. That is, A X*. We

define the characteristic function of an intuitionistic fuzzy subset A,

A : X*→ { 0,1 }, by A( x, a, b ) = 1, if A(x) = a and A(x) = b, and

A( x, a, b ) = 0, otherwise.

1.31 Definition:

Let ( L, ≤ ) be a complete lattice with an involutive order

reversing operation N : L L. An intuitionistic L-fuzzy subset (ILFS)

A in X is defined as an object of the form A = {< x, A(x), A(x) > /

xX }, where A : X L and A : X L define the degree of

membership and the degree of non-membership of the element xX

respectively and for every xX satisfying A(x) ≤ N( A(x) ).

1.20 Example:

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Let L be a complete lattice and A : X L be an intuitionistic

L-fuzzy subset A = < x, A(x), A(x) > defined as

0.7, if x (4)

A(x) = 0.4, if x (2) – (4)

0, otherwise

and

0.2, if x (4)

A(x) = 0.6, if x (2) – (4)

1, otherwise, for all x X.

1.32 Definition[2] :

Let A and B be two intuitionistic L-fuzzy subsets of a set X. We

define the following relations and operations:

(i) A B iff A(x) ≤ B(x) and A(x) ≥ B(x), for all xX.

(ii) A = B iff A(x) = B(x) and A(x) = B(x), for all xX.

(iii) Ā = { x, A(x), A(x) / xX }.

(iv) A B = { x, A(x) B(x) , A(x) B(x) / xX }.

(v) A B = { x, A(x) B(x) , A(x) B(x) / xX }.

(vi) A + B = { x, ( A(x) + B(x) – A(x) .B(x) ), ( A(x) .B(x) )

/ xX }.

(vii) A . B = { x, ( A(x) .B(x) ), ( A(x) + B(x) – A(x) .B(x) )

/ xX }.

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(viii) A @ B = { x, ( A(x) + B(x) ) / 2, ( A(x) + B(x) ) / 2

/ xX }.

(ix) A $ B = { x, , / xX }.

(x) A * B ={ x, ( A(x) + B(x) ) / 2( A(x) .B(x) + 1 ),

( A(x) + B(x) ) / 2 ( A(x) .B(x) + 1 ) / xX }.

(xi) A И B = { x, 2 ( A(x) .B(x) ) / ( A(x) + B(x) ),

2 ( A(x) .B(x) ) / ( A(x) + B(x) ) / xX }.

(xii) A B = { x, A(x) B(x) , A(x) B(x) / xX }.

(xiii) A = { x, A(x), 1–A(x) / xX }.

(xiv) A = { x, 1–A(x), A(x) / xX }, for all xX.

CHAPTER -II

INTUITIONISTIC L-FUZZY SUBGROUPS

2.1 Introduction :

This chapter contains some definitions and results in intuitionistic

L-fuzzy subgroup theory and intuitionistic L-fuzzy normal subgroup

theory, which are required in the sequel. Some Theorems and properties

of homomorphism and anti-homomorphism of intuitionistic L-fuzzy

subgroups and normals are introduced.

2.1.1 Definition :

Let G be a group. An intuitionistic L-fuzzy subset A of G is said

to be an intuitionistic L-fuzzy subgroup(ILFSG) of G if the following

conditions are satisfied:

(i) A( xy ) A(x)A(y),

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(ii) A( x -1 ) A( x ),

(iii) A( xy ) A(x) A(y),

(iv) A( x-1 ) A( x ), for all x and yG.

2.1.1 Example :

Let G = { 1, 1, i, i } be a group with multiplicative operation.

Then A = { 1, 0.7, 0.2 , 1, 0.4, 0.3 , i, 0.2, 0.4 , i, 0.2, 0.4 }

is an intuitionistic L-fuzzy subgroup of G.

2.1.2 Definition :

Let G and G׀ be any two groups. Let f : G → G׀ be any function

and let A be an intuitionistic L-fuzzy subgroup in G, V be an

intuitionistic L-fuzzy subgroup in f ( G ) = G׀, defined by

V(y) = A(x) and V (y) = A(x), for all xG and yG׀.

Then A is called a preimage of V under f and is denoted by f -1(V).

2.1.3 Definition :

Let A and B be two intuitionistic L-fuzzy subsets of sets G and

H, respectively. The product of A and B, denoted by AxB, is defined as

AxB = { ( x, y ), AxB( x, y ) , AxB( x, y ) / for all x in G and y in H },

where AxB( x, y ) = A(x) B(y) and AxB( x, y ) = A(x) B(y).

2.1.4 Definition :

Let A and B be two intuitionistic L-fuzzy subgroups of a group

G. Then A and B are said to be conjugate intuitionistic L-fuzzy

subgroups of G if for some gG , A(x) = B( g-1xg ) and

A(x) = B( g-1xg ), for every xG.

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2.1.5 Definition:

Let A be an intuitionistic L-fuzzy subset in a set S, the strongest

intuitionistic L-fuzzy relation on S, that is an intuitionistic L-fuzzy

relation on A is V given by V( x, y ) = A(x) A(y) and

V (x, y ) = A(x) A(y), for all x, yS.

2.1.6 Definition :

Let G be a group. An intuitionistic L-fuzzy subgroup A of G is

said to be an intuitionistic L-fuzzy normal subgroup(ILFNSG) of G if

the following conditions are satisfied:

(i) A(xy) = A(yx),

(ii) A(xy) = A(yx), for all x and yG.

2.1.7 Definition :

Let G be a group. An intuitionistic L-fuzzy subgroup A of G is

said to be an intuitionistic L-fuzzy characteristic subgroup(ILFCSG) of

G if the following conditions are satisfied:

(i) A(x) = A( f(x) ),

(ii) A(x) = A( f(x) ), for all xG and fAutG.

2.1.8 Definition :

An intuitionistic L-fuzzy subset A of a set X is said to be

normalized if there exist xX such that A(x) =1 and A(x) =0.

2.1.9 Definition :

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Let A be an intuitionistic L-fuzzy subgroup of a group G. For any

aG, aA defined by (aA)(x) = A( a-1x ) and (aA)(x) = A( a-1x ), for

every xG is called an intuitionistic L-fuzzy coset of the group G .

2.1.10 Definition :

Let A be an intuitionistic L-fuzzy subgroup of a group G and

H = { xG / A(x) = A(e) and A(x) = A(e) }, then O(A) ,order of A is

defined as O(A) = O(H).

2.1.11 Definition :

Let A be an intuitionistic L-fuzzy subgroup of a group G. Then

for any a,bG, an intuitionistic L-fuzzy middle coset aAb of G is

defined by (aAb)(x) = A( a-1x b-1 ) and (aAb)(x) = A( a-1x b-1 ), for

every xG.

2.1.12 Definition :

Let A be an intuitionistic L-fuzzy subgroup of a group G and

aG. Then the pseudo intuitionistic L-fuzzy coset (aA)p is defined by

( (aA)p )(x) = p(a)A(x) and ( (aA)p )(x) = p(a)A(x), for every xG and

for some pP.

2.1.13 Definition :

An intuitionistic L-fuzzy subgroup A of a group G is called a

generalized characteristic intuitionistic L-fuzzy subgroup (GCILFSG) if

for all x,y in G, (x) = (y) implies A(x) = A(y) and A(x) = A(y) .

2.2 – PROPERTIES OF INTUITIONISTIC L-FUZZY

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SUBGROUPS :

2.2.1 Theorem : If A is an intuitionistic L-fuzzy subgroup of a group

G, then A( x -1) = A(x) and A( x-1) = A(x), A(x) A(e) and

A(x) A(e), for xG and eG.

proof : For xG and eG.

Now, A(x) = A( (x-1 )-1 )

A( x -1) A( x ).

Therefore, A( x-1) = A(x).

And, A( x ) = A( (x-1)-1 )

A( x-1) A( x).

Therefore, A( x-1) = A(x).

Now, A(e) = A( xx-1)

A(x) A(x-1) } = A( x).

Therefore, A(e) A(x).

And, A(e) = A( xx-1)

A(x) A(x-1) = A(x).

Therefore, A(e) A(x).


G, then (i) A( xy -1) = A(e) gives A(x) = A(y).

(ii) A( xy-1) = A(e) gives A(x) = A(y), for xG and eG.

Proof : Let xG and e G.

Now, A(x) = A( xy-1y )

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A(xy-1) A(y)

= A(e) A(y)

= A(y)

= A( yx-1x )

A(yx-1) A(x)

= A(e) A(x)

= A(x).

Therefore, A(x) = A(y).

And, A(x) = A( xy-1y )

A( xy-1) A(y)

= A(e) A(y)

= A(y)

= A( yx-1x )

A( yx-1) A(x)

= A(e) A(x)

= A(x).


2.2.3 Theorem :A is an intuitionistic L-fuzzy subgroup of a group G iff

A( xy -1 ) A(x) A(y) and A( xy-1) A(x) A(y) , for all x G

and yG.

Proof : Assume that A is an intuitionistic L-fuzzy subgroup of

a group G.

We have, A( xy -1) A(x) A(y-1)

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A(x) A(y) , since A is an ILFSG of G.

Therefore, A( xy -1) A(x) A(y) .

And, A( xy-1) A(x) A(y-1)

A(x) A(y) , since A is an ILFSG of G.

Therefore, A( xy-1) A(x) A(y) .

Conversely,

if A( xy -1) A(x) A(y) and A( xy-1) A(x) A(y) ,

replace y by x, then

A(x) A(e) and A(x) A(e), for all xG and yG.

Now, A( x -1) = A( ex -1)

A(e) A(x)

= A(x).

Therefore, A( x -1) A(x).

It follows that,

A(xy) = A( x(y-1)-1 )

A(x) A( y -1)

A(x) A(y) .

Therefore, A(xy ) A(x) A(y) .

And, A( x-1) = A( ex-1)

A(e) A(x)

= A(x).

Therefore, A ( x-1) A(x).

Then, A( xy) = A( x(y-1)-1 )

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A(x) A(y-1)

A(x) A(y) .

Therefore, A( xy) A(x) A(y) .

Hence A is an intuitionistic L-fuzzy subgroup of a group G.

2.2.4 Theorem : Let A be an intuitionistic L-fuzzy subset of a group G.

If A(e) = 1 and A(e) = 0 and A( xy-1 ) A(x) A(y) } and

A( xy-1 ) A(x) A(y) , then A is an intuitionistic L-fuzzy subgroup

of a group G.

Proof: We verify the axioms of an intuitionistic L-fuzzy subgroup of a

group G.

(i) A(x -1) = A( ex -1)

A(e) A(x)

= 1 A(x)

= A(x).

Therefore, A( x -1) A(x).

(ii) A(x-1) = A( ex-1)

A(e) A(x)

= 0 A(x)

= A(x).

Therefore, A( x-1) A(x).

(iii) A( xy) = A( x(y-1)-1 )

A(x) A(y -1)

A(x) A(y) .

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Therefore, A(xy) A(x) A(y) .

(iv) A( xy) = A( x(y-1)-1 )

A(x) A(y-1)

A(x) A(y) .



2.2.5 Theorem : If A is an intuitionistic L-fuzzy subgroup of a group G,

then H = { x / xG : A(x) = 1, A(x) = 0 } is either empty or is

a subgroup of a group G.

proof : If no element satisfies this condition, then H is empty.

If xH and yH, then

A( xy -1) A(x) A( y-1)

A(x) A(y) , since A is an ILFSG of a group G

= 1 1 = 1.

Therefore, A( xy -1) = 1.

And, A( xy-1) A(x) A(y-1)

= A(x) A(y) , since A is an ILFSG of a group G

= 0 0 = 0.

Therefore, A( xy-1) = 0.

We get, xy -1H.

Therefore, H is a subgroup of a group G.

Hence H is either empty or is a subgroup of a group G.

22


G , then H = { x, A(x) : 0 < A(x) 1 and A(x) = 0 } is either

empty or is a L-fuzzy subgroup of G.

proof : If no element satisfies this condition, then H is empty.

If A is an intuitionistic L-fuzzy subgroup of a group G , then

A( xy-1) A(x) A(y-1)

A(x) A(y)

= 0 0 = 0.

Therefore, A( xy-1) = 0.

And, A( xy -1 ) A(x) A(y-1)

A(x) A(y) , since A is an ILFSG of a group G.


Hence H is a L-fuzzy subgroup of a group G.

Therefore, H is either empty or is a L-fuzzy subgroup of a group G.


G, then H = { x, A(x) : 0 < A(x) 1 } is a L-fuzzy subgroup of G.

proof : If A is an intuitionistic L-fuzzy subgroup of a group G , then

A( xy -1) A(x) A(y-1)




23


then H = { x, A(x) : 0 < A(x) 1} is a anti-L-fuzzy subgroup of a

group G.

proof : If A is an intuitionistic L-fuzzy subgroup of a group G , then

A( xy-1) A(x) A(y-1)



Hence H is an anti-L-fuzzy subgroup of a group G.


then H = { xG : A(x) = A(e) and A(x) = A(e) }is a subgroup of G.

proof : Given H = { xG : A(x) = A(e) and A(x) = A(e) }.

Let xG.

By Theorem 2.2.1, we have

A( x -1) = A(x) = A(e).

A( x-1) = A(x) = A(e).

Therefore, A( x -1) = A(e) and A( x-1) = A(e).

Hence x -1 H.

Now, A( xy-1) A(x) A(y -1)


= A(e) A(e)

= A(e).

Therefore, A( xy-1) A(e) ------------------------- (1).

And, A(e) = A( (xy-1)(xy-1)-1 )

24

A( xy-1) A( xy-1)-1 , since A is an ILFSG of a group G

A(xy-1) A(xy-1) }

= A( xy-1).

Therefore, A(e) A( xy-1) --------------------------- (2).

From (1) and (2), we get A(e) = A( xy-1 ).

Now, A( xy-1) A(x) A( y -1 )


= A(e) A(e)

= A(e).

Therefore, A( xy-1 ) A(e) ----------------------------- (3).

And, A(e) = A( (xy-1)(xy-1)-1 )

A( xy-1 ) A( (xy-1)-1) , since A is an ILFSG of a group G

A( xy-1 ) A( xy-1 )

= A( xy-1 ).

Therefore, A(e) A( xy-1 ) ------------------------------- (4).

From (3) and (4), we get A(e) = A( xy-1 ).

Hence A(e) = A( xy-1 ) and A(e) = A( xy-1 ).

Therefore, xy-1H.

Hence H is a subgroup of a group G.


G , then H = { x, A(x) : A(x) = A(e) and A(x) = A (e) } is a

L-fuzzy subgroup of G.

Proof : Given that A is an intuitionistic L-fuzzy subgroup of a group G.

25

Let x, y and e be elements in G.

By Theorem 2.2.9,

H = { x : A(x) = A(e) and A(x) = A(e) }is a subgroup of G.

Therefore, xy-1H.

Now, A( xy-1) A(x) A( y -1 )

A(x) A(y), since A is an ILFSG of a group G.




G , then H = { x, A(x) : A(x) = A(e) and A(x) = A(e) } is an anti-


Proof : Given that A is an intuitionistic L-fuzzy subgroup of a group G.

Let x, y and e be elements in G.

By Theorem 2.2.9,

H = { x : A(x) = A(e) and A(x) = A(e) }is a subgroup of G.

Therefore, xy-1H.

Now, A( xy-1 ) A(x) A( y -1 )


Therefore, A( xy-1 ) A(x) A(y) .

Hence H is an anti-L-fuzzy subgroup of a group G.

2.2.12 Theorem : Let A be an intuitionistic L-fuzzy subgroup of a

group G, then( i) if A( xy-1 ) = 1, then A(x) = A(y).

(ii) if A( xy-1 ) = 0, then A(x) = A(y), for xG and yG.

26

Proof: Let x and y belongs to G.

(i) Now, A(x) = A( xy-1y )

A( xy-1 ) A(y), since A is an ILFSG of a group G

= 1 A(y) = A(y)

= A( y-1 ) , since A is an ILFSG of a group G

= A( x-1xy-1 )

A( x-1 ) A( xy-1 )

= A( x-1) 1 = A( x-1) = A(x).


(ii) Now, A(x) = A( xy-1y )

A( xy-1 ) A(y) , since A is an ILFSG of a group G

= 0 A(y) = A(y)

= A( y-1), since A is an ILFSG of a group G

= A( x-1xy-1 )

A( x-1 ) A( xy-1 )

= A( x-1 ) 0 = A( x-1) = A(x).


2.2.13 Theorem : If A be an intuitionistic L-fuzzy subgroup of a group

G, then

(i) if A( xy-1 ) = 0, then A(x) A(y) but A(x) 0.

(ii) if A( xy-1 ) = 1, then A(x) A(y) but A(x) 1

for all xG and yG.

Proof: Let x G and yG.

27

(i) Now, A(x) = A( xy-1y )

A( xy-1 ) A(y) , since A is an ILFSG of a group G

= 0 A(y) = 0.

Therefore, A(x) 0 and clearly A(x) A(y).

(ii) Now, A(x) = A( xy-1y )

A( xy-1 ) A(y), since A is an ILFSG of a group G

= 1 A(y) = 1.

Therefore, A(x) 1.

2.2.14 Theorem : Let G be a group. If A is an intuitionistic L-fuzzy

subgroup of G, then A( xy ) = A(x) A(y) and A( xy ) = A(x) A(y)

for each x and y in G with A(x) A(y) and A(x) A(y).

Proof: Let xG and yG.

Assume that A(x) A(y) and A(x) A(y).

Now, A(y) = A( x-1xy )

A( x-1 ) A( xy )

A(x) A(xy)

= A(xy)

A(x) A(y)

= A(y).

Therefore, A(xy) = A(y) = A(x) A(y) .

And, A(y) = A( x-1xy )

A( x-1 ) A( xy )

A(x) A( xy )

28

= A(xy)

A(x) A(y)

= A(y).

Therefore, A( xy) = A(y) = A(x) A(y) .

2.2.15 Theorem : If A and B are two intuitionistic L-fuzzy subgroups of

a group G, then their intersection AB is an intuitionistic L-fuzzy

subgroup of G.

Proof : Let xG and yG.

Let A = { x, A(x), A(x) / xG } and

B = { x, B(x), B(x) / xG }.

Let C = AB and C = { x, C(x), C(x) / xG }.

(i) C(xy) = A(xy) B(xy)

{A(x) A(y)}{ B(x) B(y) }

{ A(x) B(x) } { A(y) B(y) }

= C(x) C(y) .

Therefore, C(xy) C(x) C(y) .

(ii) C( x-1) = A( x-1 ) B( x-1 )

A(x) B(x)

= C(x).

Therefore, C( x-1) C(x).

(iii) C(xy) = A(xy) B(xy)

{ A(x) A(y) }{ B(x) B(y) }

{ A(x) B(x) }{ A(y) B(y) }

29

= C(x) C(y) .

Therefore, C(xy) C(x) C(y) .

(iv) C(x-1) = A( x-1 ) B( x-1 )

A(x) B(x)

= C(x).

Therefore, C( x-1 ) C(x).

Hence AB is an intuitionistic L-fuzzy subgroup of a group G.

2.2.16 Theorem : The intersection of a family of intuitionistic L-fuzzy

subgroups of a group G is an intuitionistic L-fuzzy subgroup of a

group G.

Proof: Let { Ai }iI be a family of an intuitionistic L-fuzzy subgroup of

a group G and let A = Ai .

Then for x and y belongs to G,

we have

(i) A(xy) =

( ) ( )

= A(x) A(y) .


(ii) A( x-1) =

30

= A(x).


(iii) A(xy) =

( ) ( )

= A(x) A(y) .


(iv) A( x-1) =

= A(x).


Hence the intersection of a family of intuitionistic L-fuzzy subgroups of

a group G is an intuitionistic L-fuzzy subgroup of G.


G, then (i) A(xy) = A(yx) iff A(x) = A( y-1xy ).

(ii) A(xy) = A(yx) iff A(x) = A( y-1xy ), for xG and yG.

Proof : (i) Let xG and yG .

Assume that A(xy) = A(yx).

31

Now, A( y-1xy ) = A( y-1yx )

= A(ex)

= A(x).

Therefore, A(x) = A( y-1xy ).

Conversely, assume that A(x) = A( y-1xy ).

Now, A( xy ) = A( xyxx-1 )

= A(yx).

Therefore, A(xy) = A(yx).

(ii) Assume that A(xy) = A(yx).

Now, A( y-1xy ) = A( y-1yx )

= A(ex) = A(x).

Therefore, A(x) = A( y-1xy ).

Conversely, assume that A(x) = A( y-1xy ).

Now, A( xy ) = A( xyxx-1 )

= A(yx).



group G. If A(x) < A(y ) and A(x) > A(y), for some xG and yG,

then (i) A(xy) = A(x) = A(yx)

(ii) A(xy) = A(x) = A(yx) .

proof : Let A be an intuitionistic Lfuzzy subgroup of a group G.

Given A(x) < A(y) and A(x) > A(y), for some xG and yG,

A(xy) A(x) A(y) , as A is an ILFSG of G

32

= A(x) ; and

A(x) = A( xyy-1)

A(xy) A(y-1)

A(xy) A(y) , as A is an ILFSG of G

= A(xy).

Therefore, A(xy) = A(x).

And, A(yx) A(y) A(x) , as A is an ILFSG of G

= A(x) ; and

A(x) = A( y-1yx )

A( y-1) A(yx)

A(y) A(yx) , as A is an ILFSG of G

= A(yx).

Therefore, A(yx) = A(x).

Hence A(xy) = A(x) = A(yx).

Thus (i) is proved.

Now, A(xy) A(x) A(y) , as A is an ILFSG of G

= A(x) ; and

A(x) = A( x yy-1 )

A(xy) A(y-1)


= A(xy).



33

= A(x) ; and

A(x) = A( y-1yx )

A(y-1) A(yx)


= A(yx).

Therefore, A(yx) = A(x).


Thus (ii) is proved.


group G .If A(x) < A(y) and A(x) < A(y), for some xG and yG,

then (i) A(xy) = A(x) = A(yx) and

(ii) A(xy) = A(y) = A(yx).

proof : Let A be an intuitionistic L-fuzzy subgroup of a group G.

Given A(x) < A(y) and A(x) < A(y), for some xG and yG,

A( xy) A(x) A(y) , as A is an ILFSG of G

= A(x) ; and

A(x) = A( x yy-1 )

A(xy) A(y-1 )


= A(xy).



= A(x) ; and

34

A(x) = A( y-1yx )

A(y-1) A(yx)


= A(yx).


Thus (i) is proved .

Now, A(xy) A(x) A(y) , as A is an ILFSG of G

= A(y) ; and

A(y) = A( x-1xy )

A(x-1) A(xy)

A(x) A(xy) , as A is an ILFSG of G

= A(xy).

Therefore, A(xy) = A(y).

And, A(yx) A(y ) A(x) , as A is an ILFSG of G

= A(y) ; and

A(y) = A( yxx-1 )

A(yx) A(x-1)

A(yx) A(x) , as A is an ILFSG of G

= A(yx).

Therefore, A(yx) = A(y).

Hence A(xy) = A(y ) = A(yx).

Thus (ii) is proved .

35


group G. If A(x) > A(y) and A(x) > A(y), for some xG and yG,

then (i) A(xy) = A(y) = A(yx) and

(ii) A(xy) = A(x) = A(yx).

Proof : It is trivial.


group G. If A(x) > A(y) and A(x) < A(y), for some xG and yG ,

then (i) A(xy) = A(y) = A(yx) and

(ii) A(xy) = A(y) = A(yx).



group G such that Im A = { } and Im A = { }, where , L. If

A = BC, where B and C are intuitionistic L-fuzzy subgroups of a

group G, then either B C or C B.

Proof: Case (i) :

Let A = B C = { x , A(x) , A(x) / xG },

B = { x, B(x), B(x) / xG } and

C = { x, C(x), C(x) / xG }.

Assume that B(x) > C(x) and B(y) < C(y), for some x and yG.

Then, = A(x)

= BC(x)

= B(x) C(x)

36

= B(x) > C(x).

Therefore, > C(x).

And, = A(y)

= BC(y)

= B(y) C(y)

= C(y) > B(y).

Therefore, > B(y).

So that, C(y) > C(x) and B(x) > B(y).

Hence B(xy) = B(y) and C(xy) = C(x),

by Theorem 2.2.18 and 2.2.20.

But then, = A(xy) = BC(xy)

= B(xy) C(xy)

= B(y) C(x)

< --------------------------(1).

Case (ii) :

Assume that B(x) < C(x) and B(y) > C(y), for some x and yG.

Then, = A(x) = BC(x)

= B(x) C(x)

= B(x) < C(x).

Therefore, < C(x).

And, = A(y) = BC(y)

= B(y) C(y)

= C(y) < B(y).

37

Therefore, < B(y).

So that, C(y) < C(x) and B(x) < B(y).

Hence B(xy) = B(y) and C(xy) = C(x), by Theorem 2.2.18 and 2.2.20.

But then, = A(xy) = BC(xy)

= B(xy) C(xy)

= B(y) C(x)

> --------------------------(2).

It is a contradiction by (1) and (2).

Therefore, either B C or C B is true.


G, then A is an intuitionistic L-fuzzy subgroup of a group G.

Proof: Let A be an intuitionistic L-fuzzy subgroup of a group G.

That is A = { x, A(x), A(x) }, for all xG.

Let A = B ={ x, B(x), B(x) }.

Clearly, B(xy) ≥ B(x) B(y)

and B(x) = B(x-1), by the definition 1.32 (xiii).

Now, A( xy ) ≥ A(x) A(y) , since A is an ILFSG of G,

which implies that 1– B(xy) ≥ ( 1– B(x) ) ( 1– B(y) ).

That is , B(xy) ≤ 1– { ( 1– B(x) ) ( 1– B(y) ) }

≤ B(x) B(y) .

Therefore, B(xy) ≤ B(x) B(y).

And, A(x) = A( x-1),

which implies that 1– B(x) = 1– B( x-1).

38

That is, B(x) = B( x-1).

Hence B = A is an intuitionistic L-fuzzy subgroup of a group G.

Remark :

The converse of the above theorem is not true. It is shown by the

following example:

Example :

Let G be the Klein’s four group G ={ e, a, b, ab }, where

a2 = b2 = e and ab = ba.

Define A = { e, 0.5, 0.4 , a, 0.3, 0.3 , b, 0.2, 0.4 , ab, 0.2, 0.4

}.

Clearly A is not an intuitionistic L-fuzzy subgroup of a group G.

Now, A = { e, 0.5, 0.5 , a, 0.3, 0.7 , b, 0.2, 0.8 , ab, 0.2, 0.8 }

which is an intuitionistic L-fuzzy subgroup of G.


G, then A is an intuitionistic L-fuzzy subgroup of a group G.

Proof: Let A be an intuitionistic L-fuzzy subgroup of a group G.

That is A = { x, A(x), A(x) }, for all xG.

Let A = B = { x, B(x), B(x) }.

Clearly, B(xy) ≤ B(x) B(y)

and B(x) = B(x-1), by the definition 1.32 (xiv).

Now, A(xy) ≤ A(x) A(y) , since A is an ILFSG of G,

which implies that 1– B(xy) ≤ ( 1– B(x) ) ( 1– B(y) ).

That is, B(xy) ≥ 1– { ( 1– B(x) ) ( 1– B(y) ) }

39

≥ B(x) B(y) .

Therefore, B(xy) ≥ B(x) B(y) .

And, A(x) = A(x-1),

which implies that 1– B(x) = 1– B(x-1).

That is , B(x) = B(x-1).

Hence B = A is an intuitionistic L-fuzzy subgroup of a group G.

Remark : The converse of the above theorem is not true. It is shown by

the following example.

Example :

Let G be the Klein’s four group G = { e, a , b, ab }, where a2 = b2 =

e and ab = ba.

Define A = { e, 0.1, 0.1 , a, 0.3, 0.3 , b, 0.2, 0.4 , ab, 0.2, 0.4

}.

Clearly A is not an intuitionistic L-fuzzy subgroup of a group G.

Now, A = { e, 0.9, 0.1 , a, 0.7, 0.3 , b, 0.6, 0.4 , ab, 0.6, 0.4 }

which is an intuitionistic L-fuzzy subgroup of G.

Let us verify the operations in Definition 1.32 for all intuitionistic

L-fuzzy subgroups of a group.

Consider the following example:

Example :

40

Let G be the Klein’s four group G = { e, a , b, ab }, where

a2 = b2 = e and ab = ba.

Case I :

Define intuitionistic L-fuzzy subgroups A and B of a group G

such that A = { e, 0.5, 0.1, a, 0.3, 0.3 , b, 0.2, 0.4 , ab, 0.2, 0.4 }

and B = { e, 0.4, 0.2 , a, 0.1, 0.3 , b, 0.3, 0.2 , ab, 0.1, 0.3 }.

When A B.

(i) A B = { e, 0.4, 0.2 , a, 0.1, 0.3 , b, 0.2, 0.4 ,

ab, 0.1, 0.4 }.

Clearly A B is an intuitionistic L-fuzzy subgroup of a group G.

(ii) A B ={ e, 0.5, 0.1 , a, 0.3, 0.3 , b, 0.3, 0.2 ,

ab, 0.2, 0.3 }.

Now, AB (ab) A B(a) A B (b)

gives 0.2 ≥ 0.3 0.3 which is not true.

Therefore, A B is not an intuitionistic L-fuzzy subgroup of a group G.

(iii) A + B = { e, 0.7, 0.2 , a, 0.37, 0.09 , b, 0.44, 0.08 ,

ab, 0.28, 0.12 }.

Now, A + B (ab) A + B (a) A + B (b)


Therefore, A + B is not an intuitionistic L-fuzzy subgroup of a group G.

(iv) A . B = { e, 0.2, 0.28 , a, 0.03, 0.51 , b, 0.06, 0.52 ,

ab, 0.02, 0.58 }.

Now, A.B(ab) A.B (a) A.B (b)

41


Therefore, A . B is not an intuitionistic L-fuzzy subgroup of a group G.

(v) A @ B = { e, 0.45, 0.15 , a, 0.2, 0.3 , b, 0.25, 0.3 ,

ab, 0.15, 0.35 }.

Now, A @ B (ab) A @ B (a) A @ B (b)


Therefore, A @ B is not an intuitionistic L-fuzzy subgroup of a group G.

(vi) A $ B ={ e, 0.44, 0.14 , a, 0.17, 0.3 , b, 0.24, 0.28 ,

ab, 0.14, 0.34 }.

Now, A $ B (ab) A $ B (a) A $ B (b)


Therefore, A $ B is not an intuitionistic L-fuzzy subgroup of a group G.

(vii) A * B ={ e, 0.37, 0.14 , a, 0.19, 0.27 , b, 0.23, 0.27 ,

ab, 0.14, 0.31 }.

Now, A * B (ab) A * B (a) A * B (b)


Therefore, A * B is not an intuitionistic L-fuzzy subgroup of a group G.

(viii) A И B = { e, 0.44, 0.13 , a, 0.15, 0.3 , b, 0.24, 0.26 ,

ab, 0.13, 0.34 }.

Now, A И B (ab) A И B (a) A И B (b)


Therefore, A И B is not an intuitionistic L-fuzzy subgroup of a group G.

42

(ix) A B = { e, 0.4, 0.2 , a, 0.3, 0.3 , b, 0.4, 0.2 ,

ab, 0.4, 0.2 }.

Now, A B (a) A B (ab) A B (b)


Therefore, A B is not an intuitionistic L-fuzzy subgroup of

a group G.

(x) A = { e, 0.5, 0.5 , a, 0.3, 0.7 , b, 0.2, 0.8 ,

ab, 0.2, 0.8 }.

Clearly A is an intuitionistic L-fuzzy subgroup of a group G.

(xi) A = { e, 0.9, 0.1, a, 0.7, 0.3 , b, 0.6, 0.4 , ab, 0.6, 0.4

}.


(xii) Ā ={ e, 0.1, 0.5 , a, 0.3, 0.3 , b, 0.4, 0.2 , ab, 0.4, 0.2

}.

Now, Ā (a) Ā (ab) Ā (b)


Therefore , Ā is not an intuitionistic L-fuzzy subgroup of a group G.

Case II :


such that A = {e, 0.5, 0.1, a, 0.3, 0.3, b, 0.2, 0.4, ab, 0.2, 0.4 }

and B = { e, 0.4, 0.2 , a, 0.1, 0.3 , b, 0.2, 0.4 , ab, 0.1, 0.4 }.

When B A.

43

(i) A B = B = { e, 0.4, 0.2 , a, 0.1, 0.3 , b, 0.2, 0.4 ,

ab, 0.1, 0.4 }.


(ii) A B = { e, 0.5, 0.1 , a, 0.3, 0.3 , b, 0.2, 0.4 ,

ab, 0.2, 0.4 }.


(iii) A + B ={ e, 0.7, 0.02 , a, 0.37, 0.9 , b, 0.36, 0.08 ,

ab, 0.28, 0.12 }.

Now, A + B (ab) A + B (a) A + B (b)


Therefore, A + B is not an intuitionistic L-fuzzy subgroup of a group G.

(iv) A . B = { e, 0.2, 0.28 , a, 0.03, 0.51 , b, 0.04, 0.52 ,

ab, 0.02, 0.58 }.

Now, A.B (ab) A.B (a) A.B (b)


Therefore, A . B is not an intuitionistic L-fuzzy subgroup of a group G.

(v) A @ B = { e, 0.45, 0.15 , a, 0.2, 0.3 , b, 0.2, 0.3 ,

ab, 0.15, 0.35 }.

Now, A @ B (ab) A @ B (a) A @ B (b)


Therefore, A @ B is not an intuitionistic L-fuzzy subgroup of a group G.

(vi) A $ B = { e, 0.44, 0.14 , a, 0.17, 0.3 , b, 0.2, 0.28 ,

ab, 0.14, 0.34 }.

44

Now, A $ B (ab) A $ B (a) A $ B (b)


Therefore, A $ B is not an intuitionistic L-fuzzy subgroup of a group G.

(vii) A * B = { e, 0.37, 0.14 , a, 0.19, 0.27 , b, 0.19, 0.27 ,

ab, 0.14, 0.31 }.

Now, A * B (ab) A * B (a) A *B (b)


Therefore, A * B is not an intuitionistic L-fuzzy subgroup of a group G.

(viii) A И B = { e, 0.44, 0.13 , a, 0.15, 0.3 , b, 0.2, 0.26 ,

ab, 0.13, 0.34 }.

Now, A И B (ab) A И B (a) A И B (b)


Therefore, A И B is not an intuitionistic L-fuzzy subgroup of a group G.

(ix) A B = { e, 0.4, 0.2 , a, 0.3, 0.3 , b, 0.4, 0.2 ,

ab, 0.4, 0.2 }.



Therefore,A B is not an intuitionistic L-fuzzy subgroup of a group G.

Case III :


such that A = {e, 0.5, 0.1, a, 0.3, 0.3 , b, 0.2, 0.4 , ab, 0.2, 0.4 }

and B = { e, 0.5, 0.1 , a, 0.3, 0.3 , b, 0.2, 0.4 , ab, 0.2, 0.4 }.

When A = B

45

(i) A B = { e, 0.5, 0.1,a, 0.3, 0.3 , b, 0.2, 0.4 , ab, 0.2, 0.4

}.


Therefore, A A = A.

(ii) AB = {e, 0.5, 0.1 , a, 0.3, 0.3 , b, 0.2, 0.4 , ab, 0.2, 0.4

}.


Therefore, A A = A.

(iii) A + B = { e, 0.75, 0.01 , a, 0.51, 0.09 , b, 0.36, 0.16 ,

ab, 0.36, 0.16 }.

Clearly A + B is an intuitionistic L-fuzzy subgroup of a group G.

Therefore, A + A A.

(iv) A . B = { e, 0.25, 0.19 , a, 0.09, 0.51 , b, 0.04, 0.64 ,

ab, 0.04, 0.64 }.

Clearly A . B is an intuitionistic L-fuzzy subgroup of a group G.

Therefore, A . A A.

(v) A @ B = { e, 0.5, 0.1, a, 0.3, 0.3, b, 0.2, 0.4, ab, 0.2,

0.4 }.

Clearly A @ B is an intuitionistic L-fuzzy subgroup of a group G.

Therefore, A @ A =A.

(vi) A $ B ={ e, 0.5, 0.1, a, 0.3 0.3 , b, 0.2, 0.4, ab, 0.2, 0.4

}.

Clearly A $ B is an intuitionistic L-fuzzy subgroup of a group G.

46

Therefore, A $ A = A.

(vii) A * B = { e, 0.4, 0.09 , a, 0.27, 0.27 , b, 0.19, 0.34 ,

ab, 0.19, 0.34 }.

Clearly A * B is an intuitionistic L-fuzzy subgroup of a group G.

Therefore, A * A A.

(viii) A И B = {e, 0.5, 0.1, a, 0.3, 0.3, b, 0.2, 0.4, ab, 0.2, 0.4

}.

Clearly A И B is an intuitionistic L-fuzzy subgroup of a group G.

Therefore, A И A = A.

(ix) A B = { e, 0.5, 0.1 , a, 0.3, 0.3 , b, 0.4, 0.2 ,

ab, 0.4, 0.2 }.



Therefore, A B is not an intuitionistic L-fuzzy subgroup of

a group G.

In this case even though intuitionistic L-fuzzy subgroups A and B are

equal this operation is not satisfied.

2.2.25 Theorem : If A is an intuitionistic L-fuzzy subgroup of a group G and

if there is a sequence {xn} in G such that { A(xn) A(xn) } = 1 and

{ A(xn) A(xn) } = 0, then A(e) = 1 and A(e) = 0, where e is the

identity in G .

47

Proof : Let A be an intuitionistic L-fuzzy subgroup of a group G with e

as its identity element.

Let xG be an arbitrary element.

We have xG implies x-1G and hence xx-1 = e.

Then, we have

A(e) = A( xx-1 )

≥ A(x) A( x-1 )

≥ A(x) A(x) .

Therefore, for each n, we have

A(e) ≥ A(x) A(x) .

Since A(e) ≥ { A(xn ) A(xn ) } = 1.

Therefore A(e) = 1.

And, A(e) = A(xx-1)

≤ A(x) A(x-1)

= A(x) A(x) .

Therefore, for each n, we have

A(e) ≤ A(x) A(x) .

Since A(e) ≤ A(xn ) A(xn ) = 0.

Therefore, A(e) = 0.

2.2.26 Theorem : If every intuitionistic L-fuzzy subgroup of a group G is

normal , then G is a Dedekind group.

48

Proof : Suppose that every intuitionistic L-fuzzy subgroup of a group G

is normal.

Since any subgroup H of a group G can be regarded as a level subgroup

of some intuitionistic L-fuzzy subgroup A of group G.

By assumption, A is an intuitionistic L-fuzzy normal subgroup of a

group G.

Now, it is easy to deduce that H is a normal subgroup of G.

Thus G is a Dedekind group.

2.2.27 Theorem : If A and B are intuitionistic L-fuzzy subgroups of the

groups G and H, respectively, then AxB is an intuitionistic L-fuzzy

subgroup of GxH.

Proof : Let A and B be intuitionistic L-fuzzy subgroups of the groups G

and H respectively.

Let x1 and x2 be in G, y1 and y2 be in H.

Then ( x1, y1 ) and ( x2, y2 ) are in GxH.

Now,

AxB [ (x1, y1)(x2, y2) ] = AxB ( x1x2, y1y2 )

= A( x1x2 ) B( y1y2 )

{ A(x1) A(x2 ) } { B(y1) B(y2) }

= { A(x1) B(y1) } { A(x2) B(y2) }

= AxB (x1, y1) AxB (x2, y2) .

Therefore, AxB [ (x1, y1)(x2, y2) ] AxB (x1, y1) AxB (x2, y2) .

49

And,

AxB [ (x1, y1)(x2, y2) ] = AxB ( x1x2, y1y2 )

= A(x1x2) B(y1y2)

{ A(x1) A(x2) }{ B(y1) B(y2) }

= { A(x1) B(y1) } { A(x2) B(y2) }

= AxB (x1, y1) AxB (x2, y2) .

Therefore, AxB [ (x1, y1)(x2, y2) ] AxB (x1, y1) AxB (x2, y2) .

Hence AxB is an intuitionistic L-fuzzy subgroup of GxH.

2.2.28 Theorem : Let an intuitionistic L-fuzzy subgroup A of a group G

be conjugate to an intuitionistic L-fuzzy subgroup M of G and an

intuitionistic L-fuzzy subgroup B of a group H be conjugate to an

intuitionistic L-fuzzy subgroup N of H. Then an intuitionistic L-fuzzy

subgroup AxB of a group GxH is conjugate to an intuitionistic L-fuzzy

subgroup MxN of GxH.

Proof : Let A and B be intuitionistic L-fuzzy subgroups of the groups G

and H respectively.

Let x, x-1 and f be in G and y, y-1 and g be in H.

Then (x, y) ,( x-1, y-1) and (f, g) are in GxH.

Now,

since ILFSGs A and B of groups G and H are conjugate to

ILFSGs M and N of G and H

AxB ( f, g ) = A(f) B(g)

= M( xf x-1) N( yg y-1 ) ,

50

= MxN ( xf x-1, yg y-1 )

= MxN[ (x, y)(f, g)(x-1, y-1) ]

= MxN[ (x, y) (f, g)(x, y )-1 ].

Therefore, AxB ( f, g ) = MxN[ (x, y) (f, g)(x, y )-1 ].

And,

since ILFSGs A and B of groups G and H are conjugate to

ILFSGs M and N of G and H

AxB ( f, g ) = A(f) B(g)

= M( xf x-1) N( yg y-1 )

= MxN ( xf x-1, yg y-1 )

= MxN[ (x, y)(f, g)(x-1, y-1) ]

= MxN[ (x, y) (f, g)(x, y )-1 ].

Therefore, AxB ( f, g ) = MxN[ (x, y) (f, g)(x, y )-1 ].

Hence an intuitionistic L-fuzzy subgroup AxB of a group GxH is

conjugate to an intuitionistic L-fuzzy subgroup MxN of GxH.

2.2.29 Theorem : Let A and B be intuitionistic L-fuzzy subsets of the

groups G and H, respectively. Suppose that e and e׀ are the identity

element of G and H, respectively. If AxB is an intuitionistic L-fuzzy

subgroup of GxH, then at least one of the following two statements must

hold.

(i) B(e׀ ) A(x) and B(e׀ ) A(x), for all x in G,

(ii) A(e ) B(y) and A(e ) B(y), for all y in H.

51

Proof : Let AxB is an intuitionistic L-fuzzy subgroup of GxH.

By contraposition, suppose that none of the statements (i) and (ii) holds.

Then we can find a in G and b in H such that A(a ) B(e׀ ),

A(a ) B(e׀ ) and B(b) A(e ), B(b ) A(e ).

We have,

AxB ( a, b ) = A(a) B(b)

A(e) B(e׀ ),

= AxB (e, e׀ ).

And, AxB ( a, b ) = A(a) B(b)

A(e) B(e׀ ) ,

= AxB (e, e׀ ).

Thus AxB is not an intuitionistic L-fuzzy subgroup of GxH.

Hence either B(e׀ ) A(x) and B(e׀ ) A(x), for all x in G or

A(e ) B(y) and A(e ) B(y), for all y in H.

2.2.30 Theorem :Let A and B be intuitionistic L-fuzzy subsets of the

groups G and H, respectively and AxB is an intuitionistic L-fuzzy

subgroup of GxH. Then the following are true :

(i) if A(x ) B(e׀ ) and A(x ) B( e׀ ), then A is an

intuitionistic L-fuzzy subgroup of G.

(ii) if B(x ) A( e ) and B(x ) A( e ), then B is an

intuitionistic L-fuzzy subgroup of H.

52

(iii) either A is an intuitionistic L-fuzzy subgroup of G or B is an

intuitionistic L-fuzzy subgroup of H.

Proof : Let AxB be an intuitionistic L-fuzzy subgroup of GxH and

x, y in G.

Then (x, e׀ ) and (y, e׀ ) are in GxH.

Now, using the property A(x ) B(e׀ ) and A(x ) B( e׀ ),

for all x in G, we get,

A( xy-1 ) = A(xy-1) B(e׀e׀ )

= AxB ( (xy-1), (e׀e׀ ) )

= AxB [ (x, e׀ )(y-1, e׀ ) ]

AxB(x, e׀ ) AxB(y-1, e׀ )

= { A(x) B(e׀ ) } { A(y-1) B(e׀ ) }

= A(x) A(y-1)

A(x) A(y).

Therefore, A( xy-1 ) A(x) A(y) .

And,

A( xy-1 ) = A(xy-1) B(e׀e׀ )

= AxB ( (xy-1), (e׀e׀ ) )

= AxB [ (x, e׀ )(y-1, e׀ ) ]

AxB(x, e׀ ) AxB(y-1, e׀ )

= { A(x) B(e׀ ) } { A(y-1) B(e׀ ) }

= A(x) A(y-1)

53

A(x) A(y).

Therefore, A( xy-1 ) A(x) A(y).

Hence A is an intuitionistic L-fuzzy subgroup of G.

Thus (i) is proved.

Now,

using the property B(x ) A(e ) and B(x ) A( e ), for all x in G,

we get,

B( xy-1 ) = B(xy-1) A(ee )

= AxB ( (ee ) ,(xy-1) )

= AxB [ (e, x)(e, y-1 ) ]

AxB(e, x ) AxB(e, y-1 )

= { B(x) A(e ) }{ B(y-1) A(e ) }

= B(x) B(y-1)

B(x) B(y).

Therefore, B( xy-1 ) B(x) B(y) .

And,

B( xy-1 ) = B(xy-1) A(ee )

= AxB ( (ee ), (xy-1) )

= AxB [ (e, x )(e, y-1 ) ]

AxB(e, x ) AxB( e, y-1 )

= { B(x) A(e ) } { B(y-1) A(e ) }

= B(x) B(y-1)

B(x) B(y).

54

Therefore, B( xy-1 ) B(x) B(y).

Hence B is an intuitionistic L-fuzzy subgroup of H.

Thus (ii) is proved.

(iii) is clear.

2.2.31 Theorem: Let A be an intuitionistic L-fuzzy subset of a group G and V

be the strongest intuitionistic L-fuzzy relation of G. Then A is an intuitionistic

L-fuzzy subgroup of G if and only if V is an intuitionistic L-fuzzy subgroup of

GxG.

Proof: Suppose that A is an intuitionistic L-fuzzy subgroup of a group G.

Then for any x = (x1, x2) and y = (y1, y2) are in GxG.

We have, V ( x– y ) = V [ (x1, x2) – (y1, y2) ] = V ( x1-y1 , x2-y2 )

= A(x1-y1) A(x2-y2)

{ A(x1) A(y1 ) } { A(x2) A(y2) }

= { A(x1) A(x2) }{ A(y1 ) A(y2) }

= V (x1, x2) V (y1, y2) = V (x) V (y) .

Therefore, V ( x – y ) V (x) V (y), for all x, y GxG.

Also we have,

V (x – y) = V [ (x1, x2) – (y1, y2) ]

= V( x1 – y1 , x2 – y2 )

= A(x1– y1) A(x2 – y2)

{ A(x1) A(y1 ) }{ A(x2) A(y2) }

= { A(x1) A(x2) } { A(y1 ) A(y2) }

= V (x1, x2) V (y1, y2) = V (x) V (y) .

Therefore, V ( x – y ) V (x) V (y), for all x, y GxG.

55

This proves that V is an intuitionistic L-fuzzy subgroup of GxG.

Conversely assume that V is an intuitionistic L-fuzzy subgroup of GxG,

then for any x = (x1, x2) and y = (y1, y2) are in GxG, we have

min { A( x1– y1 ) , A( x2 – y2 ) } = V( x1– y1 , x2 – y2 )

= V [ ( x1, x2 ) – ( y1, y2 ) ] = V ( x – y )

V (x) V (y) = V ( x1, x2 ) V ( y1, y2 )

= { A(x1) A(x2) }{ A(y1 ) A(y2) }.

If we put x2 = y2 = 0,

We get, A( x1– y1) A(x1) A(y1 ), for all x1, y1 G.

Also we have,

A( x1 – y1 ) A( x2 – y2 ) = V ( x1 – y1 , x2 – y2 )

= V [ ( x1, x2 ) – ( y1, y2 ) ]

= V ( x – y ) V (x) V (y)

= V ( x1, x2 ) V ( y1, y2 )

= { A(x1) A(x2) }{ A(y1 ) A(y2) }.

If we put x2 = y2 = 0,

We get, A ( x1 – y1 ) A(x1) A(y1 ), for all x1, y1 G.


2.3 – INTUITIONISTIC L-FUZZY SUBGROUPS OF A

GROUP G UNDER HOMOMORPHISM AND

ANTI-HOMOMORPHISM :

56

2.3.1 Theorem : Let G and G׀ be any two groups. The homomorphic

image of an intuitionistic L-fuzzy subgroup of G is an intuitionistic L-

fuzzy subgroup of G׀.

Proof: Let G and G׀ be any two groups.

Let f : G→G׀ be a homomorphism.

That is f(xy) = f(x)f(y) for all x and yG.

Let V=f(A), where A is an intuitionistic L-fuzzy subgroup of a group G.

We have to prove that V is an intuitionistic L-fuzzy subgroup of a

group G׀.

Now, for f(x) and f(y) in G׀, we have

V( f(x)f(y) ) = V( f(xy) ), as f is a homomorphism

≥ A(xy)

≥ A(x) A(y) , as A is an ILFSG of G,

which implies that V( f(x)f(y) ) ≥ V( f(x) ) V( f(y) ) .

For f(x) in G׀, we have

V( [ f(x) ]-1 ) = V( f(x-1) ), as f is a homomorphism

≥ A( x-1)

A(x), as A is an ILFSG of G,

which implies that V( [ f(x) ]-1 ) V( f(x) ).

V( f(x)f(y) ) = V( f(xy) ), as f is a homomorphism

A(xy)

A(x) A(y) , as A is an ILFSG of G,

which implies that V( f(x)f(y) ) V( f(x) ) V( f(y) ) .

57

V( [ f(x) ]-1 ) = V( f(x-1) ), as f is a homomorphism

A(x-1)


which implies that V( [ f(x) ]-1 ) V( f(x) ).

Hence V is an intuitionistic L-fuzzy subgroup of a group G׀.


pre-image of an intuitionistic L-fuzzy subgroup of G׀ is an intuitionistic



Let f : G→G׀ be a homomorphism.

That is f(xy) = f(x)f(y), for all x and yG.

Let V=f(A), where V is an intuitionistic L-fuzzy subgroup of a group G

.׀

We have to prove that A is an intuitionistic L-fuzzy subgroup of a

groupG.

Let x and yG. Then,

A(xy) = V( f(xy) ), since A(x) = V( f(x) )

= V( f(x)f(y) ), as f is a homomorphism

V( f(x) ) V( f(y) ) , as V is an ILFSG of G׀

= A(x) A(y), since A(x) = V( f(x) ),

which implies that A(xy) ≥ A(x) A(y).

A(x-1) = V( f(x-1) ), since A(x) = V( f(x) )

= V( [f(x)]-1 )

58

V( f(x) ), as V is an ILFSG of G׀

= A(x),

which implies that A(x-1) A(x).

And, A(xy) = V( f(xy) ), since A(x) = V( f(x) )



= A(x) A(y) ,

which implies that A(xy) A(x) A(y) .

And, A(x-1) = V( f(x-1) ), since A(x) = V( f(x) )

= V( [f(x)]-1 )


= A(x),


Hence A is an intuitionistic L-fuzzy subgroup of a groupG.

2.3.3 Theorem : Let G and G׀ be any two groups. The anti-

homomorphic image of an intuitionistic L-fuzzy subgroup of G is an

intuitionistic L-fuzzy subgroup of G׀.


Let f : G → G׀ be an anti-homomorphism.

That is f(xy) = f(y)f(x), for all x and yG.

Let V = f(A), where A is an intuitionistic L-fuzzy subgroup of

a group G.

59

We have to prove that V is an intuitionistic L-fuzzy subgroup of a

group G׀.

Now, let f(x) and f(y)G׀, we have

V( f(x)f(y) ) = V( f(yx) ), as f is an anti-homomorphism

≥ A(yx)

≥ A(x) A(y) , as A is an ILFSG of G,

which implies that V( f(x)f(y) ) ≥ V( f(x) ) V( f(y) ) .

For x in G,

V( [f(x)]-1 ) = V( f(x-1) ), as f is an anti-homomorphism

≥ A( x-1)


which implies that V( [f(x)]-1 ) V( f(x) ).

And,

V( f(x)f(y) ) = V( f(yx) ), as f is an anti-homomorphism

A(yx)

A(x) A(y) , as A is an ILFSG of G,

which implies that V( f(x)f(y) ) V( f(x) ) V( f(y) ) .

Also, V([f(x)]-1) = V(f(x-1)), as f is an anti-homomorphism

A(x-1)


which implies that V( [f(x)]-1 ) V( f(x) ).

Hence V is an intuitionistic L-fuzzy subgroup of a group G׀.

60


homomorphic pre-image of an intuitionistic L-fuzzy subgroup of G׀ is

an intuitionistic L-fuzzy subgroup of G.




Let V = f(A), where V is an intuitionistic L-fuzzy subgroup of G׀.

We have to prove that A is an intuitionistic L-fuzzy subgroup of G.

Let x and yG.

Now, A(xy) = V( f(xy) ), since A(x) = V( f(x) )

= V( f(y)f(x) ), as f is an anti-homomorphism

V( f(x) ) V( f(y) ), as V is an ILFSG of G׀

= A(x) A(y) , since A(x) = V( f(x) ),

which implies that A(xy) ≥ A(x) A(y) .

And, A(x-1) = V( f(x-1) ), since A(x) = V( f(x) )

= V( [f(x)]-1 ), as f is an anti-homomorphism


= A(x) , since A(x) = V( f(x) ),


Also, A(xy) = V( f(xy) ), since A(x) = V( f(x) )

= V( f(y)f(x) ), as f is an anti-homomorphism


= A(x) A(y), since A(x) = V( f(x) ),

61

which implies that A(xy) A(x) A(y).

And also, A(x-1) = V( f(x-1) ), since A(x) = V( f(x) )

= V( [f(x)]-1 ), as f is an anti-homomorphism


= A(x), since A(x) = V( f(x) ),




NORMAL SUBGROUPS :

2.4.1 Theorem : Let G be a group. If A and B are two intuitionistic L-

fuzzy normal subgroups of G, then their intersection AB is an

intuitionistic L-fuzzy normal subgroup of G.

Proof : Let xG and yG.

Let A ={ x, A(x), A(x) / xG }and

B = { x, B(x), B(x) / xG } be an intuitionistic L-fuzzy normal

subgroups of a group G.

Let C = A B and C = { x, C(x), C(x) / xG }.

Then,

62

Clearly C is an intuitionistic L-fuzzy subgroup of a group G,

since A and B are two intuitionistic L-fuzzy subgroups of a group G.

And ,

(i) C(xy) = A(xy) B(xy) ,

= A(yx) B(yx)

= C(yx).

Therefore, C(xy) = C(yx).

(ii) C(xy) = A(xy) B(xy) ,

= A(yx) B(yx)

= C(yx).

Therefore, C(xy) = C(yx).

Hence AB is a intuitionistic L-fuzzy normal subgroup of a group G.

2.4.2 Theorem : Let G be a group. The intersection of a family of

intuitionistic L-fuzzy normal subgroups of G is an intuitionistic L-

fuzzy normal subgroup of G.

Proof:

Let { Ai }iI be a family of intuitionistic L-fuzzy normal subgroups of a

group G and let A = Ai .

Then for x and yG.

Clearly the intersection of a family of intuitionistic L-fuzzy subgroups

of a group G is an intuitionistic L-fuzzy subgroup of a group G.

(i) A( xy ) =

63

=

= A(yx).


(ii) A(xy) =

=

= A(yx).


Hence the intersection of a family of intuitionistic L-fuzzy normal

subgroups of a group G is an intuitionistic L-fuzzy normal subgroup of

a group G.

2.4.3 Theorem : If A is an intuitionistic L-fuzzy characteristic subgroup

of a group G, then A is an intuitionistic L-fuzzy normal subgroup of a

group G.

Proof : Let A be an intuitionistic L-fuzzy characteristic subgroup of a

group G and let x,yG.

Consider the map f : G → G defined by f(x) = yxy-1.

Clearly, fAutG.

Now, A(xy) = A( f(xy) )

= A( y(xy)y-1 )

= A(yx).


64

Again, A(xy) = A( f(xy) )

= A( y(xy)y-1 )

= A(yx).


Hence A is an intuitionistic L-fuzzy normal subgroup of a group G.

2.4.4 Theorem : An intuitionistic L-fuzzy subgroup A of a group G is

an intuitionistic L-fuzzy normal subgroup of G if and only if A is

constant on the conjugate classes of G.

Proof : Suppose that A is an intuitionistic L-fuzzy normal subgroup of

a group G and let x and yG.

Now, A( y-1xy ) = A( xyy-1 )

= A(x).

Therefore, A(y-1xy) = A(x ).

And, A( y-1xy ) = A( xyy-1 )

= A(x).

Therefore, A( y-1xy ) = A(x).

Hence (x) = { y-1xy / yG }.

Hence A is constant on the conjugate classes of G.

Conversely,

suppose that A is constant on the conjugate classes of G.

Then, A( xy ) = A( xyxx-1 )

= A( x(yx)x-1 )

= A(yx).


65

And, A( xy ) = A( xyxx-1 )

= A( x(yx)x-1 )

= A( yx ).



2.4.5 Theorem : Let A be an intuitionistic L-fuzzy normal subgroup of a

group G. Then for any yG we have A(yxy-1) = A( y-1xy ) and

A( yxy-1 ) = A( y-1xy ), for every xG.

Proof :Let A be an intuitionistic L-fuzzy normal subgroup of a group G.

For any yG, we have,

A( yxy-1 ) = A(x)

= A( xyy-1 )

= A( y-1xy ).

Therefore, A( yxy-1 ) = A( y-1xy ).

And, A( yxy-1 ) = A(x)

= A( xyy-1 )

= A( y-1xy ).

Therefore, A(yxy-1 ) = A( y-1xy ).

2.4.6 Theorem : An intuitionistic L-fuzzy subgroup A of a group G is

normalized if and only if A(e) = 1 and A(e) = 0, where e is the identity

element of the group G.

Proof : If A is normalized, then there exists xG such that A(x) = 1

and A(x) = 0, but by properties of a L-fuzzy subgroup A of the group G,

A(x) ≤ A(e) and A(x) ≥ A(e), for every xG.

66

Since A(x) = 1 and A(x) = 0 and A(x) ≤ A(e) and A(x) ≥ A(e),

1 ≤ A(e) and 0 ≥ A(e).

But 1 ≥ A(e) and 0 ≤ A(e).

Hence A(e) = 1 and A(e) = 0.

Conversely,

if A(e) = 1 and A(e) = 0, then by the definition of normalized

intuitionistic L-fuzzy subset, A is normalized.

2.4.7 Theorem : Let A and B be intuitionistic L-fuzzy subgroups of the

groups G and H, respectively. If A and B are intuitionistic L-fuzzy

normal subgroups, then AxB is an intuitionistic L-fuzzy normal

subgroup of GxH.

Proof : Let A and B be intuitionistic L-fuzzy normal subgroups of the

groups G and H respectively.

Clearly AxB is an intuitionistic L-fuzzy subgroup of GxH.

Let x1 and x2 be in G, y1 and y2 be in H.

Then ( x1, y1 ) and ( x2, y2 ) are in GxH.

Now,

AxB [ (x1, y1)(x2, y2) ] = AxB ( x1x2, y1y2 )

= A( x1x2 ) B( y1y2 )

= A( x2x1 ) B( y2y1 ) ,

= AxB ( x2x1, y2y1 )

67

= AxB [ (x2, y2)(x1, y1) ].

Therefore, AxB [ (x1, y1)(x2, y2) ] = AxB [ (x2, y2)(x1, y1) ].

And,

AxB [ (x1, y1)(x2, y2) ] = AxB ( x1x2, y1y2 )

= A( x1x2 ) B( y1y2 )

= A( x2x1 ) B( y2y1 ) ,

= AxB ( x2x1, y2y1 )

= AxB [ (x2, y2)(x1, y1) ].

Therefore, AxB [ (x1, y1)(x2, y2) ] = AxB [ (x2, y2)(x1, y1) ].

Hence AxB is an intuitionistic L-fuzzy normal subgroup of GxH.

2.4.8 Theorem : Let an intuitionistic L-fuzzy normal subgroup A of a

group G be conjugate to an intuitionistic L-fuzzy normal subgroup M of

G and an intuitionistic L-fuzzy normal subgroup B of a group H be

conjugate to an intuitionistic L-fuzzy normal subgroup N of H. Then an

intuitionistic L-fuzzy normal subgroup AxB of a group GxH is

conjugate to an intuitionistic L-fuzzy normal subgroup MxN of GxH.



groups G and H, respectively. Suppose that e and e׀ are the identity

element of G and H, respectively. If AxB is an intuitionistic L-fuzzy

normal subgroup of GxH, then at least one of the following two

statements must hold.

(i) B(e׀ ) A(x) and B(e׀ ) A(x), for all x in G,

68

(ii) A(e ) B(y) and A(e ) B(y), for all y in H.


2.4.10 Theorem :Let A and B be intuitionistic L-fuzzy subsets of the

groups G and H, respectively and AxB is an intuitionistic L-fuzzy

normal subgroup of GxH. Then the following are true :

(i) if A(x ) B(e׀ ) and A(x ) B( e׀ ), then A is an


(ii) if B(x ) A( e ) and B(x ) A( e ), then B is an

intuitionistic L-fuzzy normal subgroup of H.

(iii) either A is an intuitionistic L-fuzzy normal subgroup of G

or B is an intuitionistic L-fuzzy normal subgroup of H.


2.5 – INTUITIONISTIC L-FUZZY NORMAL SUBGROUPS

OF A GROUP G UNDER HOMOMORPHISM AND

ANTI-HOMOMORPHISM :


image of an intuitionistic L-fuzzy normal subgroup of G is an

intuitionistic L-fuzzy normal subgroup of G׀.


Let f : G → G׀ be a homomorphism.


Let V = f(A),where A is an intuitionistic L-fuzzy normal subgroup of G.

69

We have to prove that V is an intuitionistic L-fuzzy normal

subgroup of G׀.

Now, for f(x) and f(y)G׀, we have

clearly V is an intuitionistic L-fuzzy subgroup of a group G׀,

since A is an intuitionistic L-fuzzy subgroup of a group G.

Now, V( f(x)f(y) ) = V( f(xy) ), as f is a homomorphism

A(xy)

= A(yx), as A is an ILFNSG of G

V( f(yx) )

= V( f(y)f(x) ), as f is a homomorphism,

which implies that V( f(x)f(y) ) = V( f(y)f(x) ).

Now, V( f(x)f(y) ) = V( f(xy) )

A(xy)

= A(yx)

V( f(yx) )

= V( f(y)f(x) )


Hence V is an intuitionistic L-fuzzy normal subgroup of a group G׀.


pre-image of an intuitionistic L-fuzzy normal subgroup of G׀ is an



Let f : G → G׀ be a homomorphism.

70


Let V=f(A), where V is an intuitionistic L-fuzzy normal subgroup of G׀.

We have to prove that A is an intuitionistic L-fuzzy normal

subgroup of G.

Let x and yG. Then,

clearly A is an intuitionistic L-fuzzy subgroup of a group G,

since V is an intuitionistic L-fuzzy subgroup of a group G׀.



= V( f(y)f(x) ), as V is an ILFNSG of G׀

= V( f(yx) ), as f is a homomorphism

= A(yx), since A(x) = V( f(x) ),

which implies that A(xy) = A(yx).



= V( f(y)f(x) ), as V is an ILFNSG of G׀

= V( f(yx) ), as f is a homomorphism

= A( yx), since A(x) = V( f(x) ),

which implies that A(xy)= A(yx).


2.5.3 Theorem :Let G and G׀ be any two groups. The anti-

homomorphic image of an intuitionistic L-fuzzy normal subgroup of G

is an intuitionistic L-fuzzy normal subgroup of G׀.

71

Proof:Let G and G׀ be any two groups.



Let V=f(A), where A is an intuitionistic L-fuzzy normal subgroup of G.

We have to prove that V is an intuitionistic L-fuzzy normal

subgroup of G׀.

For f(x) and f(y)G׀,

clearly V is an intuitionistic L-fuzzy subgroup(ILFSG) of a group G׀,

since A is an intuitionistic L-fuzzy subgroup of a group G.

Now, V( f(x)f(y) ) = V( f(yx) ), as f is an anti-homomorphism

A(yx)

= A(xy), as A is an ILFNSG of G

V( f(xy) )

= V( f(y)f(x) ), as f is an anti-homomorphism,


And, V( f(x)f(y) ) = V( f(yx) ), as f is an anti-homomorphism

A(yx)

= A(xy), as A is an ILFNSG of G

V( f(xy) )

= V( f(y)f(x) )


Hence V is an intuitionistic L-fuzzy normal subgroup of a group G׀.

72

2.5.4 Theorem :Let G and G׀ be any two groups. The anti-

homomorphic pre-image of an intuitionistic L-fuzzy normal subgroup of

G׀ is an intuitionistic L-fuzzy normal subgroup of G.


Let f : G → G׀ be anti-homomorphism.


Let V=f(A) , where V is an intuitionistic L-fuzzy normal subgroup of G

.׀

We have to prove that A is an intuitionistic L-fuzzy normal

subgroup of G.

Let x and yG, we have

clearly A is an intuitionistic L-fuzzy subgroup of a group G,

since V is an intuitionistic L-fuzzy subgroup of a group G׀.


= V( f(y)f(x) ), as f is an anti homomorphism

= V( f(x)f(y) ), as V is an ILFNSG of G׀

= V( f(yx) ), as f is an anti homomorphism




= V(f(y)f(x)), as f is an anti homomorphism

= V(f(x)f(y)), as V is an ILFNSG of G׀

= V(f(yx)), as f is an anti homomorphism

73




In the following Theorem ◦ is the composition operation of

functions :

2.5.5 Theorem :Let A be an intuitionistic L-fuzzy subgroup of a group

H and f is an isomorphism from a group G onto H.

Then we have the following:

(i) A◦f is an intuitionistic L-fuzzy subgroup of the group G.

(ii) If A is an intuitionistic L-fuzzy normal subgroup of the group

H, then A◦f is an intuitionistic L-fuzzy normal subgroup of the

group G.

Proof :

Let x and yG and A be an intuitionistic L-fuzzy subgroup of a

group H.

Then we have ,

( A◦f )( xy-1 ) = A( f(xy-1) )

= A( f(x)f(y-1) ) , as f is an isomorphism

= A( f(x)(f(y)) –1 ),

≥ A( f(x) ) A( f(y) ) , as A is an ILFSG of H

≥ ( A◦f )(x) ( A◦f )(y) ,

which implies that ( A◦f )( xy-1 ) ≥ ( A◦f )(x) ( A◦f )(y) .

And,

74

( A◦f )( xy-1 ) = A( f( xy-1 ) )

= A( f(x)f(y-1) ), as f is an isomorphism

= A( f(x)(f(y)) -1 ),

≤ A( f(x)) A( f(y) )

≤ ( A◦f )(x) ( A◦f )(y) ,

which implies that ( A◦f )( xy-1)≤ ( A◦f )(x) ( A◦f )(y) .

Therefore ( A◦f ) is an intuitionistic L-fuzzy subgroup of a group G.

Thus (i) is proved.

Let x,yG and A be an intuitionistic L-fuzzy normal subgroup of a

group H.

Then we have,

( A◦f )(xy) = A( f(xy) )

= A( f(x)f(y) ), as f is an isomorphism

= A( f(y)f(x) ), as A is an ILFNSG of a group H

= A( f(yx) ), as f is an isomorphism

= ( A◦f )(yx),

which implies that ( A◦f )(xy) = ( A◦f )(yx).

Now,

( A◦f )(xy) = A( f(xy) )

= A( f(x)f(y) ), as f is an isomorphism

= A( f(y)f(x) ), as A is an ILFNSG of a group H

= A( f(yx) ), as f is an isomorphism

= ( A◦f )(yx),

75


Hence A◦f is an intuitionistic L-fuzzy normal subgroup of a group G.


H and f is an anti-isomorphism from a group G onto H. Then we have

the following:

(i) A◦f is an intuitionistic L-fuzzy subgroup of the group G.

(ii) If A is an intuitionistic L-fuzzy normal subgroup of the group

H, then A◦f is an intuitionistic L-fuzzy normal subgroup of the

group G.

Proof :Let x and yG and A be an intuitionistic L-fuzzy subgroup of a

group H.

Then we have,

( A◦f )( xy-1 ) = A( f(xy-1) )

= A( f(y-1)f(x) ), as f is an anti-isomorphism

= A( (f(y))-1f(x) ),

≥ A( f(x) ) A( f(y) ) , as A is an ILFSG of H

≥ ( A◦f )(x) ( A◦f )(y) ,

which implies that ( A◦f )( xy-1 ) ≥ ( A◦f )(x) ( A◦f )(y) .

And, ( A◦f )(xy-1) = A( f(xy-1) )

= A( f(y-1)f(x) ), as f is an anti-isomorphism

= A( (f(y))-1f(x) ),

≤ A( f(x) ) A( f(y) ), as A is an ILFSG of H

≤ ( A◦f )(x) ( A◦f )(y) ,

76

which implies that ( A◦f )(xy-1) ≤ ( A◦f )(x) ( A◦f )(y).

Therefore A◦f is an intuitionistic L-fuzzy subgroup of a group G.

Thus(i) is proved.

Let x,yG and A be an intuitionistic L-fuzzy normal subgroup of a

group H.

Then we have ,

( A◦f )(xy) = A( f(xy) )

= A( f(y)f(x) ), as f is an anti-isomorphism

= A( f(x)f(y) ), as A is an ILFNSG of a group H

= A( f(yx) ), as f is an anti-isomorphism

= ( A◦f )(yx),


Now, ( A◦f )(xy) = A( f(xy) )

= A( f(y)f(x) ), as f is an anti-isomorphism

= A( f(x)f(y) ) as A is an ILFNSG of a group H

= A( f(yx) ), as f is an anti-isomorphism

= ( A◦f )(yx),


Hence (A◦f) is an intuitionistic L-fuzzy normal subgroup of a group G.

2.6 P ROPERTIES OF INTUITIONISTIC L--FUZZY COSETS

2.6.1 Theorem : Let A be an intuitionistic L-fuzzy subgroup of a finite

group G, then O(A) / O(G).

77

Proof : Let A be an intuitionistic L-fuzzy subgroup of a finite group G

with e as its identity element.

Clearly H ={ xG / A(x) = A(e) and A(x) = A(e) } is a subgroup of

the group G for H is a (, β)-level subset of a group G where = A(e)

and β = A(e) .

By Lagranges theorem O(H) / O(G).

Hence by the definition of the order of the intuitionistic L-fuzzy

subgroup of the group G, we have O(A) / O(G).

2.6.2 Theorem :Let A and B be two intuitionistic L-fuzzy subsets of an

abelian group G. Then A and B are conjugate intuitionistic L-fuzzy

subsets of the group G if and only if A = B.

Proof : Let A and B be conjugate intuitionistic L-fuzzy subsets of group

G, then for some yG, we have

A(x) = B( y-1xy ), for every xG

= B( yy-1x ) ,since G is an abelian group

= B( ex ) = B(x).

Therefore, A(x) = B(x).

And, A(x) = B( y-1xy ), for every xG

= B( yy-1x ), since G is an abelian group

= B( ex ) = B(x).

Therefore, A(x) = B(x).

Hence A = B.

Conversely, if A = B, then for the identity element e of group G,

78

we have,

A(x) = B( e-1xe ) and A(x) = B( e-1xe ), for every xG.

Hence A and B are conjugate intuitionistic L-fuzzy subsets of the

group G.

2.6.3 Theorem : If A and B are conjugate intuitionistic L-fuzzy

subgroups of the group G, then O(A) = O(B).

Proof : Let A and B be conjugate intuitionistic L-fuzzy subgroups of the

group G.

Now,

O(A) = order of { xG / A(x) = A(e) and A(x) = A(e) }

= order of { xG / B( y-1xy) = B( y-1ey ) and B(y-1xy) = B(y-1ey)}

= order of { xG / B(x) = B(e) and B(x) = B(e) }

= O(B).

Hence O(A) = O(B).

2.6.4 Theorem : Let A be an intuitionistic L-fuzzy subgroup of a group

G, then the pseudo intuitionistic L-fuzzy coset (aA)p is an intuitionistic

L-fuzzy subgroup of a group G, for every aG.

Proof : Let A be an intuitionistic L-fuzzy subgroup of a group G.

For every x and y in G, we have,

( (aA)p )( xy-1 ) = p(a)A( xy-1 )

≥ p(a) ( A(x) A(y) )

= p(a)A(x) p(a)A(y)

79

= ( (aA)p )(x) ( (aA)p )(y).

Therefore, ( (aA)p )( xy-1 ) ≥ ( (aA)p )(x) ( (aA)p )(y).

And,

( (aA)p )( xy-1 ) = p(a)A( xy-1)

≤ p(a) ( A(x) A(y) )

= p(a)A(x) p(a)A(y)

= ( (aA)p )(x) ( (aA)p )(y).

Therefore, ( (aA)p )( xy-1) ≤ ( (aA)p )(x) ( (aA)p )(y), for every x

and yG .

Hence (aA)p is an intuitionistic L-fuzzy subgroup of a group G.

2.6.5 Theorem :If A is an intuitionistic L-fuzzy subgroup of a group G ,

then for any aG the intuitionistic L-fuzzy middle coset aAa-1 of G is

also an intuitionistic L-fuzzy subgroup of G.

Proof : Let A be an intuitionistic L-fuzzy subgroup of a group G and

aG.

To prove aAa-1 = ( x, aAa-1, aAa-1 ) is an intuitionistic L-fuzzy subgroup

of a group G .

Let x and yG. Then,

( aAa-1 )( xy-1) = A( a-1xy-1a ), by the definition

= A( a-1xaa-1y-1a )

= A( (a-1xa)(a-1ya) -1 )

≥ A( a-1xa ) A( (a-1ya ) -1 )

≥ A( a-1xa ) A( a-1ya ), since A is an ILFSG of G

80

= ( aAa-1 )(x ) ( aAa-1 ) (y).

Therefore, ( aAa-1 )( xy-1) ≥ ( aAa-1 )(x) ( aAa-1 )(y).

And,

( aAa-1 )( xy-1 ) = A( a-1xy-1a ), by the definition

= A( a-1xaa-1y-1a )

= A( ( a-1xa) (a-1ya) -1 )

≤ A( a-1xa ) A( (a-1ya ) -1 )

≤ A( a-1xa ) A(a-1ya )

= ( aAa-1 )(x ) ( aAa-1 )(y).

Therefore, ( aAa-1 )( xy-1 ) ≤ ( aAa-1 )(x ) ( aAa-1 )(y).

Hence aAa-1 is an intuitionistic L-fuzzy subgroup of a group G.


G and aAa-1 be an intuitionistic L-fuzzy middle coset of G, then

O(aAa-1) = O(A), for any aG.

Proof :Let A be an intuitionistic L-fuzzy subgroup of a group G and

aG.

By Theorem 2.6.5, the intuitionistic L-fuzzy middle coset aAa-1 is an


Further by the definition of an intuitionistic L-fuzzy middle coset of a

group G, we have,

( aAa-1 )(x) = A( a-1x a ) and ( aAa-1 )(x) = A( a-1xa ) , for every xG.

81

Hence for any aG , A and aAa-1 are conjugate intuitionistic L-fuzzy

subgroups of a group G as there exists aG such that

( aAa-1 )(x) = A( a-1xa ) and ( aAa-1 )(x) = A( a-1xa ), for every xG.

By Theorem 2.6.3,

O( aAa-1 ) = O(A), for any aG.


G and B be an intuitionistic L-fuzzy subset of a group G. If A and B are

conjugate intuitionistic L-fuzzy subsets of the group G, then B is an

intuitionistic L-fuzzy subgroup of a group G.

Proof : Let A be an intuitionistic L-fuzzy subgroup of a group G and B

be an intuitionistic L-fuzzy subset of a group G.

And, let A and B be conjugate intuitionistic L-fuzzy subsets of the

group G.

To prove B is an intuitionistic L-fuzzy subgroup of the group G.

Let x and yG. Then xy-1G.

Now, B( xy-1 ) = A( g-1xy-1g ), for some gG

= A( g-1xgg-1y-1g )

= A( ( g-1xg) (g-1yg) -1 )

≥ A( g-1xg ) A( (g-1yg ) -1 )

≥ A( g-1xg ) A( g-1yg ), since A is an ILFSG of G

= B(x) B(y).

Therefore, B( xy-1 ) ≥ B(x ) B(y ).

And , B( xy-1 ) = A( g-1xy-1g ), for some gG

82

= A( g-1xgg-1y-1g )

= A( ( g-1xg) ( g-1yg ) -1 )

≤ A( g-1xg ) A( ( g-1yg ) -1 )

≤ A( g-1xg ) A( g-1yg ), since A is an ILFSG of G

= B(x) B(y).

Therefore, B(xy-1) ≤ max { B(x ), B(y ) }.

Hence B is an intuitionistic L-fuzzy subgroup of the group G.


G. Then xA = yA, for x and yG if and only if

A( x-1y ) = A( y-1x ) = A(e) and A( x-1y ) = A( y-1x ) = A(e).

Proof : Let A be an intuitionistic L-fuzzy subgroup of a group G.

Let xA = yA , for x and yG.

Then ,

xA(x) = yA(x) , xA(x) = yA(x)

and xA(y) = yA(y) , xA(y) = yA(y),

which implies that

A(x-1x) = A(y-1x) , A(x-1x) = A(y-1x)

and A(x-1y) = A(y-1y) , A(x-1y) = A(y-1y).

Hence A(e) = A(y-1x) , A(e) = A(y-1x)

and A(x-1y) = A(e) , A(x-1y) = A(e).

Therefore, A(x-1y) = A(y-1x) = A(e) and A(x-1y) = A(y-1x) = A(e).

Conversely,

let A(x-1y) = A(y-1x) = A(e) , for x and yG.

83

For every gG and we have,

xA(g) = A( x-1g )

= A( x-1yy-1g )

≥ A( x-1y ) A( y-1g )

= A(e) A( y-1g )

= A( y-1g )

= yA(g).

Therefore, xA(g) ≥ yA(g) ----------------------(1).

And ,

yA(g) = A( y-1g )

= A( y-1xx-1g )

≥ A( y-1x ) A( x-1g )

= A(e) A( x-1g )

= A( x-1g )

= xA(g) .

Therefore, yA(g) ≥ xA(g) ----------------------(2).

From (1) and (2) we get,

xA(g) = yA(g) --------------------------(3).

Now,

let A( x-1y ) = A( y-1x ) = A(e) , for x and yG.

xA(g) = A( x-1g )

= A( x-1yy-1g )

≤ A( x-1y ) A( y-1g )

84

= A(e) A( y-1g )

= A( y-1g )

= yA(g) .

Therefore, xA(g) ≤ yA(g) ----------------------(4).

And ,

yA(g) = A( y-1g )

= A( y-1xx-1g )

≤ A( y-1x ) A( x-1g )

= A(e) A( x-1g )

= A( x-1g )

= xA(g) .

There fore, yA(g) ≤ xA(g) ----------------------(5).

From (4) and (5) we get ,

xA(g) = yA(g) --------------------------(6) .


xA = yA .


G and xA = yA, for x and yG. Then A(x) = A(y) and A(x) = A(y).


xA = yA, for x and yG.

Now, A(x) = A( yy-1x )

≥ A(y) A( y-1x )

= A(y) A(e), by Theorem 2.6.8

85

= A(y).

Therefore, A(x) ≥ A(y) ----------------------(1).

And , A(y) = A( xx-1y )

≥ A(x) A( x-1y )

= A(x) A(e), by Theorem 2.6.8

= A(x) .

Therefore, A(y) ≥ A(x) ----------------------(2).


A(x) = A(y) --------------------------(3).

Now,

A(x) = A( yy-1x )

≤ A(y) A( y-1x )

= A(y) A(e), by Theorem 2.6.8

= A(y) .

Therefore, A(x) ≤ A(y) ----------------------(4).

And,

A(y) = A( xx-1y )

≤ A(x) A( x-1y )

= A(x) A(e), by Theorem 2.6.8

= A(x).

Therefore, A(y) ≤ A(x) ----------------------(5).


A(x) = A(y) --------------------------(6) .

86


A(x) = A(y) and A(x) = A(y).


G and

x = y , for x and yG ,(α,β)L. Then A(x) = A(y)

and A(x) = A(y).


x = y , for x and yG ,(α,β)L .

But y-1x and x-1y .

Now ,

A(x) = A( yy-1x )

≥ A(y) A( y-1x )

= A(y).

Therefore, A(x) ≥ A(y) ----------------------(1).

And ,

A(y) = A( xx-1y )

≥ A(x) A( x-1y )

= A(x).

Therefore, A(y) ≥ A(x) ----------------------(2).


A(x) = A(y) --------------------------(3).

Now,

A(x) = A( yy-1x )

87

≤ A(y) A( y-1x )

= A(y).

Therefore, A(x) ≤ A(y) ----------------------(4).

And ,

A(y) = A( xx-1y )

≤ A(x) A( x-1y )

= A(x).

Therefore, A(y) ≤ A(x) ----------------------(5).


A(x) = A(y) --------------------------(6) .


A(x) = A(y) and A(x) = A(y).

2.6.11 Theorem :If A is an intuitionistic L-fuzzy normal subgroup of a

group G, then the set G / A = { xA : xG } is a group with the operation

(xA)(yA) = (xy)A.

Proof :

Let x and yG , xA and yA G / A.

Clearly y-1G. Therefore, y-1A G / A.

Now , (xA)( y-1A ) = ( xy-1 )A G / A.

Hence G / A is a group.

2.6.12 Theorem :Let f : G→H be a homomorphism of groups and let B

be an intuitionistic L-fuzzy normal subgroup of H and A = f -1(B). Then

88

: G / A → H / B such that ( xA ) = f(x)B, for every xG, is an

isomorphism of group.

Proof :Clearly is onto.

Let xA and yA G / A.

Now , (xA) = (yA),

which implies that, f(x)B = f(y)B,

by Theorem 2.6.8,

B( f(x )-1f(y) ) = B( f(y )-1f(x) ) = B( f(e) ),

which implies that B( f(x-1)f(y) ) = B( f(y-1)f(x) ) = B( f(e) ),

which implies that B( f(x-1y) ) = B( f(y-1x) ) = B( f(e) ).

And ,

B( f(x )-1f(y) ) = B( f(y )-1f(x) ) = B( f(e) ),


which implies that B( f ( x-1y ) ) = B( f ( y-1x ) ) = B( f (e) )

Using Theorem 2.6.8, we get,

xA = yA.

Hence is one-one .

Now ,

( (xA)(yA) ) = ( (xy)A) )

= f(xy)B

= ( f(x)f(y) )B, since f is homomorphism

= ( f(x)B ) ( f(y)B )

= (xA) (yA).

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Therefore, ( (xA)(yA) ) = (xA) (yA).

Hence is an isomorphism.

2.6.13 Theorem :Let f : G → H be an anti-homomorphism of groups

and let B be an intuitionistic L-fuzzy normal subgroup of H and

A = f -1(B). Then : G / A → H / B such that ( xa ) = f(x)B, for every

xG, is an anti-isomorphism of group.

Proof : Clearly is onto.

Let xA and yA G / A.

Now ,

(xA) = (yA),

which implies that f(x)B = f(y)B,

by Theorem 2.6.8

B( f(x )-1f(y) ) = B( f(y )-1f(x) ) = B( f(e) ),


which implies that B( f(yx-1) ) = B( f(xy-1) ) = B( f(e) ).

And ,

B( f(x )-1f(y) ) = B( f(y )-1f(x) ) = B( f(e) ),


which implies that B( f(yx-1) ) = B( f(xy-1) ) = B( f(e) ).

Using Theorem 2.6.8, we get,

xA = yA.

Hence is one-one.

Now ,

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( (xA)(yA) ) = ( (xy)A) )

= f(xy)B

= ( f(y)f(x) )B, since f is an anti-homomorphism

= ( f(y)B )( f(x)B )

= ( yA ) ( xA ).

Therefore, ( (xA)(yA) ) = ( yA ) ( xa ).

Hence is an anti-isomorphism.

In the following proposition ◦ is the composition operation of

functions :


H and f is an isomorphism from a group G onto H. Then we have the

following :

i) If A is a generalized characteristic intuitionistic L-fuzzy

subgroup(GCILFSG) of H , then A◦f is a generalized

characteristic intuitionistic L-fuzzy subgroup of G.

ii) If A is a generalized characteristic intuitionistic L-fuzzy

subgroup(GCILFSG) and f is an automorphism on G, then

A◦f = A.

Proof :Let A be a generalized characteristic intuitionistic L-fuzzy

subgroup(GCILFSG) of H.

Then from Theorem 2.5.5,

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A◦f is an intuitionistic L-fuzzy subgroup of G.

Let x and yG and (x) = (y).

Then we have ,

( A◦f ) (x) = A( f(x) )

= A( f(y) ), as ( f(x) ) = ( f(y) )

= ( A◦f )(y), as f is an isomorphism,

which implies that ( A◦f )(x) = ( A◦f )(y).

And,

( A◦f ) (x) = A( f(x) )

= A( f(y) ), as ( f(x) ) = ( f(y) )

= ( A◦f )(y), as f is an isomorphism,


Therefore,

A◦f is a generalized characteristic intuitionistic L-fuzzy subgroup of G.

Thus (i) is proved.

(ii) Clear.


group H and f is an anti-isomorphism from a group G onto H. Then we

have the following :

(i) If A is a generalized characteristic intuitionistic L-fuzzy

subgroup(GCILFSG) of H , then A◦f is a generalized

characteristic intuitionistic L-fuzzy subgroup of G.

(ii) If A is a generalized characteristic intuitionistic

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L-fuzzy subgroup(GCILFSG) and f is an anti-automorphism on

G, then A◦f = A.

Proof : Let A be a generalized characteristic intuitionistic L-fuzzy

subgroup(GCILFSG) of H.

Then from Theorem 2.5.6,

A◦f is an intuitionistic L-fuzzy subgroup of G.

Let x and yG and (x) = (y).

Then we have,

( A◦f )(x) = A( f(x) )

= A( f(y) ), as ( f(x) ) = ( f(y) )

= ( A◦f )(y), as f is an anti-isomorphism,


And,

( A◦f )(x) = A( f(x) )

= A( f(y) ), as ( f(x) ) = ( f(y) )

= ( A◦f )(y), as f is an anti-isomorphism,


Therefore, A◦f is a generalized characteristic intuitionistic L-fuzzy

subgroup of G.

Thus (i) is proved.

(ii) Clear.

CHAPTER-III

93

INTUITIONISTIC L-FUZZY LEVEL SUBSETS

3.1 Introduction :In this chapter, the basic definitions and properties of

the intuitionistic L-fuzzy level subsets of an intuitionistic L-fuzzy subset

are discussed. Using these concepts, some results are established.

3.1.1 Definition:Let A be an intuitionistic L-fuzzy subset of X. For ,

L, the level subset of A is the set,A ( , ) = { x X : A(x) ≥ and

A(x) ≤ }. This is called an intuitionistic L-fuzzy level subset of A.

3.2 PROPERTIES OF INTUITIONISTIC L-FUZZY

LEVEL SUBSETS :


G. Then for , L such that ≤ A(e) and ≥ A(e), A( , ) is a level

subgroup of G.

Proof: For all x, y in A( , ),

we have, A(x) ≥ and A(x) ≤ and A(y) ≥ and A(y) ≤ .

Now, A( x – y ) ≥A(x) A(y) ≥ = .

Which implies that, A( x – y ) ≥ .

And also, A( x – y ) ≤ A(x) A(y) ≤ = .

Which implies that, A( x – y ) ≤ .

Therefore, A( x – y ) ≥ and A( x – y ) ≤ .

Hence x – y A( , ).

Hence A( , ) is a level subgroup of a group G.

94

3.2.1 Definition: Let A be an intuitionistic L-fuzzy subgroup of a group

G. The level subgroup A( , ), for , L and A(e) and A(e)

are called intuitionistic L-fuzzy level subgroup of A.


G. Then two intuitionistic L-fuzzy level subgroups A( 1, 1 ), A( 2, 2 ) and

1, 2, 1, 2 L and 1, 2 ≤ A(e) and 1, 2 ≥ A(e) with 2 < 1 and 1

< 2 of A are equal iff there is no x in G such that

1 > A(x) > 2 and 1 < A(x) < 2.

Proof: Assume that A( 1, 1 ) = A( 2, 2 ).

Suppose there exists x G such that 1 > A(x) > 2 and

1 < A(x) < 2.

Then A ( 1, 1 ) A ( 2, 2 ) which implies that x belongs to A ( 2, 2 ),

but not in A ( 1, 1 ).

This is contradiction to A ( 1, 1 ) = A ( 2, 2 ).

Therefore there is no x G such that 1 > A(x) > 2 and

1 < A(x) < 2.

Conversely,

if there is no x G such that 1 > A(x) > 2 and 1 < A(x) < 2.

Then A( 1, 1 ) = A( 2, 2 ) .

3.2.3 Theorem : Let G be a group and A be an intuitionistic L-fuzzy

subset of G such that A( , ) be a subgroup of G. If , L satisfying ≤

A(e) and ≥ A(e), then A is an intuitionistic L-fuzzy subgroup of G.

Proof: Let G be a group and x, y in G.

95

Let A(x) = 1 and A(y) = 2 , A(x) = 1 and A(y) = 2.

Case (i): If 1 < 2 and 1 > 2, then x, y A( 1, 1 ).

As A( 1, 1 ) is a subgroup of G, x – yA( 1, 1 ).

Now, A( x –y ) ≥ 1 = 1 2 = A(x) A(y)

which implies that A( x – y) ≥ A(x) A(y) , for all xG and y G.

And , A( x– y) ≤ 1 = 1 2 = A(x) A(y)

which implies that A( x– y ) ≤ A(x) A(y), for all xG and y G.

Case (ii): If 1 < 2 and 1 < 2, then x, y A( 1, 2 ).

As A( 1, 2 ) is a subgroup of G, x –yA( 1, 2 ).

Now, A( x –y) ≥ 1 = 1 2 = A(x) A(y)

which implies that A( x–y ) ≥ A(x) A(y), for all xG and y G.

And, A( x– y ) ≤ 2 = 2 1 = A(y) A(x)

which implies that A( x– y ) ≤ A(y) A(x), for all xG and y G.

Case (iii): If 1 > 2 and 1 > 2, then x, yA( 2, 1 ).


Now, A( x– y ) ≥ 2 = 2 1 = A(y) A(x)

which implies that A( x–y ) ≥ A(x) A(y), for all x, y G.

And, A( x– y ) ≤ 1 = 1 2 = A(x) A(y)

which implies that A( x– y ) ≤ A(x) A(y) , for all x, y G.

Case (iv): If 1 > 2 and 1 < 2, then x, y A( 2, 2 ).


Now, A( x– y ) ≥ 2 = 2 1 = A(y) A(x)

which implies that A( x– y ) ≥ A(y) A(x), for all x, yG.

96

And, A( x– y ) ≤ 2 = 2 1 = A(y) A(x)

which implies that A( x– y ) ≤ A(y) A(x) , for all x, yG.

Case (v): If 1 = 2 and 1 = 2.

It is trivial.

In all the cases, A is an intuitionistic L-fuzzy subgroup of a group G.

3.2.4 Theorem: Let A be an intuitionistic L-fuzzy subgroup of a group

G. If any two level subgroups of A belongs to G, then their intersection

is also level subgroup of A in G.

Proof :Let 1, 2, 1, 2 L and 1, 2 ≤ A(e) and 1, 2 ≥ A(e).

Case (i):

If 1 < A(x) < 2 and 1 > A(x) > 2, then A( 2, 2 ) A( 1, 1 ).

Therefore, A( 1, 1 ) A( 2, 2 ) = A( 2, 2 ), but A( 2, 2 ) is a level subgroup

of A.

Case (ii):

If 1 > A(x) > 2 and 1 < A(x) < 2, then A( 1, 1 ) A( 2, 2 ).

Therefore , A( 1, 1 ) A( 2, 2 ) = A( 1, 1 ), but A( 1, 1 ) is a level subgroup

of A.

Case (iii):

If 1 < A(x) < 2 and 1 < A(x) < 2, then A( 2, 1 ) A( 1, 2 ).


of A.

Case (iv):

If 1 > A(x) > 2 and 1 > A(x) > 2, then A( 1, 2 ) A( 2, 1 ).

97

Therefore, A( 1, 2 ) A( 2, 1) = A( 1, 2 ), but A( 1, 2 ) is a level subgroup

of A.

Case (v):

If 1 = 2 and 1 = 2, then A( 1, 1 ) = A( 2, 2 ).

In all cases,

intersection of any two level subgroup is a level subgroup of A.


G. If i, jL, i ≤ A(e), j ≥ A(e) and A( i, j ), i, jI, is a collection of

level subgroup of A, then their intersection is also a level subgroup of

A.

Proof: It is trivial.


G. If any two level subgroups of A belongs to G, then their union is also

a level subgroup of A in G.

Proof: Let 1, 2, 1, 2 L and 1, 2 ≤ A(e) and 1, 2 ≥ A(e).

Case (i):

If 1 < A(x) < 2 and 1 > A(x) > 2, then A( 2, 2 ) A( 1, 1 ).


of A.

Case (ii):

If 1 > A(x) > 2 and 1 < A(x) < 2, then A( 1, 1 ) A( 2, 2 ).

Therefore , A( 1, 1 ) A( 2, 2 ) = A( 2, 2 ), but A( 2, 2 ) is a level subgroup

of A.

98

Case (iii):

If 1 < A(x) < 2 and 1 < A(x) < 2, then A( 2, 1 ) A( 1, 2 ).


of A.

Case (iv):

If 1 > A(x) > 2 and 1 > A(x) > 2, then A( 1, 2 ) A( 2, 1 ).


of A.

Case (v):

If 1 = 2 and 1 = 2, then A( 1, 1 ) = A( 2, 2 ).

In all cases,

union of any two level subgroup is also a level subgroup of A.


G. If i, jL, i ≤ A(e) and j ≥ A(e) and A( i, j ), i, jI, is a

collection of level subgroups of A, then their union is also a level

subgroup of A.

Proof: It is trivial.


G. If A is an intuitionistic L-fuzzy characteristic subgroup of G, then

each level subgroup of A is a characteristic subgroup of G.

Proof: Let A be an intuitionistic L-fuzzy characteristic subgroup of a

group G.

Let x, y G and Im A , Im A; f Aut(G) and x A( , ).

99

Now, A(f(x) ) = A(x) ≥ .

Therefore, A( f(x) ) ≥ .

And, A( f(x) ) = A(x) ≤ .

Therefore, A( f(x) ) ≤ .

Therefore, f(x) A (, ).

Hence, f(A(, ) ) A(, ) ------------------------------------(1).

For the reverse inclusion, let x f(A(, ) ) and let y in G be such that f(y)

= x.

Then, A(y) = A( f(y) ) = A(x) ≥ .

And, A(y) = A(f(y)) = A(x) ≤ .

Therefore, A(y) ≥ and A(y) ≤ .

Hence, y A( , ), when x f( A( , ) ).

Hence, f(A( , ) ) A( , ) ------------------------------------ (2).

From (1) and (2),

we get A( , ) is a characteristic subgroup of a group G.

3.2.9 Theorem : Any subgroup H of a group G can be realized as a

level subgroup of some intuitionistic L-fuzzy subgroup of G.

Proof: Let A be the intuitionistic L-fuzzy subset of a group G defined

by

if x H, 0 < < 1

A(x) = 0 if x H and

if x H, 0 < < 1

100

A(x) = 0 if x H

and + ≤ 1, where H is subgroup of a group G.

We claim that A is an intuitionistic L-fuzzy subgroup of a group G.

Let x, yG.

If x, y H, then x – y H.

Since H is a subgroup of G,

A( x– y ) = , A(x) = , A(y) = .

So, A( x– y ) ≥ A(x) A(y).

Also, A( x–y ) = , A(x) = , A(y) = .

So, A(x– y) ≤ A(x) A(y).

If x H, y H, then x– yH.

Then, A( x– y ) = 0, A(x) = , A(y) = 0.

Therefore, A( x– y ) ≥ A(x) A(y).

And A( x– y ) = 0, A(x) = , A(y) = 0.

Therefore, A( x – y ) ≤ A(x) A(y) .

If x, y H, then x – y may or may not belong to H.

Clearly A( x – y ) ≥ A(x) A(y).

Also, A( x–y ) ≤ A(x) A(y) .

In any case,

A( x–y ) ≥ A(x) A(y) and A( x–y ) ≤ A(x) A(y) .

Thus in all the cases, A is an intuitionistic L-fuzzy subgroup of G.

3.2.10 Theorem : Let f be any mapping from a group G1 to G2 and let A

be an intuitionistic L-fuzzy subgroup of G1. Then for , L , we have

101

f( A( , ) ) =

Proof: Suppose that , L and y = f(x)G2.

If yf( A( , ) ) , then

f(A)(y) = and f(A)(y) = .

Therefore, for every real number 1, 2 > 0, there exist x0 f-1(y) such

that A(x0) > - 1 and A(x0) < + 2.

So, for every 1, 2 > 0, y = f(x0) and hence

Therfore, f( A( , ) )

……………(1)

Conversely, , then for each 1, 2 > 0 we have

y and there exist x0 such that y = f(x0).

Therefore for each 1, 2 > 0, there exist x0 f-1(y) and A(x0) - 1

and A(x0) + 2.

Hence, f(A)(y) = and

f(A)(y) = .

So, y f(A(, )).

Therefore, f(A(, ))

……………..(2).

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From (1) and (2) we get, f(A(, )) =

3.2.11 Theorem : Let A be an intuitionistic L-fuzzy subset of a set X.

Then A(x) = { / x A } and A(x) = { β / x A }, where x

X.

Proof: Let 1 = { / x A } and > 0 be arbitrary.

Then 1 - < { / x A },

which implies that 1 - < , for some such that x A .

That is, 1 - < A(x), since A(x) ≥ .

Therefore, 1 ≤ A(x), since > 0 is arbitrary --------------- (1).

Now, assume that A(x) = s.

Then x A and so s{ / x A }.

Hence s ≤ { / x A }, where A(x) ≤ 1 ----------------- (2).

From (1) and (2), we get

A(x) = 1 = { / x A }.

And, let 2 = { β / x A } and > 0 be arbitrary.

Then 2 + > { β / x A },

which implies that 2 + > β , for some β such that x A .

That is, 2 + > A(x) since A(x) ≤ β.

Therefore, 2 ≥ A(x), since > 0 is arbitrary ------------------- (3).

Now, assume that A(x) = t.

Then x A and so t { β / x A }.

103

Hence t ≤ { β / x A }, implies A(x) ≥ 2 --------------------- (4).

From (3) and (4), we get,

A(x) = 2 = { β / x A }.

3.2.12 Theorem : Two different intuitionistic L-fuzzy subgroups of a

group may have identical family of level subgroups.

Proof: We consider the following example :

Let G be Klein’s four group:

G = { e, a, b, ab }, where a2 = e = b2, ab = ba.

Define intuitionistic L-fuzzy subsets A and B of G by

A = { e, 0.7, 0.1 , a , 0.6, 0.2 , b, 0.4, 0.3 , ab, 0.4, 0.3 } and

B = { e, 0.8, 0.2 , a, 0.7, 0.3 , b, 0.5, 0.4 , ab, 0.5, 0.4 }.

Clearly A and B are intuitionistic L-fuzzy subgroups of G.

And, Im A = { 0.7, 0.6, 0.4}, Im A = { 0.1, 0.2, 0.3 }.

The level subgroups of A are A0.7 = {e}, A0.6 = {e, a},

A0.4 = { e, a, b, ab }= G.

And, Im B = { 0.8, 0.7, 0.5 }, Im B = { 0.2, 0.3, 0.4 }.

The level subgroups of B are B0.8 = {e}, B0.7 = { e, a },

B0.5 = {e, a, b, ab}= G.

Thus the two intuitionistic L-fuzzy subgroups A and B have the same

family of level subgroups.

But A B, because A B.

3.2.13 Theorem : Let I be the subset of L and let G be a group with

subgroups { Hi}, iI such that Hi = G and i < j implies that Hi

104

Hj. Then an intuitionistic L-fuzzy subset A of G defined by A(x) ={ i /

x Hi } and A(x) = { i / x Hi }is an intuitionistic L-fuzzy subgroup

of G.

Proof: Let A be an intuitionistic L-fuzzy subset of G defined by

A(x) = { i / xHi } and A(x) = { i / x Hi }, where iI L.

Let x and yG and A(x) = m1 and A(y) = n1.

If A(xy) = { i / xyHi } < m1 n1, then there exists j such that x and y

are elements of Hj, but xy is not an element of Hj, since Hj is a subgroup

of G.

This is a contradiction.

Therefore, A(xy) ≥ m1 n1,

which implies that A(xy) ≥ A(x) A(y).

Clearly A(x-1) = A(x).

Also, A(x) = m2 and A(y) = n2.

If A(xy) = { i / xyHi } m2 n2, then there exists j such that x and y

are elements of Hj, but xy is not an element of Hj,

since Hj is a subgroup of G.

This is a contradiction.

Therefore, A(xy) ≤ m2 n2,

which implies that A(xy) ≤ A(x) A(y).

Clearly A(x-1) = A(x).

Hence A is an intuitionistic L-fuzzy subgroup of G.

105

3.2.14 Theorem : If A is an intuitionistic L-fuzzy normal subgroup of a

group G, then for each level subgroup ,(,β)L, ≤ A(e) and β

≥ A(e) is a normalsubgroup of G.

Proof: Let A be an intuitionistic L-fuzzy normal subgroup of a group G.

Let be any level subgroup of A.

To prove that is normal in G.

Let x and gG.

Then, A(x) ≥ and A(x) ≤ β.

Now, A( g-1xg ) = A( xgg-1 ), since A is an ILFNSG of G

= A(x) ≥ .

And, A( g-1xg ) = A( xgg-1 ), since A is an ILFNSG of G

= A(x) ≤ β.

Hence A( g-1xg ) ≥ and A( g-1xg ) ≤ β.

Therefore, g-1xg and hence is a normal subgroup of G.


sets G and H, respectively, and let (,β) L .

Then = x

Proof : Let (x,y)

AxB ( x, y ) and AxB ( x, y ) β

A(x) B (y ) and A(x) B (y ) β

A( x) and B (y) and A( x) β and B (y) β

A( x) and A( x) β and B (y) and B (y) β

x and y

106

(x,y) x

Therefore, = x .

3.2.16 Theorem : Let G be a finite group and A be an intuitionistic L-

fuzzy subgroup of G. If , are elements of the image set of A

such that = , then need not be and .

Proof: We consider the following example :

Let G be Klein’s four group.

G = { e, a, b, ab }, where a2 = e = b2, ab = ba.

Define intuitionistic L-fuzzy subgroup A by

A = { e, 0.5, 0.4 , a , 0.4, 0.5 , b, 0.3, 0.6 , ab, 0.3, 0.6 }.

If = 0.5, = 0.6 and = 0.4, = 0.4 are in Im A , then =

{ e }, = { e } are the level subgroups of G.

Clearly and .

Result : In a fuzzy group, if ti, tj are elements of the image set of A such

that = , then ti = tj.


group G .Then for every aG and ( L.

Proof : Let A be an intuitionistic L-fuzzy subgroup of a group G and let

xG.

Now , x ( aA )(x) ≥ α and ( aA )(x) ≤ β

A(a-1x) ≥ α and A(a-1x) ≤ β

a-1x

x a .

107

Therefore, a = for every xG.

3.3 – HOMOMORPHISM AND ANTI-HOMOMORPHISM

OF LEVEL SUBGROUPS OF INTUITIONISTIC

L-FUZZY SUBGROUPS

3.3.1 Theorem : Let (G,•) and (G׀, •) be any two groups.If

f : (G,•) → (G׀, •) is a homomorphism,then the homomorphic image of

a level subgroup of an intuitionistic L-fuzzy subgroup of a group G is a

level subgroup of an intuitionistic L-fuzzy subgroup of a group G׀.

Proof: Let (G,•) and (G׀, •) be any two groups.

Let f : (G,•) → (G׀, •) be a homomorphism.

That is, f(xy) = f(x)f(y) for all xG and y G.

Let V = f(A), where A is an intuitionistic L-fuzzy subgroup of

a group G.

Clearly V is an intuitionistic L-fuzzy subgroup of a group G׀.

Let x and y G, implies f(x) and f(y) in G׀.

Let is a level subgroup of A.

That is, A(x) ≥ and A(x) ≤ β,

A(y) ≥ and A(y) ≤ β,

A(xy-1) ≥ and A(xy-1) ≤ β

We have to prove that f ( ) is a level subgroup of V.

108

Now,

V( f(x) ) ≥ A(x) ≥ , implies that V( f(x) ) ≥

V( f(y) ) ≥ A(y) ≥ , implies that V( f(y) ) ≥ and

V( f(x)(f(y))-1 ) = V( f(x)f(y-1) ), as f is a homomorphism

= V( f(xy-1) ), as f is a homomorphism

≥ A( xy-1 ) ≥

which implies that V( f(x)(f(y))-1 ) ≥ .

And,

V( f(x) ) ≤ A(x) ≤ β, implies that V( f(x) ) ≤ β

V( f(y) ) ≤ A(y) ≤ β, implies that V( f(y) ) ≤ β and

V( f(x)(f(y))-1 ) = V( f(x)f(y-1) ), as f is a homomorphism

= V( f(xy-1) ), as f is a homomorphism

≤ A( xy-1 ) ≤ β,

which implies that V( f(x)(f(y))-1 ) ≤ β .

Therefore, V( f(x)(f(y))-1 ) ≥ and V( f(x)(f(y))-1 ) ≤ β.

Hence f ( ) is a level subgroup of an intuitionistic L-fuzzy

subgroup V of a group G׀.


f : (G,•) → (G׀, •) is a homomorphism,then the homomorphic pre-

image of a level subgroup of an intuitionistic L-fuzzy subgroup of a

group G׀ is a level subgroup of an intuitionistic L-fuzzy subgroup of a

group G.


109

Let f : (G,•) → (G׀, •) be a homomorphism.

That is, f(xy) = f(x)f(y) for all xG and y G.

Let V = f(A), where V is an intuitionistic L-fuzzy subgroup of

a group G׀.


Let f(x) and f(y) G׀, implies xG and y G.

Let f( ) is a level subgroup of V.

That is, V( f(x) ) ≥ and V( f(x) ) ≤ t ;

V( f(y) ) ≥ and V( f(y) ) ≤ t ;

V( f(x)(f(y))-1 ) ≥ and V( f(x)(f(y))-1 ) ≤ t.

We have to prove that is a level subgroup of A.

Now,

A(x) = V( f(x) ) ≥ , implies that A(x) ≥ β

A(y) = V( f(y) ) ≥ , implies that A(y) ≥ β and

A( xy-1 ) = V( f(xy-1) ),

= V( f(x)f(y-1) ), as f is a homomorphism

= V( f(x)(f(y))-1 ), as f is a homomorphism

≥ ,

which implies that A(xy-1) ≥ .

And,

A(x) = V( f(x) ) ≤ β, implies that A(x) ≤ β

A(y) = V( f(y) ) ≤ β, implies that A(y) ≤ β and

A( xy-1 ) = V( f(xy-1) ),

110

= V( f(x)f(y-1) ), as f is a homomorphism

= V( f(x)(f(y))-1 ), as f is a homomorphism

≤ β

which implies that A(xy-1) ≤ β.

Therefore, A(xy-1) ≥ and A(xy-1) ≤ β.

Hence is a level subgroup of an intuitionistic L-fuzzy subgroup A

of G.


f : (G,•) → (G׀, •) is a homomorphism,then the anti-homomorphic


group G is a level subgroup of an intuitionistic L-fuzzy subgroup of a

group G׀.


Let f : (G,•) → (G׀, •) be an anti-homomorphism.

That is, f(xy) = f(y)f(x) for all xG and y G.

Let V = f(A), where A is an intuitionistic L-fuzzy subgroup of G.

Clearly V is an intuitionistic L-fuzzy subgroup of G׀.

Let x and y G, implies f(x) and f(y) in G׀.

Let is a level subgroup of A.

That is, A(x) ≥ and A(x) ≤ β;

A(y) ≥ and A(y) ≤ β

A( y-1x ) ≥ and A( y-1x ) ≤ β.

We have to prove that f ( ) is a level subgroup of V.

111

Now,

V( f(x) ) ≥ A(x) ≥ , implies that V( f(x) ) ≥

V( f(y) ) ≥ A(y) ≥ , implies that V( f(y) ) ≥ and

V( f(x)(f(y))-1 ) = V( f(x)f(y-1) ), as f is an anti-homomorphism

= V( f( y-1x )), as f is an anti-homomorphism

≥ A( y-1x) ≥

which implies that V( f(x)(f(y))-1 ) ≥ .

And,

V( f(x) ) ≤ A(x) ≤ β, implies that V( f(x) ) ≤ β

V( f(y) ) ≤ A(y) ≤ β, implies that V( f(y) ) ≤ β and

V( f(x) ( f(y) )-1 ) = V( f(x) f(y-1) )

= V( f( y-1x ) ), as f is an anti-homomorphism

≤ A( y-1 x) ≤ β,

which implies that V( f(x) ( f(y) )-1 ) ≤ β.

Therefore, V( f(x) ( f(y) )-1 ) ≥ and V( f(x) ( f(y) )-1 ) ≤ β.

Hence f( ) is a level subgroup of an intuitionistic L-fuzzy

subgroup V of G׀.

3.3.4 Theorem :Let (G,•) and (G׀, •) be any two groups.If

f : (G,•) → (G׀, •) is a homomorphism,then the anti-homomorphic pre-


group G׀ is a level subgroup of an intuitionistic L-fuzzy subgroup of a

group G.


112

Let f : (G,•) → (G׀, •) be an anti-homomorphism.

That is, f(xy) = f(y)f(x), for all xG and y G.

Let V = f(A), where V is an intuitionistic L-fuzzy subgroup of

a group G׀.


Let f(x) and f(y) G׀, implies x and y in G.

Let f( ) is a level subgroup of V.

That is, V( f(x) ) ≥ and V( f(x) ) ≤ β

V( f(y) ) ≥ and V( f(y) ) ≤ β

V( ( f(y) )-1 f(x) ) ≥ and V( ( f(y) )-1 f(x) ) ≤ β.

We have to prove that is a level subgroup of A.

Now,

A(x) = V( f(x) ) ≥ , implies that A(x) ≥

A(y) = V( f(y) ) ≥ , implies that A(y) ≥ and

A( xy-1 ) = V( f(xy-1) ),

= V( f(y-1)f(x) ), as f is an anti-homomorphism

= V( ( f(y) )-1 f(x) )

≥ ,

which implies that A(xy-1) ≥ .

And,

A(x) = V( f(x) ) ≤ β, implies that A(x) ≤ β

A(y) = V( f(y) ) ≤ β, implies that A(y) ≤ β and

A( xy-1 ) = V( f(xy-1) ),

113

= V( f(y-1)f(x) ), as f is an anti-homomorphism

= V( ( f(y) )-1 f(x) )

≤ β,

which implies that A(xy-1) ≤ β.

Therefore, A( xy-1 ) ≥ and A( xy-1 ) ≤ β.

Hence is a level subgroup of an intuitionistic L-fuzzy

subgroup A of G.

CHAPTER- IV

INTUITIONISTIC L-FUZZY TRANSLATION

4.1 Introduction : This chapter contains the intuitionistic L-fuzzy

translation of intuitionistic L-fuzzy subgroup. These concepts are used

in the development of some important results and theorems.

4.1.1 Definition: Let A be an intuitionistic L-fuzzy subset of X and

, [ 0, 1– Sup{ A(x) + A(x) : xX, 0 A(x) + A(x) 1 }]. A

mapping T = : X L is called an intuitionistic L-fuzzy translation

of A if T = (x) = A(x) + , T = (x) = A(x) + ,

+ 1– Sup{ A(x) + A(x) : xX, 0 A(x) + A(x) 1 }, for all

xX.


TRANSLATION :

114

4.2.1 Theorem :If T is an intuitionistic L-fuzzy translation of an

intuitionistic L-fuzzy subgroup A of a group G, then T(x -1) = T(x) and

T(x-1) = T(x), T(x) T(e) and T(x) T(e), for all x, eG.

Proof: Let x and e be elements of G.

Now, T(x) = A(x) +

= A( (x-1 )-1 ) +

≥ A(x -1) +

= T( x-1)

= A(x -1) +

≥ A(x) +

= T(x).

Therefore, T(x) = T(x-1).

And T(x) = A(x) +

= A( (x-1 )-1 ) +

≤ A(x -1) +

= T(x-1)

= A(x -1) +

≤ A(x) +

= T(x).

Therefore, T(x) = T(x-1).

Now, T(e) = A(e) +

= A(xx-1) +

{ A(x) A(x-1) }+

115

= A(x) +

= T(x).

Therefore, T(e) T(x).

And T(e) = A(e) +

= A(xx-1) +

{ A(x) A(x-1) }+

= A(x) +

= T(x).

Therefore, T(e) T(x).


intuitionistic L-fuzzy subgroup A of a group G, then

(i) T( xy -1) = T(e) implies T(x) = T(y).

(ii) T( xy-1) = T(e) implies T(x) = T(y) , for all x, y, eG.

Proof: Let x, y and e be elements of G.

Now, T(x) = A(x) +

= A(xy-1y) +

{ A(xy-1) A(y) }+

= ( A(xy-1) + ) ( A(y) + )

= T(xy-1) T(y)

= T(e) T(y)

= T(y) = A(y) +

= A(yx-1x) +

{ A(yx-1) A(x) }+

116

= ( A(yx-1) + ) ( A(x) + )

= T(yx-1) T(x)

= T(e) T(x)

= T(x).

Therefore, T(x) = T(y).

And T(x) = A(x) +

= A(xy-1y) +

{ A(xy-1) A(y) }+

= ( A(xy-1) + ) ( A(y) + )

= T(xy-1) T(y)

= T(e) T(y)

= T(y) = A(y) +

= A(yx-1x) +

{ A(yx-1) ( A(x) }+

= ( A(yx-1) + ) ( A(x) + )

= T(yx-1) T(x)

= T(e) T(x)

= T(x).



intuitionistic L-fuzzy subgroup A of a group G, then T is an

intuitionistic L-fuzzy subgroup of G, for all x, yG.

117

Proof: Assume that T is an intuitionistic L-fuzzy translation of an

intuitionistic L-fuzzy subgroup A of a group G. Let x, yG.

We have, T(xy -1) = A(xy-1) +

{ A(x) A(y-1) }+

= { A(x) A(y) }+

= ( A(x) + ) ( A(y) + )

= T(x) T(y) .

Therefore, T(xy -1) T(x) T(y) .

And T(xy-1) = A(xy-1) +

{ A(x) A(y-1) }+

= { A(x) A(y) }+

= ( A(x) + ) ( A(y) + )

= T(x) T(y).

Therefore, T(xy-1) T(x) T(y).

Hence T is an intuitionistic L-fuzzy subgroup of G.



H = { xG : T(x) = T(e) and T(x) = T(e) } is a subgroup of G.

Proof:Let x, y and e be elements of G.

Given H = { xG : T(x) = T(e) and T(x) = T(e) }.

Now, T(x -1) = T( x ) = T(e) and T(x-1) = T(x) = T(e).

Therefore, T( x -1 ) = T(e) and T( x-1 ) = T(e).

Therefore, x -1 H.

118

Now, T( xy-1) T(x) T(y)

= T(e) T(e)

= T(e)

and T(e) = T( (xy-1)(xy-1)-1 )

T(xy-1) T(xy-1)

= T(xy-1).

Therefore, T(e) = T(xy-1).

Now, T(xy-1) T(x) T(y)

= T(e) T(e)

= T(e)

and T(e) = T( (xy-1)(xy-1)-1 )

T(xy-1) T(xy-1)

= T(xy-1).

Therefore, A(e) =A(xy-1).

Therefore, xy-1H. Hence H is a subgroup of G.



H = { < x, T(x) > : T(x) = T(e) and T(x) = T(e) } is a L-fuzzy

subgroup of G.

Proof:Let x, y and e be elements of G.

By Proposition 4.2.4 , xy-1H.

Therefore, T(xy-1) = T(e) and T(xy-1) = T(e).

But, T( xy-1) T(x) T(y -1)

119

= T(x) T(y).

Hence H is a L-fuzzy subgroup of G.



H = {< x, T(x) > : T(x) = T(e) and T(x) = T(e) } is an antiL-fuzzy

subgroup of G.

Proof: It is trivial .

4.2.7 Theorem :Let T be an intuitionistic L-fuzzy translation of an

intuitionistic L-fuzzy subgroup A of a group G. If T(xy-1) = 1, then

T(x) = T(y) and if T(xy-1) = 0, then T(x) = T(y).

Proof: Let xG and y G.

Now, T(x) = T( xy-1y)

T(xy-1) T(y)

= 1 T(y)

= T(y) = T(y-1)

= T( x-1xy-1)

T(x-1) T(xy-1)

= T(x) T(xy-1)

= T(x) 1 = T(x).


Now, T(x) = T( xy-1y)

T(xy-1) T(y)

= 0 T(y)

120

= T(y) = T(y-1)

= T( x-1xy-1)

T(x-1) T(xy-1)

= T(x) T(xy-1)

= T(x) 0 = T(x).


4.2.8 Theorem :Let G be a group. If T is an intuitionistic L-fuzzy

translation of an intuitionistic L-fuzzy subgroup A of G, then T(xy) =

T(x) T(y) and T(xy) = T(x) T(y), for each x,y in G with T(x)

T(y) and T(x) T(y).

Proof: Let x and y be elements of G.

Assume that T(x) T(y) and T(x) T(y).

Then, T(y) = T( x-1xy )

T(x-1) T(xy)

= T(x) T(xy)

= T(xy)

T(x) T(y) = T(y).

Therefore, T(xy) = T(y) = T(x) T(y).

Then, T(y) = T( x-1xy )

T(x-1) T(xy)

= T(x) T(xy)

= T(xy)

T(x) T(y) = T(y).

121

Therefore, T(xy) = T(y) = T(x) T(y).

4.2.9 Theorem: Let G and G׀ be any two groups. The homomorphic

image of an intuitionistic L-fuzzy translation of an intuitionistic L-fuzzy

subgroup A of G is an intuitionistic L-fuzzy subgroup of G׀.

Proof: Let G and G׀ be any two groups and f: G → G׀ be a

homomorphism.

That is f(x y) = f(x) f(y), for all x and yG.

Let V = f( ), where is an intuitionistic L-fuzzy translation of

an intuitionistic L-fuzzy subgroup A of G.

We have to prove that V is an intuitionistic L-fuzzy subgroup of G׀.


V[ f(x) ( f(y) -1 ) ] = V[ f(x) f(y -1) ]

= V [f(x y -1)]

≥ ( x y -1 )

= A ( x y -1) +

≥ { A(x) A( y -1) } +

≥ { A(x) A( y ) } +

= ( A(x) + ) ( A( y) + )

= ( x ) ( y )

which implies that V[ f(x) ( f(y) -1 ) ] ≥ V( f(x) ) V( f(y) ).

And

V[ f(x) ( f(y) -1 ) ] = V[ f(x) f(y -1) ]

= V [ f(x y -1) ]

122

(x y -1)

= A ( x y -1) +

{ A(x) A( y -1) } +

{ A(x) A( y ) } +

= ( A(x) + ) ( A(y) + )

= (x) (y)

which implies that V[ f(x) ( f(y) -1 ) ] V( f(x) ) V( f(y) ).

Therefore, V is an intuitionistic L-fuzzy subgroup of a group G׀.

Hence the homomorphic image of an intuitionistic L-fuzzy translation of

A of G is an intuitionistic L-fuzzy subgroup of G׀.


pre-image of an intuitionistic L-fuzzy translation of an intuitionistic L-

fuzzy subgroup V of G׀ is an intuitionistic L-fuzzy subgroup of G.

Proof:Let G and G׀ be any two groups and f: G → G׀ be a

homomorphism.


Let T = = f(A), where is an intuitionistic L-fuzzy translation

of intuitionistic L-fuzzy subgroup V of G׀.


Let x and yG. Then,

A(x y-1) = T( f(xy-1) )

= T( f(x)f(y-1) )

= T[ f(x) ( f(y) )-1 ]

123

= V[ f(x) ( f(y) )-1 ] +

{ V( f(x) ) V( f(y) ) }+

= ( V( f(x) ) + ) ( V( f(y) ) + )

= T( f(x) ) T( f(y) )

= A(x) A(y)

which implies that A(x y-1) ≥ A(x) A(y).

And

A(x y-1) = T( f(xy-1) )

= T( f(x)f(y-1) )

= T[ f(x) ( f(y) )-1 ]

= V[ f(x) ( f(y) )-1 ] +

{ V( f(x) ) V( f(y) ) }+

= ( V( f(x) ) + ) ( V( f(y) ) + )

= T( f(x) ) T( f(y) )

= A(x) A(y)

which implies that A(x y-1) A(x) A(y).

Therefore, A is an intuitionistic L-fuzzy subgroup of G.

Hence the homomorphic pre-image of an intuitionistic L-fuzzy

translation of an intuitionistic L-fuzzy subgroup V of G׀ is an


4.2.11 Theorem: Let G and G׀ be any two groups. The anti-

homomorphic image of an intuitionistic L-fuzzy translation of an

124

intuitionistic L-fuzzy subgroup A of G is an intuitionistic L-fuzzy

subgroup of G׀.

Proof: Let G and G׀ be any two groups and f : G → G׀ be an

anti-homomorphism.

That is f(x y) = f(y) f(x), for all x and yG.


an intuitionistic L-fuzzy subgroup A of G.

We have to prove that V is an intuitionistic L-fuzzy subgroup of G׀.


V[ f(x) ( f(y) -1 ) ] = V[ f(x) f(y -1) ]

= V [ f(y -1x) ]

≥ (y -1x)

= A (y -1x) +

≥ { A(x) A( y -1) } +

≥ { A(x) A( y ) } +

= ( A(x) + ) ( A( y) + )

= ( x ) ( y )

which implies that V[ f(x) ( f(y) -1 ) ] ≥ V( f(x) ) V( f(y) ) .

And

V[ f(x) ( f(y) -1 ) ] = V[ f(x) f(y -1) ]

= V [ f(y -1x) ]

(y -1x)

= A (y -1x) +

125

{ A(x) A( y -1) } +

{ A(x) A( y ) } +

= ( A(x) + ) ( A(y) + )

= (x) (y)

which implies that V[ f(x) ( f(y) -1 ) ] V( f(x) ) V( f(y) ) .

Therefore, V is an intuitionistic L-fuzzy subgroup of a group G׀.

Hence the anti-homomorphic image of an intuitionistic L-fuzzy

translation of A of G is an intuitionistic L-fuzzy subgroup of G׀.

4.2.12 Theorem :

Let G and G׀ be any two groups. The anti-homomorphic

pre-image of an intuitionistic L-fuzzy translation of an intuitionistic

L-fuzzy subgroup V of G׀ is an intuitionistic L-fuzzy subgroup of G.

Proof: Let G and G׀ be any two groups and f : G → G׀ be an

anti-homomorphism.



of an intuitionistic L-fuzzy subgroup V of G׀.


Let x and yG. Then,

A(x y-1) = T( f(xy-1) )

= T( f(y-1) f(x) )

= T[ ( f(y) )-1 f(x) ]

= V[ (f (y) )-1 f(x) ] +

126

{ V( f(x) ) V( f(y) ) }+

= ( V( f(x) ) + ) ( V( f(y) ) + )

= T( f(x) ) T( f(y) )

= A(x) A(y)

which implies that A(x y-1) ≥ A(x) A(y) .

And

A(x y-1) = T( f(xy-1) )

= T( f(y-1) f(x) )

= T[( f(y) )-1 f(x)]

= V[ (f(y) )-1 f(x) ] +

( V( f(x) ) V( f(y) ) ) +

= ( V( f(x) ) + ) ( V( f(y) ) + )

= T( f(x) ) T( f(y) )

= A(x) A(y)

which implies that A(x y-1) A(x) A(y) .

Therefore, A is an intuitionistic L-fuzzy subgroup of G.

Hence the anti-homomorphic pre-image of an intuitionistic L-fuzzy

translation of an intuitionistic L-fuzzy subgroup V of G׀ is an


4.2.13 Theorem :Let G and G׀ be any two groups. The homomorphic

image of an intuitionistic L-fuzzy translation of an intuitionistic L-fuzzy

normal subgroup A of G is an intuitionistic L-fuzzy normal

subgroup of G׀.

127

Proof: Let G and G׀ be any two groups and f : G → G׀ be a

homomorphism.


Let V = f( ), where T = is an intuitionistic L-fuzzy translation

of an intuitionistic L-fuzzy normal subgroup A of G.

We have to prove that V is an intuitionistic L-fuzzy normal subgroup

of G׀.

Now, for f(x) and f(y) in G׀,

clearly V is an intuitionistic L-fuzzy subgroup of G׀ , by Proposition 2.1.

We have

V( f(x) f(y) ) = V( f(xy) ),

≥ T(xy)

= A(xy) + ,

= A(yx) +

= T(yx)

V( f(yx) )

= V( f(y) f(x) )

which implies that V( f(x)f(y) ) = V( f(y) f(x) ).

And

V( f(x) f(y) ) = V( f(xy) ),

T(xy)

= A(xy) + ,

= A(yx) +

128

= T(yx)

≥ V( f(yx) )

= V( f(y) f(x) )


Therefore, V is an intuitionistic L-fuzzy normal subgroup of a group G׀.

Hence the homomorphic image of an intuitionistic L-fuzzy translation of

A of G is an intuitionistic L-fuzzy normal subgroup of G׀.


pre-image of an intuitionistic L-fuzzy translation of an intuitionistic

L-fuzzy normal subgroup V of G׀ is an intuitionistic L-fuzzy normal

subgroup of G.

Proof:

Let G and G׀ be any two groups and f: G → G׀ be a

homomorphism.



of intuitionistic L-fuzzy normal subgroup V of G׀.

We have to prove that A is an intuitionistic L-fuzzy normal subgroup

of G.

Let x and yG. Then,

129

clearly A is an intuitionistic L-fuzzy subgroup of G,

A(xy) = T( f(xy) )

= V( f(xy) ) +

= V( f(x)f(y) ) +

= V( f(y)f(x) ) +

= V( f(yx) ) +

= T( f(yx) )

= A(yx)


And

A(xy) = T( f(xy) )

= V( f(xy) ) +

= V( f(x)f(y) ) +

= V( f(y)f(x) ) +

= V( f(yx) ) +

= T( f(yx) )

= A(yx)


Therefore, A is an intuitionistic L-fuzzy normal subgroup of G.

Hence the homomorphic pre-image of an intuitionistic L-fuzzy

translation of an intuitionistic L-fuzzy normal subgroup V of G׀ is an


130

4.2.15 Theorem: Let G and G׀ be any two groups. The anti-

homomorphic image of an intuitionistic L-fuzzy translation of an

intuitionistic L-fuzzy normal subgroup A of G is an intuitionistic

L-fuzzy normal subgroup of G׀.

Proof:

Let G and G׀ be any two groups and f : G → G׀ be an anti-

homomorphism.



an intuitionistic L-fuzzy normal subgroup A of G.

We have to prove that V is an intuitionistic L-fuzzy normal subgroup

of G׀.

Now, for f(x) and f(y) in G׀,

clearly V is an intuitionistic L-fuzzy subgroup of G׀ , by Proposition 2.3.

We have

V( f(x) f(y) ) = V( f(yx) )

≥ T(yx)

= A(yx) +

= A(xy) +

= T(xy)

V( f(xy) )

= V( f(y) f(x) )


131

And

V( f(x) f(y) ) = V( f(yx) )

T(yx)

= A(yx) +

= A(xy) +

= T(xy)

≥ V( f(xy) )

= V( f(y) f(x) )


Therefore, V is an intuitionistic L-fuzzy normal subgroup of a group G׀.

Hence the anti-homomorphic image of an intuitionistic L-fuzzy

translation of A of G is an intuitionistic L-fuzzy normal subgroup of G׀.


homomorphic pre-image of an intuitionistic L-fuzzy translation of an

intuitionistic L-fuzzy normal subgroup V of G׀ is an intuitionistic L-

fuzzy normal subgroup of G.

Proof: Let G and G׀ be any two groups and f : G → G׀ be an anti-

homomorphism.



of an intuitionistic L-fuzzy normal subgroup V of G׀.

We have to prove that A is an intuitionistic L-fuzzy normal subgroup

of G.

132

Let x and yG. Then,

clearly A is an intuitionistic L-fuzzy subgroup of G,

A(xy) = T( f(xy) )

= V( f(xy) ) +

= V( f(y)f(x) ) +

= V( f(x)f(y) ) +

= V( f(yx) ) +

= T( f(yx) )

= A(yx)


And

A(xy) = T( f(xy) )

= V( f(xy) ) +

= V( f(y)f(x) ) +

= V( f(x)f(y) ) +

= V( f(yx) ) +

= T( f(yx) )

= A(yx)


Therefore, A is an intuitionistic L-fuzzy normal subgroup of G.

Hence the anti-homomorphic pre-image of an intuitionistic L-fuzzy

translation of an intuitionistic L-fuzzy normal subgroup V of G׀ is an


133

BIBLIOGRAPHY

1. Atanassov.K.T., Intuitionistic fuzzy sets, fuzzy sets and

systems, 20(1), 87-96 (1986).

2. Atanassov.K.T., Intuitionistic fuzzy sets theory and

applications, Physica-Verlag, A Springer-Verlag company,

April 1999, Bulgaria.

3. Azriel Rosenfeld, Fuzzy Groups, Journal of mathematical

analysis and applications, 35, 512-517 (1971).

4. Anthony.J.M. and Sherwood.H, Fuzzy groups Redefined,

Journal of mathematical analysis and applications, 69,124 -130

(1979 ).

5. Asok Kumer Ray, On product of fuzzy subgroups, fuzzy sets

and sysrems, 105 , 181-183 (1999 ).

6. Banerjee.B and D.K.Basnet,Intuitionistic fuzzy subrings and

ideals, J.Fuzzy Math.11, no.1, 139-155 (2003).

7. Biswas.R, Fuzzy subgroups and Anti-fuzzy subgroups, Fuzzy

sets and systems, 35,121-124 ( 1990 ).

8. Choudhury.F.P. and Chakraborty.A.B. and Khare.S.S. , A note

on fuzzy subgroups and fuzzy homomorphism, Journal of

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

134

9. Dixit.V.N., Rajesh Kumar, Naseem Ajmal., Level subgroups

and union of fuzzy subgroups, Fuzzy Sets and Systems, 37,

359-371 (1990).

10. De, K., Biswas, R, Roy, A.R, On intuitionistic fuzzy sets,

Notes on Intuitionistic Fuzzy Sets, 3(4), (1997).

11. De, K., Biswas, R, Roy, A.R, On intuitionistic fuzzy sets,

Notes on Intuitionistic Fuzzy Sets , 4(2), (1998).

12. Chakrabarty, K., Biswas, R., Nanda, A note on union and

intersection of Intuitionistic fuzzy sets , Notes on Intuitionistic

Fuzzy Sets , 3(4), (1997).

13. Gopalakrishnamoorthy.G., Ph.D Thesis, Alagappa university,

karaikudi, Tamilnadu, India, May ( 2000 ).

14. Hur.K, Jang.S.Y and Kang.H.W, Intuitionistic fuzzy ideals of

a ring, J.Korea Soc. Math.Educ.Ser.B: pure Appl.Math. 12, no.

3, 193-209 (2005).

15. Hur.K, Kang.H.W and H.K.Song, Intuitionistic fuzzy

subgroups and subrings, Honam Math. J. 25 no.1, 19-41

(2003).

16. Jun.Y.B, Ozturk.M.A and Park.C.H, Intuitionistic nil radicals

of Intuitionistic fuzzy ideals and Euclidean Intuitionistic fuzzy

ideals in rings, Inform.Sci. 177, 4662-4677 (2007).

17. Kumbhojkar.H.V., and Bapat.M.S., Correspondence theorem

for fuzzy ideals, Fuzzy sets and systems, (1991)

135

18. Mustafa Akgul, Some properties of fuzzy groups, Journal of

mathematical analysis and applications, 133, 93-100 (1988).

19. Mohamed Asaad , Groups and fuzzy subgroups, fuzzy sets and

systems (1991), North-Holland.

20. Palaniappan.N & Muthuraj.R, The homomorphism,

anti-homomorphism of a fuzzy and an anti-fuzzy groups,

Varahmihir Journal of Mathematical Sciences, Vol. 4 No.2,

387-399 (2004).

21. Prabir Bhattacharya, Fuzzy Subgroups: Some

Characterizations, Journal of Mathematical Analysis and

Applications, 128, 241-252 (1987).

22. Rajesh Kumar, Fuzzy Algebra, Volume 1, University of Delhi

Publication Division, July -1993.

23. Rajesh Kumar, Fuzzy irreducible ideals in rings, Fuzzy Sets

and Systems, 42, 369-379 (1991).

24. Salah Abou-Zaid, On generalized characteristic fuzzy

subgroups of a finite group, fuzzy sets and systems , 235-241

(1991).

25. Sidky.F.I and Atif Mishref.M ,Fuzzy cosets and cyclic and

Abelian fuzzy subgroups , fuzzy sets and systems , 43(1991)

243-250.

136

26. Sivaramakrishna das.P, Fuzzy groups and level subgroups,

Journal of Mathematical Analysis and Applications, 84,

264-269 (1981).

27. Vasantha kandasamy .W.B , Smarandache fuzzy algebra,

American research press , Rehoboth -2003.

28. ZADEH.L.A , Fuzzy sets , Information and control ,Vol.8,

338-353 (1965).

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