Vol 7, No 5 (2016)

(1)

Available online throug

ISSN 2229 – 5046

INTERVAL VALUED INTUITIONISTIC ANTI FUZZY PRIMARY IDEAL

M. PALANIVELRAJAN*

1

_{, K. GUNASEKARAN}

2

_{AND E. ADILAKSHMI}

3

1

_{Department of Mathematics,}

Government Arts College, Paramakudi – 623 707, Tamilnadu, India.

2,3

_{Ramanujan Research Centre,}

PG and Research Department of Mathematics,

Government Arts College (Autonomous), Kumbakonam-612 001, Tamilnadu, India.

(Received On: 09-05-16; Revised & Accepted On: 30-05-16)

ABSTRACT

T

he concept of interval valued intuitionistic anti fuzzy primary ideals and interval valued intuitionistic anti fuzzy semiprimary ideals is defined. Various properties of interval valued intuitionistic anti fuzzy primary ideals and interval valued intuitionistic anti fuzzy semiprimary ideals are discussed. Finally, interval valued intuitionistic anti fuzzy primary ideals and interval valued intuitionistic anti fuzzy semiprimary ideals is defined, some properties are established.

Keywords: Intuitionistic fuzzy set, Intuitionistic anti fuzzy primary ideal, Intuitionistic anti fuzzy semi-primary ideal,

Interval valued intuitionistic anti fuzzy primary ideals, Interval valued intuitionistic anti fuzzy semi primary ideals.

1. INTRODUCTION

After on introduction of fuzzy sets by A.Zadeh[8], the fuzzy concept has introduced almost all branches of mathematics. The notion of intuitionistic fuzzy set and its operations were introduced by Atanassov [1], as a generalization of the notion of fuzzy set. Atanassov [2, 3] discussed the operators over interval valued intuitionistic fuzzy sets. Palanivelrajan and Nandakumar [6] introduced the definition and some properties of intuitionistic anti fuzzyprimary and semiprimary ideals and interval valued intuitionistic fuzzy primary ideals. In this paper interval valued intuitionistic anti fuzzy primary ideals and intuitionistic anti fuzzy semiprimary ideal are established.

2. PRELIMINARIES

In this section, some basic definitions that are essential for this paper are assembled.

Definition 2.1: Let

S

be any nonempty set. A mapping

µ

:

S

→

[0,1]

is called a fuzzy subset of

S

_.

Definition 2.2: A fuzzy subset

µ

of a ring

R

_{is called anti fuzzy ideal if for all}

x y

,

∈

R

the following conditions are satisfied

( )

i

µ

(

x

−

y

)

≤

max

(

µ

( )

x

,

µ

( )

y

)

( )

ii

µ

( )

xy

≤

min

(

µ

( )

x

,

µ

( )

y

)

Definition 2.3: A fuzzy ideal

µ

of a ring

R

is called anti fuzzy primary ideal, if for all

a b

,

∈

R

either

( )

µ a b µ a

=

or else

µ ab

( )

≥

µ b

( )

m for some

m

∈

Z

+.

Corresponding Author: M. Palanivelrajan*

1

_{Department of Mathematics, Government Arts College,}

(2)

Definition 2.4: A fuzzy ideal

µ

of a ring

R

is called anti fuzzy semiprimary ideal, if for all

a

,

b

∈

R

either

( )

n

µ ab

≥

µ a

, for some

n

∈

Z

+ , or else

µ ab

( )

≥

µ b

( )

m for some

m

∈

Z

+

.

Definition 2.5: An intuitionistic fuzzy set (IFS)

A

in

X

is defined as an object of the form

( ) ( )

,

{

_A _A

|

}

A

= 〈

x µ

x

γ

x

〉 ∈

x

X

, where

µ

_A

:

X

→

[ ]

0,1

and

γ

_A

:

X

→

[ ]

0,1

are the degree of membership

and the degree of non-membership of the element

x

∈

X

,

respectively, for every

x

∈

X

,

satisfying

( )

0 ≤

µ x

_A

( )

+

γ

_A

y

≤

1.

Definition 2.6: A fuzzy subset

µ

of a ring

R

is called intuitionistic anti fuzzy ideal if for all

x y

,

∈

R

, the following conditions are satisfied,

( )

i

µ

A

(

x

−

y

)

≤

ma

(

x

µ

A

( ) ( )

x µ

,

A

y

)

( )

ii

µ

A

( )

xy

≤

min µ

(

A

( ) ( )

x µ

,

A

y

)

( )

iii

γ

_A

(

x

−

y

)

≥

min

,

(

γ

_A

( ) ( )

x

γ

_A

y

)

( )

iv

γ

_A

( )

xy

≥

max

(

γ

_A

( ) ( )

x

,

γ

_A

y

)

Definition 2.7: An intuitionistic anti fuzzy ideal

R

of a ring

R

is called Intuitionistic anti fuzzy primary ideal if for

all

a b

,

∈

R

either

µ

_A

( )

a

b µ

=

_A

( )

a

and

γ

_A

( )

ab

=

γ

_A

( )

a

,

or

µ

A

(

ab

)

≥

µ

A

(

b

m

)

and

( )

m A

ab

A

b

γ

≤

γ

,

for some

m

∈

Z

+

.

Example 2.1:

0

4 ( )

0.3 2 ~ 4

A

if x

x

if x

µ

_{= }



∈< >

∈< > < >



1

8 ( )

0.6 2 ~ 8

A

if x

x

if x

γ

_{= }



∈< >

∈< > < >



Definition 2.8: An intuitionistic anti fuzzy ideal

A

of a ring

R

is called intuitionistic anti fuzzy semiprimary ideal if

for all

a

,

b

∈

R

either µ_A(ab)≥µ a_A( n) and

γ

_A

(

ab

)

≤

γ

_A

(

a

n

)

, for some

n

∈

Z

+ or else

µ

_A

(

ab

)

≥

µ

_A

(

b

m

)

and

γ

_A

(

ab

)

≤

γ

_A

(

b

m

)

for some

m

∈

Z

+

.

Definition 2.9: An interval valued fuzzy set

A

is specified by a function

M

_A

:

E

→

D

[ ]

0,1 ,

where

D

[ ]

0,1

is the

set of all intervals with in [0, 1], for all

x

∈

E M

,

_A

( )

x

is an interval

[ ]

a b

,

where

0 ≤ ≤ ≤

a

b

1.

Remark 2.1: An interval valued intuitionistic fuzzy set. A over E is defined as an object of the form

( )

,

{

_A _A

|

}

A

=

x M

x

N

x

∈

E

where

M

_A

( )

x

⊆

[ ]

0,1

and

N

_A

( )

x

⊆

[ ]

0,1

are interval and for all

x

∈

E

,

( )

1.

A A

rmaxM

x

+

rmaxN

x

≤

Definition 2.10: Let A be an interval valued intuitionistic fuzzy sets. A fuzzy ideal A of a ring

R

is said to be interval

valued intuitionistic anti fuzzy primary ideal of

R

if for all

a b

,

∈

R

then either

( ) ( ) ( )

A ab A ab A

µ =rmaxM =rmaxM a

=

µ a

A

( )

and

γ

A

(

a

b

)

=

rm

axN

A

(

ab

)

=

rmaxN

A

( )

a

=

γ

A

( )

a

or

(

)

A

ab

γ

r

max

M

_A

(

ab

)

≥

r

max

M

A

(

b

n

)

=

µ

A

(

b

n

)

and

γ

A

(ab)

=

rmaxN

A

(

ab

)

(

)

n A

rmaxN b

≤

=

γ

A

(

b

n

),

for some

n

∈

Z

+

.

(3)

(

)

A

µ ab

=

rmaxM

A

(

ab

)

≥

rmaxM

A

(a )

n

( )

n A

µ

a

=

and

( )

A

ab

rmax N

A

ab

γ

=

≤

rmax

N

A

(

a

n

)

(

)

n A

a

γ

=

, for some

n

∈

Z

+ or

(

)

A

ab

µ

=

rmax

M

_A

( )

ab

≥

rmax

µ

_A

(

b

m

)

=

µ

_A

(

b

m

)

and

(

)

A

ab

γ

=

rmax N

_A

(

ab

)

≤

rma

x

N (

A

b

m

)

=

γ

A

(

b

m

)

for some

m

Z

.

+

∈

Example 2.2:

( )

A A

µ

x

=

rmaxM

x

=

[0, 0]

0 [0.2, 0.3]otherwise

if x

=







max

( )

A

x

r

N

A

x

γ

=

[1,1]

0 [0.4, 0.5]otherwise

if x

=







Definition 2.12: Let

A

and

B

are interval valued intuitionistic fuzzy set then A B is an interval valued intuitionistic fuzzy set E. If the following conditions are satisfied

A

∩ =

B

{

,

〈

x min

[

(

rmin M

,

_A

( )

x

rmin

M

_B

( )),

x

max rmin

(

N

_A

( )

x

,

r

min

N

_B

(

x

))

,

〉

/ ,

x

∈

E

}

Definition 2.13: Let A and B are interval valued intuitionistic fuzzy set then

A

∪

B

is an interval valued intuitionistic fuzzy set E. If the following conditions are satisfied

{

,

[

(

_A

( )

,

A

∪ = 〈

B

x max r

minM

x

rmin

M

_B

( ))

x

,

min rminN

(

_A

( )

x

,

rmi N

n

_B

( ) ,

x

)

〉

/ ,

x

∈

E

}

Definition 2.14: Let A and B are interval valuedintuitionistic fuzzy set then

A

+

B

{

,

A

(

x

)

B

( )

–

A B

+ =

〈

x rminM

+

rminM

x

r

min

M

_A

(

x

)

.

rminMB(x),

r

min

N

A

(

x

).

r

min

N

B

(

x

)

〉 ∈

| ,

x

E

}

A B

.

{

( )

.

,

_A

.

_B

(

),

A B

=

〈

x rminM

x

rm

i

nM

x

rm

in

N

_A

(

x

)

+

rmin

_B

( )

x

–

Nr

min

M

_A

(

x

)

.

rmi

n

N

_B

( )

x

,

〉

| ,

x E

}

A

@

B

is an interval valued intuitionistic fuzzy set E. If the foll0owing conditions are satisfied

{

min

,

2 ( )

min

( )

@

,

r

M

A

x

r

M

B

x

A

B

= 〈

x

+

rminN

A

( )

rminN

B

( )

2 x

+

x

}

,

>

/ .

x

∈

E

A B

$

{

min

( )

$

,

r

M

A

x r

. min

M

B

(

)

,

A B

= 〈 



x

r

min

N

A

( ). min

x r

N

B

(

x

)

〉 ∈



|

x

E

}.

Definition 2.18: Let A and B are interval valued intuitionistic fuzzy set then A#B is an interval valued intuitionistic fuzzy set E. If the following conditions are satisfied

A#B = { x,

(

AA BB

)

(x)

( )

2.rminM

.rminM

rminM

( )

rminM

( )

x

+

x

,

(

)

A B

(x)

( )

2.rminN

.rminN

rminN

( )

rminN

( )

x

+

x

〉

|

x

∈

E

}

Definition 2.19: Let A and B are interval valued intuitionistic fuzzy set then A B is an interval valued intuitionistic fuzzy set E. If the following conditions are satisfied

A B=

{

(

AA BB

)

rminM

2. r

( )

,

(x)

minM

.rmi

nM

( )

1 x

x

+



〈 



,

(

)

A B

rminN

2. rminN

. rminN

( )

(x)

( ) 1

x

+





(4)

3. INTERVAL VALUED INTUITIONISTIC ANTI FUZZY PRIMARY IDEALS AND INTERVAL VALUED INTUITIONISTIC ANTI FUZZY SEMIPRIMARY IDEALS

Theorem 3.1: If A and B are interval valued intuitionistic anti fuzzy semiprimary ideal of R, then

A

∩

B

is an

interval valued intuitionistic anti fuzzy semiprimary ideal of R.

Proof: Let Abe an interval valued intuitionistic anti fuzzy semiprimary ideal of a ring R,then

(

)

(

)

A A

µ

xy

=

rmaxM

xy

≥

rmaxM

_A

(

x

n

)

=

µ

_A

(xy)

and

(

)

A

xy

γ

=

rmaxN

_A

(

x

y

)

≤

rmax

N

_A

(

x

n

)

=

γ

_A

(

x

n

)

for some n ∈ Z+ and x, y ∈ R.

Let B be an interval valued intuitionistic anti fuzzy semiprimary ideal of the ring R, then

(

)

(

)

B B

µ

xy

=

rmaxµ

xy

≥

rmaxM

_B

(

x

n

)

=

µ x

_B

(

n

)

a d

n

γ

_B

(

x

y

)

=

rmaxN

_B

(

xy

)

≤

rmax

N

_B

(

x

n

)

=

γ

_B

(

x

n

)

,

for some n ∈ Z+and x, y ∈ R.

Consider

x y

,

∈

R

, then

x y

,

∈ ∩

A

B

implies

x y

,

∈

A and x y

,

∈

B

Consider

ma

(

)

x

(

)

A B A B

µ

_∩

xy

=

r

M

_∩

xy

max( max

(

), max

(

))

max( max

(

), max

ma

(

)

(

)

x

)

n A

A B

n n

A B

B n A B

r

M

xy r

M

xy

r

M

x

r

M

x

r

M

x

µ

∩ ∩

=

≥

Therefore,

µ

A B∩

(

xy

)

≥

γ

A B∩

(

x

n

).

Consider

ma

)

(

x

(

A B

x

y

r

N

A B

x

y

γ

_∩

=

_∩

min( max

(

), max

(

))

min( max

(

), max

ma

(

))

(

x

)

n A

A B

n n

A B

B n A B

r

N

xy r

N

xy

r

N

x

r

N

x

r

N

x

γ

∩ ∩

=

≤

Therefore,

γ

_{A B}_∩

(

xy

)

≤

γ

_{A B}_∩

(

x

n

).

Therefore

A

∩

B

is an interval valued intuitionistic anti fuzzy semiprimary ideal of R.

Theorem 3.2: IfAandBare interval valued intuitionistic anti fuzzy semiprimary ideal ofR, then

A

∪

B

(

)

(

)

A A

µ

xy

=

rmaxM

xy

≥

rmaxM

_A

(

x

n

)

=

µ x

_A

(

n

)

and

(

)

A

xy

γ

=

rmaxN

A

(

x

y

)

≤

rmaxN

A

(

x

n

)

=

γ

A

(

x

n

)

for some n ∈ Z

+ _and_x_,_y_∈_R_.

(

)

M

(

)

B B

µ

xy

=

rmax

xy

(

n

)

B

(

)

n B

rmaxM

x

µ x

≥

=

_and

(

)

B

xy

γ

=

rmaxN

B

(

xy

)

≤

rmax

N

B

(

x

n

)

=

γ

B

(

x

n

)

,

for some n ∈ Z

+_and_x_,_y_∈_R_.

(5)

Consider

ma

)

(

x

(

A B A B

µ

_∪

x

y

=

r

M

_∪

x

y

min( max

(

), max

(

))

min( max

(

), max

ma

(

))

(

x

)

n A

A B

n n

A B

B n A B

r

M

xy r

M

xy

r

M

x

r

M

x

r

M

x

µ

∩ ∪

=

≥

Therefore,

(

)

(

n

).

A B

xy

A B

x

µ

_∪

≥

µ

_∪

Consider

ma

)

(

x

(

A B

x

y

r

N

A B

x

y

γ

_∪

=

_∪

max( max

(

), max

(

))

max( max

(

), max

ma

(

))

(

x

)

n A

A B

n n

A B

B n A B

r

N

xy r

N

xy

r

N

x

r

N

x

r

N

x

γ

∪ ∪

=

≤

Therefore,

γ

_{A B}_∪

(

xy

)

≤

γ

_{A B}_∪

(

x

n

).

Therefore

A

∪

B

Theorem 3.3: IfAandBare interval valued intuitionistic anti fuzzy semiprimary ideal ofR, then

A

+

B

(

)

(

)

A A

µ

xy

=

rmaxM

xy

≥

rmaxM

A

(

x

n

)

=

µ x

A

(

n

)

_and

(

)

A

xy

γ

=

(

)

(

n

)

(

n

)

A A A

rmaxN

x

y

≤

rmax

N

x

=

γ

x

(

)

M

(

)

B B

µ

xy

=

rmax

xy

≥

rmaxM

_B

(

x

n

)

=

µ x

_B

(

n

)

and

(

)

B

xy

γ

=

rmax N

,

_B

(

x

y

)

≤

rma

x

N

_B

(

x

n

)

=

γ

_B

(

x

n

)

Consider

x y

,

∈

R

, then

x y

,

∈ +

A

B

implies

x y

,

∈

A and x y

,

∈

B

consider

ma

)

(

x

(

A B A B

µ

₊

x

y

=

r

M

₊

x

y

( max

(

)

max

(

)

max

(

). max

(

))

( max

(

)

max

(

)

max

(

). max

(

))

(

)

(

)

(

).

(

)

(

).

A B A B

n n n n

A B A B

n n n n

A B A B

n A B

r

M

xy

r

M

xy

r

M

xy r

M

xy

r

M

x

r

M

x

r

M

x

r

M

x

µ

₊

=

+

−

≥

+

−

=

+

−

=

;

Therefore,

(

)

(

n

)

A B A B

(6)

Consider

ma

)

(

x

(

A B

x

y

r

N

A B

x

y

γ

₊

=

₊

max

(

). max

(

)

max

(

). max

(

)

(

).

(

)

(

).

A B

n n

A B

n n

A B

n A B

r

N

xy r

N

xy

r

N

x

r

N

x

γ

₊

=

≥

=

Therefore,

(

)

(

n

)

A B

xy

A B

x

γ

₊

≤

γ

₊

Therefore,

A

+

B

Theorem 3.4: IfAandBare interval valued intuitionistic anti fuzzy semiprimary ideal of R, then A.Bis an interval valued intuitionistic anti fuzzy semiprimary ideal of R.

(

)

(

)

A A

µ

xy

=

rmaxM

xy

≥

(

n

)

(

n

)

A A

rmax M

x

=

µ x

and

(

)

A

xy

γ

=

(

)

(

n

)

(

n

)

A A A

rmaxN

xy

≤

rmaxN

x

=

γ

x

(

)

M

(

)

B B

µ

xy

=

rmax

xy

≥

rmaxM

_B

(

x

n

)

=

µ x

_B

(

n

)

and

(

)

B

xy

γ

=

rmaxN

_B

(

xy

)

≤

rmax

N

_B

(

x

n

)

=

γ

_B

(

x

n

)

,

Consider

x y

, ,

∈

R

then

x y

, .

∈

A B

implies

x y

,

∈

A and x y

,

∈

B

Consider

.

(

)

max

.

(

)

A B A B

µ

xy

=

r

M

xy

.

max

(

). max

(

))

max

(

). max

(

))

(

).

(

)

(

).

A B

n n

A B

n A B

A B

r

M

xy r

M

xy

r

M

x

r

x

M

x

µ

=

≥

=

Therefore,

µ

_{A B}_.

(

xy

)

≥

µ

_{A B}_.

(

x

n

)

Consider

.

(

)

max

.

(

)

A B

x

y

r

N

A B

x

y

γ

=

.

( max

(

)

max

(

)

max

(

). max

(

))

( max

(

)

max

(

)

max

(

). max

(

))

(

).

(

)

(

).

(

)

(

).

A B A B

n n n n

A B A B

n n n n

A B A B

n A B

r

N

xy

r

N

xy

r

N

xy r

N

xy

r

N

x

r

N

x

r

N

x

r

N

x

γ

=

+

−

≤

+

−

=

−

=

Therefore,

γ

_{A B}_.

(

xy

)

≤

γ

_{A B}_.

(

x

n

)

Therefore,

A B

.

(7)

(

)

(

)

A A

µ

xy

=

rmaxM

xy

≥

(

n

)

(

n

)

A A

rmaxM

x

=

µ x

and

(

)

A

xy

γ

=

rmaxN

A

(

x

y

)

≤

rmax

N

A

(

x

n

)

=

γ

A

(

x

n

)

for some n ∈ Z

+ _and_x_,_y_∈_R_.

(

)

M

(

)

B B

µ

xy

=

rmax

xy

≥

rmaxM

_B

(

x

n

)

=

µ x

_B

(

n

)

and

(

)

B

xy

γ

=

rmax N

,

_B

(

x

y

)

≤

rma

x

N

_B

(

x

n

)

=

γ

_B

(

x

n

)

Consider

x y

, ,

∈

R

then

x y

, @

∈

A

B

implies

x y

,

∈

A and x y

,

∈

B

Consider

@

(

)

max

@

(

)

A B

x

y

r

M

A B

x

y

µ

=

@

max

(

)

max

(

)

2 max

(

)

(

)

(

)

ma

)

x

(

)

2

A B

n n

A B

n n

A A

n A B

r

M

xy

r

M

xy

r

M

x

r

M

x

µ

=

+

=

+

≥

Therefore,

µ

_A_@_B

(

xy

)

≥

µ

_A_@_B

(

x

n

)

Consider

@

(

)

max

@

(

)

A B

x

y

r

N

A B

x

y

γ

=

@

max

(

)

max

(

)

2 max

(

)

(

)

(

)

ma

)

x

(

)

2

A B

n n

A B

n n

A A

n A B

r

N

xy

r

N

xy

r

N

x

r

N

x

γ

µ

=

+

=

+

≤

Therefore,

γ

_A_@_B

(

xy

)

≤

γ

_A_@_B

(

x

n

)

Therefore,

A

@

B

Theorem 3.6: IfAandBare interval valued intuitionistic anti fuzzy semiprimary ideal ofR, then A$Bis an interval valued intuitionistic anti fuzzy semiprimary ideal of R.

(

)

(

)

A A

µ

xy

=

rmaxM

xy

≥

(

n

)

(

n

)

A A

rmax M

x

=

µ x

and

(

)

A

xy

γ

=

(

)

(

n

)

(

n

)

A A A

rmaxN

x

y

≤

rmax N

x

=

γ

x

(

)

M

(

)

B B

µ

xy

=

rmax

xy

≥

rmaxM

_B

(

x

n

)

=

µ x

_B

(

n

)

and

(

)

B

xy

(8)

Consider

x y

, ,

∈

R

then

x y

,

∈

A B

$

implies

x y

,

∈

A and x y

,

∈

B

Consider

$

(

)

max

$

(

)

A B

x

y

r

M

A B

x

y

µ

=

$

max

(

) max

(

)

max

(

)

(

) ma

)

(

)

(

x

A B

n n

A B

n n

A n A

B

r

M

xy r

M

xy

r

M

x r

M

x

µ

=

≥

=

Therefore,

µ

A B_$

(

xy

)

≥

µ

A B_$

(

x

n

)

.

Consider

$

(

)

max

_{A B}

(

)

A B

x

y

r

N

x

y

γ

=

$

max

(

) max

(

)

max

(

) max

(

)

(

)

(

)

(

)

A B

n n

A B

n n

A B

n A B

r

N

xy r

N

xy

r

N

x r

N

x

γ

=

≤

=

Therefore,

$

(

)

_{A B}

(

)

n A B

xy

N

x

γ

≤

Therefore,

A B

$

Theorem 3.7: IfAandBare interval valued intuitionistic anti fuzzy semiprimary ideal ofR, then A#Bis an interval valued intuitionistic anti fuzzy semiprimary ideal of R.

(

)

(

)

A A

µ

xy

=

rmaxM

xy

≥

rmaxM

A

(

x

n

)

=

µ x

A

(

n

)

and

(

)

A

xy

γ

=

rmaxN

_A

(

x

y

)

≤

rmax

N

_A

(

x

n

)

=

γ

_A

(

x

n

)

(

)

M

(

)

B B

µ

xy

=

rmax

xy

(

)

B

(

)

n n

B

rmaxM

x

µ x

and

≥

=

(

)

B

xy

γ

=

rmaxN

B

(

xy

)

≤

rmax

N

B

(

x

n

)

=

γ

B

(

x

n

)

,

for some n ∈ Z

+_and_x_,_y_∈_R_.

Consider

x y

, ,

∈

R

, then

x y

,

∈

A B

#

implies

x y

,

∈

A and x y

,

∈

B

#

(

)

max

#

(

)

A B

x

y

r

M

A B

x

y

µ

=

#

(

)

(

)

2 max

max

(

)

max

(

)

2 max

max

(

)

max

(

)

(

2 (

)

(

)

(

)

(

)

.

(

)

(

)

A B

n n

A B

n n

A B

n n

A B

n n

A B

n A B

xy

x

r

M

r

M

xy

r

M

r

M

xy

r

M

x r

M

x

r

M

x

r

M

x

y

x

µ

=

+

≥

+

=

(9)

Therefore,

µ

_{A B}_#

(

xy

)

≥

µ

_{A B}_#

(

x

n

)

Consider

#

(

)

max

#

(

)

A B

x

y

r

N

A B

x

y

γ

=

#

(

)

(

)

2 max

max

(

)

max

(

)

2 max

max

(

)

max

(

)

(

2 (

)

(

)

(

)

(

)

.

(

)

(

)

A B

n n

A B

n n

A B

n n

A B

n n

A B

n A B

xy

x

r

N

r

N

xy

r

N

r

N

xy

r

N

x r

N

x

r

N

x

r

N

x

y

x

γ

=

+

≤

+

=

+

Therefore, _#

(

)

_#

(

n

)

A B

xy

A B

x

γ

≤

γ

Therefore,

A B

#

Theorem 3.8: IfAandBare interval valued intuitionistic anti fuzzy semiprimary ideal ofR, then A*Bis an interval valued intuitionistic anti fuzzy semiprimary ideal of R.

(

)

(

)

A A

µ

xy

=

rmaxM

xy

≥

(

n

)

(

n

)

A A

rmaxM

x

=

µ x

and

(

)

A

xy

γ

=

rmaxN

A

(

x

y

)

≤

r

max

N

A

(

x

n

)

=

γ

A

(

x

n

)

for some n ∈ Z

+ _and_x_,_y_∈_R_.

(

)

M

(

)

B B

µ

xy

=

rmax

xy

≥

rmaxM

_B

(

x

n

)

=

µ x

_B

(

n

)

and

(

)

B

xy

γ

(

)

(

n

)

(

n

)

,

B B B

rmaxN

xy

rmax

N

x

γ

x

=

≤

=

Consider

x y

, ,

∈

R

then

x y

,

∈

A B

*

implies

x y

,

∈

A and x y

,

∈

B

Consider

*

(

)

max

*

(

)

A B

x

y

r

M

A B

x

y

µ

=

*

max

(

)

2( max

max

(

) 1)

max

(

)

2( max

ma

(

)

(

)

(

)

(

)

x

(

) 1)

(

)

(

)

2(

(

)

(

) 1

(

)

A B

n n

A B

n n

A B

n n

A B

n n

A B

n A B

r

M

r

M

xy

r

M

r

M

xy

r

M

x

r

M

x

r

M

x r

xy

M

x

µ

+

=

+

≥

+

=

+

Therefore,

µ

_{A B}_*

(

xy

)

≥

µ

_{A B}_*

(

x

n

)

Consider

*

(

)

max

*

(

)

A B

xy

r

N

A B

xy

γ

≥

max

(

)

2( max

max

(

)

(

)

(

)

1)

A B

n n

A B

r

N

r

N

xy

r

N

x r

N

x

xy

+

=

(10)

*

max

(

)

2( max

max

(

) 1)

(

)

(

)

(

)

(

)

2 ((

)

(

)

(

)

1)

.

n n

A B

n n

A B

n n

A B

n n

A B

n A B

r

N

x

r

N

x

r

N

x r

N

x

γ

+

≤

+

=

Therefore,

γ

A B_*

(

x

y

)

≤

γ

A B_*

(

x

n

)

Therefore,

A B

*

Theorem 3.9: IfAand Bare interval valued intuitionistic anti fuzzy primary ideal of R then

A

∩

B

is an interval valued intuitionistic anti fuzzy primary ideal of R.

Theorem 3.10: If A andB are interval valued intuitionistic anti fuzzy primary ideal of R then

A

∪

B

Theorem 3.11: IfA and B are interval valued intuitionistic anti fuzzy primary ideal ofR then

A

+

B

is an interval valued intuitionistic anti fuzzy primary ideal of R

Theorem 3.12: If A and B are interval valued intuitionistic anti fuzzy primary ideal of R then

A B

.

Theorem 3.13: IfA andB are interval valued intuitionistic anti fuzzy primary ideal ofRthen

A

@

B

is an interval

valued intuitionistic anti fuzzy primary ideal of R.

Theorem 3.14: IfA andB are interval valued intuitionistic anti fuzzy primary ideal ofR thenA$Bis an interval valued intuitionistic anti fuzzy primary ideal of R.

Theorem 3.15: IfA andB are interval valued intuitionistic anti fuzzy primary ideal ofRthenA#B is an interval valued intuitionistic anti fuzzy primary ideal of R.

Theorem 3.16: If Aand B are interval valued intuitionistic anti fuzzy primary ideal of R thenA B is an interval valued intuitionistic anti fuzzy primary ideal of R.

REFERENCES

1. Atanassov, K. T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, Vol. 20, 1986, No. 1, 87–96.

2. Atanassov, K., Operators over interval valued intuitionistic fuzzy sets, Fuzzy Sets andSystems, Vol. 64, 1994,

No. 2, 159–174.

3. Atanassov, K., Intuitionistic Fuzzy Sets, Springer, Heidelberg, 1999.

4. Atanassov, K. T, G. Gargov. Interval valued intuitionistic fuzzy sets, Fuzzy Sets andSystems, Vol. 31, 1989,

343–349.

5. Chakrabarty, K., R. Biswas, S. Nanda, A note on union and intersection of intuitionistic fuzzy sets, Notes on

Intuitionistic Fuzzy Sets, 3(4), (1997).

6. Palanivelrajan.M, S.Nandakumar some properties of intuitionistic fuzzy primary and semiprimary ideals, Notes

on Intuitionistic Fuzzy Sets, Vol.18, 2012, No.3, 68-74.

7. Rajesh, K. Fuzzy semiprimary ideals of rings, Fuzzy Sets and Systems, Vol. 42, 1991, 263–272.

8. Zadeh, L. A., Fuzzy sets, Information and Control, Vol. 8, 1965, 338–353.

Source of support: Nil, Conflict of interest: None Declared