Some Aspects of Intuitionistic fuzzy ideals, quasi ideals and bi-ideals in near-ring

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Some Aspects of Intuitionistic fuzzy ideals, quasi ideals

and bi-ideals in near-ring

By

Dr. P.K. Sharma

Associate Professor in Mathematics

D.A.V. College, Jalandhar, Punjab

Email Id :[email protected]

At

24th International Conference on

“Near-rings, Near-fields and related topics”

From July 05-12, 2015

Manipal Institute of Technology (Manipal University),

Karnataka-576104

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Some Aspects of Intuitionistic fuzzy ideals, quasi ideals and bi-ideals

in near-ring

Abstract Result concerning various ideals in near rings (as defined by Yakabe, Chelvam and

Ganesan, Kim and Zhan and Xueling as well as some generalization thereof) that have been

established over the past three decays, will be discussed. These include the fuzzy ideals, fuzzy

quasi ideals and fuzzy bi-ideals in near rings. In this talk, we extend these notion to intuitionistic

fuzzy ideals, intuitionistic fuzzy quasi-ideals, intuitionistic fuzzy bi-ideals in near-rings. A

relationship between various types of ideals has been established.

Introduction and Preliminaries

The concept of near-rings was introduced by Pilz [18] in 1977 and that of quasi- ideal in near

ring was introduced by Yakabe [16] in 1983. The notion of bi-ideals was introduced by Chelvam

and Ganesan [17] in 1987. As we know, near-rings are a generalization of rings, and bi-ideal are

a generalization of quasi-ideals and ideals in near-rings. Thereafter many types of ideals on the

algebraic structures were characterized by several authors. The concept of quasi-ideals play an

important role in studying the structure of near-ring. Thus the notion of bi-ideals is an important

and useful generalization of quasi-ideals of near-rings. Therefore, we will study bi-ideals of near

rings in the same way as of quasi-ideals of near rings.

After the introduction of the concept of fuzzy sets by Zadeh [9] in 1965. Various

mathematician are engaged in the fuzzification of the some of the algebraic structure. The notion

of fuzzy subgroup was made by Rosenfeld [10] in 1971. In [11] Liu introduced the notion of

fuzzy ideals of a ring. The notion of fuzzy near-ring, fuzzy ideal and fuzzy R-subgroup of a near

ring was introduced by Salah Abou-Zaid [12] and it has been studied by many authors like S.D

Kim and H.S Kim etc. The notion of fuzzy bi-ideals of near rings was introduced by

T.Manikantan in 2009.

After the introduction of notion of intuitionistic fuzzy sets by Atanassov [1] in 1986.

Many researchers are engaged in applying this notion to algebra and Biswas [2] introduced the

notion of intuitionistic fuzzy subgroups in 1989. The notion of an intuitionistic fuzzy R-

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subgroups of a near-ring is given by Jun, Cho in 2005. Narmada and Mahesh Kumar [6] has

studied intuitionistic fuzzy bi-ideals in near ring in 2011.

By a near-ring, we mean a non-empty set N with two binary operations “+” and “.”

such that

(i) ( N,+ ) is a group not necessarily abelian

(ii) ( N, ) is a semi-group

(iii) (x + y).z = x.z + y.z , for all x, y, zN.

Precisely speaking it is a right near-ring because it satisfies the right distributive property.

Similarly, we can have the left near-ring. In this talk, N will denote right distributive near-ring,

unless otherwise specified. For the basic terminology and notation we refer to Pilz [16] and

Abou-Zaid [1].

We denote xy instead of x.y. We note that 0.x = 0; x N and (-x)y = -xy ; x,y N but in

general x.0 0 for some xN. The set N0 = {nN | n0 = 0} is called the zero symmetric part of

N and the set Nc= {nN | n0 = n} is called the constant part of N. The near-ring N is called zero

symmetric if N = N0 and N is called constant if N = Nc.

If A and B are two non-empty subset of near ring N. We define two types of products:

AB = {ab | aA, bB} and AB = {a(a+b) - aa | a,aA, bB}.

A subgroup S of (N, +) is called left (right) N-subgroup of N if NS S (SN S);

A subgroup M of (N, +) is called sub near-ring of N if MM M;

A sub near-ring M is called invariant in N if MN NM M.

Note that N0 and Nc are subnear-ring of N

Definition1[18] Let (N, +, .) is a near-ring. A subset I of N is said to be ideal of N if

(i) (I, +) is a normal subgroup of (N, +)

(ii) IN I i.e., inI for all nN and iI

(iii) NI I i.e., n1(n2 + i) - n1n2I for all iI and n1,n2N

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Note that I is a right ideal of N if I satisfies (i) and (ii), and I is a left ideal of N if I satisfies (i)

and (iii).

Definition2[16] A subgroup Q of (N, +) is called a quasi-ideal of N if QN NQ NQ Q.

Examples The right ideals, lefts ideals and invariant sub near-rings of N are quasi-ideal of

N. In particular, the zero symmetric part N0 and the constant part Nc of N are quasi-ideal.

Proposition3[16] Intersection of a quasi-ideal Q and a subnear-ring M of a near ring is a quasi-

ideal of N; i.e., Q M is quasi-ideal of N.

Proposition4 [16]Let N be a zero symmetric near-ring. Then the subgroup Q of (N, +) is a

quasi-ideal of N if and only if QN NQ Q.

Proof Let nN, qQ be any element. As N is zero symmetric near-ring.

nq = n(0 + q) - n0NQ NQ NQ so that NQ NQ = NQ.

Therefore, from definition 2, we have QN NQ Q.

Definition5[17] A subgroup B of (N, +) is called a bi-ideal of N if BNB (BN)B B.

Proposition6[17] Let N be a zero symmetric near-ring. A subgroup B of N is a bi-ideal if and

only if BNB B.

Proof Since N is a zero symmetric NB NB [see proposition 4]

BNB = BNB BNB BNB (BN)B B.

Proposition7[17] If B be a bi-ideal of a near ring N and S is a subnear-ring of N, then B S is a

bi-ideal of S.

Lemma Every quasi-ideal in a zero symmetric near-ring is bi-ideal.

Proof. Let Q be a quasi-ideal in a zero symmetric near-ring N. Then (Q,+) is a subgroup of N

and QN NQ Q. Now, QNQ QN and QNQ NQ. Thus, QNQ QN NQ Q.

Hence Q is bi-ideal of N.

Remark[17] The converse of above Lemma need not be true.

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For example: Let N = {0, a, b, c} be the near-ring defined by the Cayley table (see the following

Clearly, I ={0, a} is bi-ideal as NIN ={0} I, but I is not a quasi-ideal of N, for, aI = {0, b} I

Note: Every one-sided ideals, N-subgroup and an invariant subnear-ring are quasi-ideal and

so they are also bi-ideals.

We summarize the above results in the following diagram:

Right Ideal Left Ideal

Quasi-Ideal

Bi-Ideal

Ideal

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Definition8[10] A fuzzy subset of a group (G, +) is said to be a fuzzy subgroup of G if

x, y G

(i) (x + y) min{(x), (y)}

(ii) (-x) = (x)

Equivalently, (x - y) min {(x), (y)}; x, y G. Note that if is a fuzzy subgroup of G,

then (0) (x), xG.

Definition9[10] A fuzzy subgroup of a group (G, +) is called a fuzzy normal subgroup of G

if x, y G

(i) (x - y) min{(x), (y)}

(ii) (y + x - y) ≥ (x)

Let and be two fuzzy subsets of a near-ring N. We define the product of and as:

(o)(x) =

{min{ ( ), ( )}} , , N

0

x yz

Sup y z if x yz for y z

otherwise

It is easy to verify that the product of fuzzy subsets is associative. If S N, then the

characteristic function on S is denoted by S and is defined as 1 ;

( ) .0 ; \

S

if x Sx

if x N S

The characteristic function on N is N and N(x) = 1 xN.

Definition10[13] A fuzzy subgroup of a near-ring N is called a fuzzy subnear-ring of N if

oN No

Note that ( o )( ) {min{ ( ), ( )}} {min{ ( ),1}} ( )

( o )( ) ( ). Similarly, we have ( )( ) ( ).

Thus ( o )( ) ( ) min{( o )( ), ( )( )} ( )

i.e., min{

N Nz xy z xy

N N

N N N N

z Sup x y Sup x x

xy x o xy y

o xy xy xy o xy xy

( ), ( )} ( ). Thus ( ) min{ ( ), ( )}x y xy xy x y

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Another definition: A fuzzy subset of N is called a fuzzy subnear-ring of N if x, y N

(i) (x - y) min{(x), (y)}

(ii) (xy) min{(x), (y)}

Definition11[13]A fuzzy subgroup of a near-ring N is called fuzzy right (left) N-subgroup

of N if (oN) [ No ]

Another definition A fuzzy subset of N is called fuzzy right (left) N-subgroup of N if

(i) (x - y) min{(x), (y)}

(i) (xy) (x) [ (xy) (y) ]

If µ is both left and right fuzzy N−subgroup of N, then it is called a fuzzy N−subgroup of N.

Definition12[13] Let and be two fuzzy subsets of near-ring N. We define a fuzzy subset

of N by

( )

{min{ ( ), ( ), ( )}} ; ( ) ; , ,( )( )

0 ;otherwise

x a b i ab

Sup a b i if x a b i ab a b i Nx

Theorem13 [15]If N is a zero symmetric and (0) (x) xN, then o

Proof. Let x = ab = a(0 + b) – a0 N be any element, then

(0 ) 0

{min{ ( ), (0), ( )}} {min{ (( ) ( )

( )

), ( )}

( )

x a b a x ab

Sup a b Supx a xb

Definition14 [12] A fuzzy subgroup of a near-ring N is called a fuzzy quasi-ideal of N if

(oN) (No) (N)

( ) ( )

Remark 15 Note that ( )( ) {min{ ( ), ( ), ( )}} {min{1,1, ( )}} ( ).

Thus ( )( ( ) ) ( ).

N N Nz x y i xy z x y i xy

N

z Sup x y i Sup i i

x y i xy i

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Another definition A fuzzy subset of a near-ring N is called a fuzzy quasi-ideal of N if

(i) (x - y) min{(x), (y)}

(ii) (xy) min{(x), (y)}

(iii) ( x(y + i) - xy) ≥ (i), x, y, i N.

Theorem 16 [12]Every fuzzy quasi ideal in a zero symmetric near-ring is a fuzzy subnear-

ring

Proof. Let be fuzzy quasi-ideal in a zero symmetric near-ring N. Then by Theorem13, we have

( ) ( ) and so, ( ) ( ) ( ).

Thus, ( ) ( ) ( ) ( ) ( ) .

Hence is fuzzy subnear-ring of N.

N N N N N

N N N N N

Definition 17[12] A fuzzy normal subgroup of a near-ring N is called a fuzzy right (left)

ideal of N if ( o ) [ ( ) ]N N

Another Definition Let µ be a fuzzy subset of N. µ is called a fuzzy ideal of N if for all x, y, i

in N;

(1) µ(x − y) ≥ min{µ(x), µ(y)},

(2) µ(y + x − y) ≥ µ(x),

(3) µ(xy) ≥ µ(x),

(4) µ(x(y + i) − xy) ≥ µ(i) .

If µ satisfies (1), (2) and (3), then it is called a fuzzy right ideal of N. If µ satisfies

(1), (2) and (4), then it is called a fuzzy left ideal of N.

Note that every fuzzy right ideal of N is a fuzzy right N-subgroup of N.

Theorem 18 Let be a fuzzy subset of N. If is a fuzzy right ideal (left ideal, N-subgroup,

subnear-ring) of N, then is a fuzzy quasi-ideal of N

Proof. Let be a fuzzy subset of N, then we have

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Case (i) When is fuzzy right ideal of N, then is fuzzy normal subgroup of N and

( ) .N

Now ( ) ( ) ( )N N N N is fuzzy quasi-ideal of N

Case (ii) When is fuzzy left ideal of N, then is fuzzy normal subgroup of N and

( ) .N

Again ( ) ( ) ( )N N N N is fuzzy quasi-ideal of N

Case (iii) When is fuzzy right (left) N-subgroup of N, then is fuzzy subgroup of N and

N ( )N . In both cases, we get ( ) ( ) ( )N N N

is fuzzy quasi-ideal of N

Case (iv) When is fuzzy subnear-ring of N, then ( ) ( )N N

( ) ( ) ( ) ( ) ( ) .N N N N N

Hence is fuzzy quasi-ideal of N.

Theorem 19 [13] Let I be a non-empty subset of N and I be a characteristic function on I. Then

the following conditions are equivalent

(i) I is a right (left) ideal of N

(ii) I is a fuzzy right (left) ideal of N

Proof (ii) (i) Let I is a fuzzy right (left) ideal of N. let x, yN. Then I(x) = I(y) = 1. Then

I(x - y) min {I(x), I(y)} = 1 and so I(x - y) = 1 x – y I. Therefore I is additive subgroup

of N. let xN and yI. Then I(xy) I(y) = 1 and hence I(xy) = 1. So xyI, proving that I is a

right ideal of N ( If x, yN and iI, then I(x(y + i) - xy) I(i) = 1 implies that x(y + i) - xyI,

proving that I is a left ideal of N).

(i) (ii) Let I is a right (left) ideal of N. To show I is right (left) fuzzy ideal of N.

Let x, y N be any element. If x – yI, then I(x - y) = 1 min {I(x), I(y)} and if x – yI,

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then either xI or y I, i.e., either I(x) = 0 or I(y) = 0 so that min {I(x), I(y)}= 0.

Again, we have I(x - y) = 0 0 = min {I(x), I(y)}. Thus I is a fuzzy subgroup of N.

Similarly, we can show that I is also a fuzzy normal subgroup of N.

Next, let i I and xN be any element. As I is right ideal of N, we have ixI. So, we have

I(ix) = 1 ≥ I(x) implies that I is fuzzy right ideal of N [ let x, y N and i I be any elements.

As I is left ideal of N, we have x(y + i) – xyI so that I( x(y + i) – xy) = 1≥ 1= I(i) proving that

I is fuzzy left ideal of N]

Definition 20[15] A fuzzy subgroup of a near-ring N is called a fuzzy bi-ideal of N if

( ) (( ) )N N

Note : ( )( ) {min{ ( ), ( ), (c)}} {min{ ( ), (c)}}[ ( ) 1]

( )( ) min{ ( ), (c)}

( ) ( )( ) ( )

( ) ( )( ) min{ ( ), (c)}, also,

[( )

N N Nx abc x abc

N

N N

N

N

x Sup a b Sup a b

abc a

abc abc

abc abc a

1 2 1 2

1 2

( )

1 1 1 2 1 2( )

1

]( ) {min{( )( ), ( )( ), ( )}}

{min{ ( ), ( ), ( )}}, where ,

min{ ( ),

N Nx p q r pq

x p q r pq p p p q q q

p p p

x Sup p q r

Sup Sup p Sup q r p p p q q q

Sup p S

1 2

1 2 1 2

1

1 1

( ), ( )}

( ) [ ( ) ] min{ ( ), ( ), ( )}

q q q

Np p p q q q

up q r

p q r pq Sup p Sup q r

Another definition Let be a fuzzy subset of a near ring N, then is called a fuzzy bi-ideal of

N if

(i) µ(x − y) ≥ min{µ(x), µ(y)}

(ii) (xyz) min{µ(x), µ(z)}

(iii) 1 2 1 2

1 1[ ( ) ] min{ ( ), ( ), ( )}p p p q q q

p q r pq Sup p Sup q r

x, y, z, p1, p2, q1, q2, r N

Theorem 21[15] Let N be a zero symmetric near-ring. If is a fuzzy bi-ideal of N, then

( )N

Proof Let be fuzzy bi-ideal of N. Then ( ) (( ) )N N ………………….(1)

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Since is a fuzzy subgroup of N and (0) (1) xN and N(x) = 1, xN, we have

(N)(0) (N)(x) xN.

Since N is zero symmetric ( ) (( ) )N N the result follows from (1)

Another definition A fuzzy set of a zero-symmetric near-ring N is called a fuzzy bi-ideal of

N if

(i) µ(x − y) ≥ min{µ(x), µ(y)}

(ii) (xyz) min{µ(x), µ(z)} x, y, zN

Theorem 22[15] Let I be a non-empty subset of N and I be a characteristic function on I. Then

the following conditions are equivalent:

(i) I is a bi-ideal of N,

(ii) I is a fuzzy bi-ideal of N.

Theorem 23 [15]Every fuzzy right N-subgroup of N is a fuzzy bi-ideal of N.

Proof Let be a fuzzy right N-subgroup of N. Choose a, b, c, x, y, i, b1, b2, x1, x2, y1, y2 in N

such that a = bc = x(y + i) – xy, b = b1b2, x = x1x2, y = y1y2. Then

(( ) (( ) ))( ) min{( ) )( ), (( ) )( )}

= min{ min{( )( ), ( )}, (( ) )( ( ) )}

= min{ min{

N N N N

N Na bc

a bc b b

a a a

Sup b c x y i xy

Sup Sup

1 2

1 2

1 2

1

min{ ( ), ( )}, (c)}, (( ) )( ( ) )}

= min{ min{ ( ), (c)}, (( ) )( ( ) )

N Nb

Na bc b b b

b b x y i xy

Sup Sup b x y i xy

1 2 1 2 1 [Since is a fuzzy right N-subgroup of N, we have ( ) = (( ) ) = ( ( )) ( )]

min{ {min{ ( ), ( )}, ( ( ) )}

N Na bc

bc b b c b b c b

Sup bc c x y i xy

= min{ ( ), ( ( ) )}

= ( ) = ( ).

So μ is a fuzzy bi-ideal oThus ( ) (( ) ) . . f N

N

N N

bc x y i xy

bc a

Theorem 24[15] Every fuzzy left N-subgroup of N is a fuzzy bi-ideal of N.

Proof Let A be a fuzzy left N-subgroup of N. Choose a, b, c, x, y, z, i, c1, c2, x1, x2, y1, y2 in N

such that a = bc = x (y + i) – xy, c = c1c2, x = x1x2 , y = y1y2. Then

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(( ) (( ) ))( ) min{( ( )( ), (( ) )( )}

= min{ min{ ( ), ( )( )}, (( ) )( ( ) )}

= min{ min{ ( ),

N N N N

N Na bc

a bc

a a a

Sup b c x y i xy

Sup b S

1 2

1 2

1 2

2

min{ (c ), (c )}}, (( ) )( ( ) )}

= min{ min{ ( ), (c )}, (( ) )( ( ) )

N Nc c c

Na bc c c c

up x y i xy

Sup b Sup x y i xy

1 2 1 2 2 [Since is a fuzzy left N-subgroup of N, we have ( ) = ( (c c )) = (( c )c ) (c )]

min{ {min{ (b), ( )}, ( ( ) )}

=

N Na bc

bc b b

Sup bc x y i xy

min{ ( ), ( ( ) )}

= ( ) = ( ).

Thus ( ) ( So μ is a fuzzy bi-ideal of N( ) ) . .

N

N N

bc x y i xy

bc a

Theorem 25 [15]Every fuzzy two-sided N-subgroup of N is a fuzzy bi-ideal of N

Proof The proof is straight forward from previous Theorem 23 and Theorem 24

Theorem 26 [15]Every fuzzy right ideal of N is a fuzzy bi-ideal of N

Proof. Since every fuzzy right ideal is a fuzzy right N-subgroup of N. Therefore, the result

follows from Theorem 23

Theorem 27 [15]Every fuzzy left ideal of N is a fuzzy bi-ideal of N

Proof Let be a fuzzy left ideal of N. Choose a, b, c, x, y, z, i, x1, x2, y1, y2 in N such that a = bc

= x (y + i) – xy, x = x1x2 , y = y1y2. Then

1 2( )

(( ) (( ) ))( )= min{ min{( )( ), ( )}, (( ) )( ( ) )}

= min{ min{( )(( )), ( )}, {( )( ), ( )( ), ( )}}

[Since

N N N Na bc

N N Na bc a x y i xy

N N

a Sup b c x y i xy

Sup b b c Sup x y i

and also is a fuzzy left N-subgroup of N, ( ( ) ) ( )] x y i xy i

1 2( )

min{ min{ ( ), (c)}, {( )( ), ( )( ), ( ( ) )}

= ( ( ) ) = ( )

Therefore ( ) (( ) ) Henc. e

N N N Na bc a x y i xy

N N

Sup b b Sup x y x y i xy

x y i xy a

μ is a fuzzy bi-ideal of N.

Theorem 28 [15]Every fuzzy left ideal of N is a fuzzy bi-ideal of N

Proof The Proof is straight forward from Theorem 26 and Theorem 27

Remark The converse of Theorem 25 and Theorem 28 are not necessarily true as shown by the

following example.

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Example Let N = {0, a, b, c} be the near ring with ( N, + ) as the Klein’s four group and ( N, . )

as defined below [see , Pilz, p.408]

Define a fuzzy subset : N[0,1] by (0) = 0.9, (a) = 0.4, (b) = 0.4 and (c) = 0.7. Then

( )(0) 0.9, ( )( ) 0.7, ( )( ) 0.4, ( )( ) 0.7.

Also,( ) )(0) 0.9,( ) )( ) 0,( ) )( ) 0.7,( ) )( ) 0.

N N N N

N N N N

a b c

a b c

Therefore, is a fuzzy bi-ideal of N. Since (a) = (ca) (c) and (a) = (a0) (0), is not

a two-sided N-subgroup of N. Also, (a) = (c0) min{(c), (0), is not a fuzzy subnear-ring

of N and so is not a fuzzy ideal of N.

We summarize the above results in the following diagram:

Fuzzy Ideal

Right fuzzy Ideal Left fuzzy Ideal

Fuzzy Quasi-Ideal

Fuzzy Bi-Ideal

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Definition 29 [1]An intuitionistic fuzzy set (IFS) A in a non-empty set X is an object having the

form A = {(x, A(x), A(x)) | xX}, where the functions A: X [0, 1] and A: X [0, 1] denote

the degree of membership and the degree of non-membership of each element xX to the set A,

respectively, and 0 A(x) +A(x) 1 for all xX.

For the sake of simplicity, we shall use the symbol A = (A, A) for the intuitionistic

fuzzy set A = {(x, A(x), A(x)) | xX}.

Definition 30 [2]An IFS A = (A,A) of a group (G, +) is said to be an intuitionistic fuzzy

subgroup (IFSG) of G if x, yG

(i) A(x + y) min{A(x), A(y)} (ii) A(x + y) max{A(x), A(y)}

(iii) A(-x) = A(x) (iv) A(-x) = A(x)

Equivalently, A(x - y) min{A(x), A(y)} and A(x - y) min{A(x), A(y)} x, yG.

Note that if A is an IFSG of G, then A(0) A(x) and A(0) A(x) xG.

Definition 31[2] An IFS A = (A, A) of a group (G, +) is called an intuitionistic fuzzy Normal

subgroup (IFSNG) of G if x, yG

(i) A(x - y) min{A(x), A(y)} (ii) A(x - y) max{A(x), A(y)}

(iii) A(y + x - y) A(x) (iv) A(y + x - y) A(x)

Let A = (A, A) and B = (B, B) be two intuitionistic fuzzy subset (IFS) of a near-ring N.

We define the product of A and B as AB = ( , ),where AB AB

Sup{min{ ( ), ( )}}; if = Inf{max{ ( ), ( )}}; if = ( ) , ( )

0; otherwise 1; otherwise

A B A Bx yz x yz

AB AB

y z x yz y z x yzx x

.

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It is easy to verify that the product of IFSs is associative. If SN, then, we define the

characteristic function s on N is defined as (1,0) S

( ) .(0,1) N \ S

S

if xx

if x

The characteristic function on N is N and N(x) = (1, 0) xN.

Definition 32 [3] An intuitionistic fuzzy subgroup(IFSG) A of a near-ring N is called an

intuitionistic fuzzy subnear-ring of N if (A ) ( A) AN N

. It is easy to check that (A )( ) A( ) and ( A)( ) A( )

Thus, (A ) ( A) ( ) A( ) A( ) A( )

( ) min{ ( ), ( )} and ( ) max{ ( ), ( )}

N N

N N

A A A A A A

Note xy x xy y

xy x y xy

xy x y xy x y

Another definition An IFS A = (A, A) is a near-ring N is called an intuitionistic fuzzy

subnear-ring of N if for all x, yN

(i) A(x - y) min{A(x), A(y)} (ii) A(x - y) max{A(x), A(y)}

(iii) A(xy) min{A(x), A(y)} (iv) A(xy) max{A(x), A(y)}

Definition 33 [3]An intuitionistic fuzzy subgroup (IFS) A of a near-ring N is called an

intuitionistic fuzzy right (left) N-subgroup of N if AoN A ( No A A )

Another definition An IFS A = (A, A) is a near-ring N is called An intuitionistic fuzzy right

(left) N-subgroup of N if for all x, y N

(i) A(x - y) min{A(x), A(y)} (ii) A(x - y) max{A(x), A(y)}

(iii) A(xy) A(x) (A(xy) A(y)) (iv) A(xy) A(x) (A(xy) A(y))

Note that every intuitionistic fuzzy N-subgroup of a near-ring N is an intuitionistic fuzzy sub-

near ring of N. But the converse is not true in general as seen in the following examples

Example Let N = {a, b, c, d} be a set with two binary operations as follows

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Then (N, +, .) is a near-ring.

(1) We define an IFS A = (A, A) in N by A = {< a, 0.7, 0.2 >, < b, 0.5, 0.3 >, < c, 0.4, 0.5 >}.

Then A is an intuitionistic fuzzy subnear-ring of N. But A is not an intuitionistic fuzzy right N-

subgroup of N since A (b.c) = A(c) = 0.4 < 0.5 = A(b) and/or A (b.c) = A(c) = 0.5 > 0.3 =

A(b).

(2) Let B = (B, B) be an IFS of ring of real number R. ((R, +, .) is a near-ring) defined by

1 0 1 0

0.5 Z \{0} 0.25 Z \{0}( ) , ( )

0.25 Q \ Z 0.5 Q \ Z

0 R \ Q 1 R \ Q

B B

if r if r

if r if rr r

if r if r

if r if r

For rR, where Q and Z are rings of rational number and the integers respectively. Then B is an

intuitionistic fuzzy subnear-ring of R. But B (2.1/3) = B(2/3) = 0.25 < 0.5 = B(2) and/or

B (2.1/3) = B(2/3) = 0.5 > 0.25 = B(2). Hence B is not an intuitionistic fuzzy right R-subgroup

of R.

Definition 34 [6]Let A and B be two intuitionistic fuzzy subsets of near-ring N. We define an

intuitionistic fuzzy subset AB of N by AB = (AB, AB), where

( )

{min{ ( ), ( ), ( )}} ; ( ) ; , ,( )

0 ;otherwise

A A Bx a b i ab

A B

Sup a b i if x a b i ab a b i Nx

17

( )

{max{ ( ), ( ), ( )}} ; ( ) ; , ,( )

1 ;otherwise

A A Bx a b i ab

A B

Inf a b i if x a b i ab a b i Nx

Theorem 35 [6]If N is a zero symmetric and A(0) A(x) and A(0) A(x) xN, then

AoB AB.

Proof. Let x = ab = a(0 + b) – a0 N be any element, then

(0 ) 0

{min{ ( ), (0), ( )}} {min{ ( ), ( )}

. .   Hence * .

A B A A B A B A Bx a b a x ab

A B A B A B A B

Sup ax x

x x Similarly we can sh

b

ow that x x Ao

b

B B

Sup a

A

Definition 36[5] An intuitionistic fuzzy subgroup (IFSG) A of a near-ring N is called an

intuitionistic fuzzy quasi-ideal of N if (AoN) (NoA) (NA) A

( ) ( )

( ) ( )

Remark 36 Now ( )( ) {min{ ( ), ( ), ( )}} {min{1,1, ( )}} ( )

( )( ) {max{ ( ), ( ), ( )}} {max{0,0, ( )}} ( ).

Thus ( )

N A N N A A Az x y i xy z x y i xy

N A N N A A Az x y i xy z x y i xy

N

z Sup x y i Sup i i

z Inf x y i Inf i i

A

( ( ) ) ( ), ( ) ( ).A Ax y i xy i i A i

Another definition An intuitionistic fuzzy subset A of a near-ring N is called an intuitionistic

fuzzy quasi-ideal of N if x, y, i N

(i) A(x - y) min{A(x), A(y)} (ii) A(x - y) max{A(x), A(y)}

(iii) A(xy) min{A(x), A(y)} (iv) A(xy) max{A(x), A(y)}

(ii) A( x(y + i) - xy) ≥ A(i) (vi) A( x(y + i) - xy) A(i)

Theorem 37[5] Every intuitionistic fuzzy quasi-ideal in a zero symmetric near-ring is an

intuitionistic fuzzy subnear-ring.

Proof Let A be an intuitionistic fuzzy quasi-ideal in a zero-symmetric near-ring N. Then by

Theorem 38 [5]we have (NoA) (NA) and so (NoA) (NA) = (NoA).

Thus (AoN) (NoA) = (AoN) (NoA) (NA) A. Hence A is an intuitionistic fuzzy

subnear-ring of N.

18

Definition 39[5] An intuitionistic fuzzy normal subgroup A of a near-ring N is called an

intuitionistic fuzzy right (left) ideal of N if (AoN) A [(NA) A]

Another definition Let A be an IFS of a near-ring N. Then A is called an intuitionistic fuzzy

ideal of N if for all x, y, i in N;

(i) A(x - y) min{A(x), A(y)} (ii) A(x - y) max{A(x), A(y)}

(iii) A(y + x - y) ≥ A(x) (iv) A(y + x - y) A(x)

(v) A(xy) A(x) (vi) A(xy) A(x)

(vii) A(x(y + i) - xy) ≥ A(i) (viii) A( x(y + i) - xy) A(i)

If A satisfies (i), (ii), (iii), (iv), (v), (vi), then it is called an intuitionistic fuzzy right ideal of N.

If A satisfies (i), (ii), (iii), (iv), (vii), (viii), then it is called an intuitionistic fuzzy left ideal of N.

Note that every intuitionistic fuzzy right ideal of N is an intuitionistic fuzzy right N-subgroup of

N and every intuitionistic fuzzy left ideal of N is an intuitionistic fuzzy left N-subgroup of N.

Theorem 40 Let A an IFS on N. If A is an intuitionistic fuzzy right ideal (left ideal, N-

subgroup, N-subnear-ring) of N, then A is an intuitionistic fuzzy quasi-ideal of N

Proof Let A be an IFS of N, then we have

Case (i) When A is an intuitionistic fuzzy right ideal of N, then A is an IFNSG of N and

(AoN) A. Now (AoN) (NoA) (NA) (AoN) A.

A is an intuitionistic fuzzy quasi-ideal of N

Case (ii) When A is an intuitionistic fuzzy left ideal of N, then A is IFNSG of N and (NA) A

Again (AoN) (NoA) (NA) (NA) A.

A is an intuitionistic fuzzy quasi-ideal of N

Case (iii) When A is an intuitionistic fuzzy right (left) N-subgroup of N, then A is an IFSG of N

and (AoN) A [(NoA) A]. In both cases, we get (AoN) (NoA) (NA) A.

A is an intuitionistic fuzzy quasi-ideal of N.

19

Case (iv) When A is an intuitionistic fuzzy subnear-ring of N, then (AoN) (NoA) A.

(AoN) (NoA) (NA) (AoN) (NoA) A.

Hence A is an intuitionistic fuzzy quasi-ideal of N.

Definition 41 [6]An intuitionistic fuzzy subgroup-of a near-ring N is called an intuitionistic

fuzzy bi-deal of N if (A o N o A) ((A o N) A) A.

Note:- (A A)( ) ( ( ), ( )), where

( ) Sup{min{ ( ), ( ), (c)} (Sup{min{ ( ), (c)}} and

( ) Inf {max{ ( ), ( ), (c)}} Inf {max{ ( ), (c)}}

N N

N

N

N A A A A

A A A N A A Ax abc x abc

A A A N A A Ax abc x abc

A

x x x

x a b a

x a b a

( ) ( )

( )(

( ) min{ ( ), (c)} and ( ) max{ ( ), (c)},

Thus, (A A) A ( ) min{ ( ), (c)} and ( ) max{ ( ), (c)}

[(A ) A)]( ) ( ( ), ( ))

( ) = Sup

N N

N N

N

A A A A A A A

N A A A A A A

N A A A A

A Ax p q

abc a abc a

abc a abc a

x x x

x

1 2 1 2

)

1 1( )

( )( )

{min{( )( ), ( )( ), ( )}}

= Sup {min{ Sup ( ), Sup ( ), ( )}} and

( ) = Inf {max{( )( ), ( )( ), ( )}}

N

A N A N Ar pq

A A Ax p q r pq p p p q q q

A A A N A N Ax p q r pq

p q r

p q r

x p q r

1 2 1 2

1 2 1 2

1 2 1 2

1 1( )

( ) 1 1

( ) 1 1

= Inf {max{ ( ), ( ), ( )}}

( ( ) ) min{ Sup ( ), Sup ( ), ( )}and

( ( ) ) max{ ( ), ( ), ( )}

Now

N

N

A A Ax p q r pq p p p q q q

A A A A Ap p p q q q

A A A A Ap p p q q q

Inf p Inf q r

p q r pq p q r

p q r pq Inf p Inf q r

1 2 1 2

1 2 1 2

1 1

1 1

, (A ) A A ( ( ) ) min{ Sup ( ), Sup ( ), ( )}and

( ( ) ) max{ ( ), ( ), ( )}.

N A A A Ap p p q q q

A A A Ap p p q q q

p q r pq p q r

p q r pq Inf p Inf q r

Another definition Let A be an IFS of a near-ring N, then A is called an intuitionistic fuzzy bi-

ideal of N if

(i) A(x - y) min{A(x), A(y)} (ii) A(x - y) max{A(x), A(y)}

(iii) A(xyz) min{A(x), A(z)} (iv) A(xyz) max{A(x), A(z)}

(v) A(x(y + i) - xy) ≥ min{1 2x x x

Sup

A(x1), 1 2y y y

Sup

A(y1), A(i)}

20

(vi) A( x(y + i) - xy) max{1 2x x x

Inf

A(x1), 1 2y y y

Inf

A(y1), A(i)}

Theorem 42 Let N be a zero symmetric near-ring and if A is an intuitionistic fuzzy bi-ideal

of N, then (A o N o A) A.

Proof Let A be an intuitionistic fuzzy bi-ideal of N. Then (A o N o A) ((A o N) A) A...(1)

Since A is an IFSG of N and A(0) A(x), A(0) A(x) xN and N(x) = 1, xN, we have

(A o N)(0) (A o N)(x) and (A o N)(0) (A o N)(x) xN. Since N is zero symmetric

(A o N o A) ((A o N) A), so the result follows from (1)

Another definition An IFS A of a zero symmetric ring is called an intuitionistic fuzzy bi-

ideal of N if x, y, zN

(i) A(x - y) min{A(x), A(y)} (ii) A(x - y) max{A(x), A(y)}

(ii) A(xyz) min{A(x), A(z)} (iv) A(xyz) max{A(x), A(z)}

Theorem 43 Every intuitionistic fuzzy right N-subgroup of N is an intuitionistic fuzzy bi-ideal

of N.

Proof let A = (A, A) be an intuitionistic fuzzy right N-subgroup of N. Choose a, b, c, x, y, i, b1,

b2, x1, x2, y1, y2 in N such that a = bc = x(y + i) – xy, b = b1b2, x = x1x2, y = y1y2. Then

21

(A A) ((A ) A) (A A) ((A ) A)

(A A) ((A ) A) ( ) ( ) (A A) ((A ) A) ( ) ( )

(

[(A A) ((A ) A)]( ) ( ), ( ) , where

( ) min{ ( ), ( ) and ( ) max{ ( ), ( )})

N N N N

N N N N N N N N

N N

A A A A A A A A

A

a a a

a a a a a a

1 2

1 2

) 1 2

1

( ) = min{( )( ), ( )}= min{ min{ ( ), ( )}, ( )}

= min{ ( ), ( )}

N A A N A A N Aa bc a bc b b b

A Aa bc b b b

a Sup b c Sup Sup b b c

Sup Sup b c

1 2 1 2 1

[Since A is an intuitionistic fuzzy right N-subgroup of N,

we have ( ) = (( )c) = ( ( c )) ( )]

{min{ ( ), (c)} = min{ ( )} = ( ) = ( ).

Therefor

A A A A

A N A A Aa bc

bc b b b b b

Sup bc bc bc a

e, ( ) ( ). Similarly, we can show that ( ) ( ).N NA A A A A Aa a a a

( )

( ) ( )

( ) {( ) }( ) {( ) }( ( ) )

= ( ) ( ( ) ) ( )

i.e., ( ) ( ). Similarly, ( ) ( ).

So, A A A and (A ) A A implies that (A

N

N N

A A A N A A N A

A A A

A A A A A A

N N

a a x y i xy

i x y i xy a

a a a a

A) ((A ) A) AN N

Theorem 44 Every intuitionistic fuzzy left N-subgroup of N is an intuitionistic fuzzy bi-ideal of

N.

Proof let A = (A, A) be an intuitionistic fuzzy left N-subgroup of N. Choose a, b, c, x, y, i, b1,

b2, x1, x2, y1, y2 in N such that a = bc = x(y + i) – xy, b = b1b2, x = x1x2, y = y1y2. Then

(A A) ((A ) A) (A A) ((A ) A)

(A A) ((A ) A) ( ) ( ) (A A) ((A ) A) ( ) ( )

(

[(A A) ((A ) A)]( ) ( ), ( ) , where

( ) min{ ( ), ( ) and ( ) max{ ( ), ( )})

N N N N

N N N N N N N N

N N

A A A A A A A A

A

a a a

a a a a a a

1 2

1 2

) 1 2

2

( ) = min{ ( ), ( )( )}= min{ ( ), min{ ( ), ( )}}

= min{ ( ), ( )}

N A A N A A N Aa bc a bc c c c

A Aa bc c c c

a Sup b c Sup b Sup c c

Sup b Sup c

1 2 1 2 2[Since A is an intuitionistic fuzzy left N-subgroup of N, we have ( ) = ( ( ))= (( ) ) ( )]

{min{ ( ), ( )} = min{ ( )} = ( ) = ( ).

Therefore,

A A A A

N A A A Aa bc

bc b c c bc c c

Sup b bc bc bc a

( ) ( ). Similarly, we can show that ( ) ( ).N NA A A A A Aa a a a

( )

( ) ( )

( ) {( ) }( ) {( ) }( ( ) )

= ( ) ( ( ) ) ( )

i.e., ( ) ( ). Similarly, ( ) ( ).

So, A A A and (A ) A A implies that (A

N

N N

A A A N A A N A

A A A

A A A A A A

N N

a a x y i xy

i x y i xy a

a a a a

A) ((A ) A) AN N

22

Theorem 45 Every intuitionistic fuzzy two sided N-subgroup of N is an intuitionistic fuzzy bi-

ideal of N.

Proof. The proof is straight forward from previous Theorems 43 and Theorem 44

Theorem 46 Every intuitionistic fuzzy right ideal of N is an intuitionistic fuzzy bi-ideal of N.

Proof. Since every intuitionistic fuzzy right ideal is an intuitionistic fuzzy right N-subgroup of

N. Therefore, the result follows from Theorems 44

Theorem 47 Every intuitionistic fuzzy left ideal of N is an intuitionistic fuzzy bi-ideal of N.

Proof. Let A be an intuitionistic fuzzy left ideal of N. Choose a, b, c, x, y, i, b1, b2, x1, x2, y1, y2 in

N such that a = bc = x(y + i) – xy, b = b1b2, x = x1x2, y = y1y2. Then

Intuitionistic Fuzzy Ideal

Right intuitionistic fuzzy Ideal Left intuitionistic fuzzy Ideal

Intuitionistic Fuzzy Quasi-Ideal

Intuitionistic Fuzzy Bi-Ideal

23

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