Intuitionistic Fuzzy Subsemiring

International Journal of Mathematics and Computer Application Research (IJMCAR) ISSN (P): 2250-2378; ISSN (E): 2319-4510 Vol. 9, Issue 2, Dec 2019, 73–82 © TJPRC Pvt. Ltd.

INTUITIONISTIC FUZZY SUBSEMIRING B. ANITHA Annamalai University, Annamalainagar,Tamilnadu, India.

ABSTRACT: In this paper, we introduce KEYWORDS:

-intuitionistic fuzzy set,

-intuitionistic fuzzy set &

-intuitionistic fuzzy subsemiring. Also we study their properties.

-intuitionistic fuzzy subsemiring

Received: Jun 13, 2019; Accepted: Jul 03, 2019; Published: Oct 26, 2019; Paper Id.: IJMCARDEC20197

1. INTRODUCTION Zadeh [13] in 1965 introduced fuzzy sets after which several researchers explored on the generalizations of the notion of fuzzy sets and its application to many mathematical branches. Atanassov introduced intuitionistic fuzzy set which constitute a genralization of the notion of fuzzy sets [1, 2]. A Solairaju and R. Nagarajan [8, 9, 10] have introduced and -fuzzy subgroups. S. Hemalatha, et.al. [3] introduced the concept of

subring of a ring and established some results. A study on anti Vanathi, et.al. [11]. Some theorems in

-fuzzy subsemiring of a semiring has been introduced by

-intuitionistic fuzzy subsemiring of a semiring has been introduced by Vanathi

et.al. [12]. O. Ratnabala Devi [7] introduced the concept of intuitionistic introduce the concept of

-fuzzy

-intuitionistic fuzzy subset,

rings, semirings, near-rings are studied with

-fuzzy ideals of Near-rings. In this paper we −fuzzy subsets of

-intuitionistic fuzzy subsemiring. So far all

as a set only. In this paper we introduce the concept of

-intuitionistic

fuzzy subset of a semiring where ( , . ) is a semigroup.

2. PRELIMINARIES Definition 2.1 Let

be a non empty set and

be a non empty set. A

-fuzzy subset

of

is a function

:

×

→

[0,1].

Definition 2.2 Let ( , +,⋅) be a semiring. A

-fuzzy subset

of

is said to be a

-fuzzy subsemiring of

if it satisfies

the following conditions: [(i)] •

.

( + , ) ≥ min{ (

•

, ) ≥ min{

Definition 2.3 A ∈

and

( , ),

( , ),

( , )}

( , )}, for all ,

-intuitionistic fuzzy subset

∈ }, where

:

non-membership of the element

× in

in

is defined as an object of the form

→ [0,1] and " : and

in

∈ ×

= {〈( , ),

( , ), " ( , )〉/

→ [0,1] define the degree of membership and the degree of

respectively and for every

in

and

in

satisfying 0 ≤

( , ) + " ( , ) ≤ 1.

Definition 2.4 Let ( , +,⋅) be a semiring. A subsemiring of

www.tjprc.org

-intuitionistic fuzzy subset

of

is said to be a

-intuitionistic fuzzy

if it satisfies the following conditions: [(i)]

editor@tjprc.org

Original Article

defined a new algebraic structure called

74

B. Anitha

3.

•

( + , ) ≥ min{

•

(

, ) ≥ min{

( , ),

( , ),

( , )} ( , )}

•

" ( + , ) ≤ max{" ( , ), " ( , )}

•

" (

, ) ≤ max{" ( , ), " ( , )}, ∀ ,

∈

∈ .

and

-INTUITIONISTIC FUZZY SUBSEMIRING

Definition 3.1 Let ( , +,⋅) be a semiring. Let ( ,⋅) be a semigroup. A map of

:

×

→ [0,1] is said to be a

-fuzzy subset

.

Definition 3.2 Let ( , +, ,⋅) be a semiring. Let ( ,⋅) be a semigroup. An ( , ),

( , ), " ( , ) >/ ∈ ,

∈ } where the function

:

×

-intuitionistic fuzzy set is defined as → [0,1] and " : ∈ ,

degree of membership and degree of non-membership for each element ( , ) + " ( , ) ≤ 1, for each

0≤

∈ ,

∈

×

= {<

→ [0,1] denote the

to the set

respectively and

∈ .

Definition 3.3 Let ( , +,⋅) be a semiring. A

-fuzzy subset

of

is said to be a

-fuzzy subsemiring of

if it satisfies

the following conditions: •

( + , ) ≥ min{

•

(

•

( , ,

, ) ≥ min{ +)

( , ),

( , ),

≥ min{

( ,

( , )}

( , )} , ),

( ,

+ )}, ∀

Definition 3.4 Let ( , +,⋅) be a semiring. A

,

∈

-fuzzy subset

and , of

,, +

in

.

is said to be an anti

-fuzzy subsemiring of

if it

satisfies the following conditions: •

( + , ) ≤ max{

•

(

•

( , ,

, ) ≤ max{ +)

( , ),

( , ),

≤ max{

( ,

( , )} ( , )}

, ),

( ,

Definition 3.5 Let ( , +,⋅) be a semiring. A

+ )}

-fuzzy subset

of

is said to be a

-intuitionistic fuzzy subsemiring of

if it satisfies the following conditions: •

( + , ) ≥ min{

•

(

•

( , .

, ) ≥ min{ +)

≥ min{

( , ),

( , ), ( ,

( , )} ( , )}

, ),

( ,

+ )}

•

" ( + , ) ≤ max{" ( , ), " ( , )}

•

" (

•

" ( , .

, ) ≤ max{" ( , ), " ( , )} +)

≤ max{" ( ,

4. SOME THEOREMS OF

, ), "

+ )}∀

,

∈

and ,

,, +

∈ .

-INTUITIONISTIC FUZZY SUBSEMIRING

Theorem 4.1 Intersection of any two subsemiring of

( ,

-intuitionistic fuzzy subsemiring of a semiring

is a

-intuitionistic fuzzy subsemirings of a semiring

and ,

-intuitionistic fuzzy

.

Proof. Let

and - be any two

Impact Factor (JCC): 6.9195

∈

and ,

,, +

∈

NAAS Rating: 3.76

Intuitionistic Fuzzy Subsemiring

75

. = {(( , ),

Let also 2 =

( , ), " ( , ))/ ∈ ./0 ∈ } and -={(( , ),

∩ - = {(( , ),

where min{ 4(

Now,

4(

( , ),

, ), "4 ( , ))/ ∈ 1(

, )} =

+ , ) = min{

4(

4(

, ) and max{" ( , ), "1 ( , )} = "4 ( , ).

( + , ),

1(

+ , )}

≥ min{min{

( , ),

= min{min{

( , ),

+ , ) ≥ min{

4(

and

4(

, ) = min{

, ),

(

4(

, ),

( , )}, min{ 1(

, )}∀ ,

1(

4(

,

4(

,. +)

, ) ≥ min{

( , ),

= min{min{

( , ),

= min{

( ,

, ),

, . + ),

4(

1(

,

∈

4(

, .

+)

, )}∀ ,

, ),

= min{min{

( ,

, ),

,

, ),

, )}}

( , ),

1(

, )}}

, ),

1(

, )}}

( , ),

1(

, )}}

, )}

1(

, )}, min{

4(

, ),

and

4(

, )}

∈

, . + )}

( ,

4(

1(

∈

and

1(

∈

≥ min{min{

≥ min{

4(

( , )}, min{

( , 1(

= min{ ∴

, ),

, ),

, )}

≥ min{min{

4(

1(

, )}, min{

4(

= min{ ∴

, ), "1 ( , ))/ ∈ and ∈ } and

∈ },

and

= min{ ∴

1(

4(

,

4(

+ )}∀

∈

+ )}, min{ 1 (

,

, )}, min{

,

, ),

and

4(

,

,, +

,

, ),

1(

,

+ )}}

( ,

+ ),

1(

,

+ )}}

+ )}

∈

Now "4 ( + , ) = max{" ( + , ), "1 ( + , )} ≤ max{max{" ( , ), " ( , )}, max{"1 ( , ), "1 ( , )}} = max{max{" ( , ), "1 ( , )}, max{" ( , ), "1 ( , )}} = max{"4 ( , ), "4 ( , )} ∴ "4 ( + , ) ≤ max{"4 ( , ), "4 ( , )}∀ , Now "4 (

, ) = max{" (

, ), "1 (

∈

∈

and

, )}

≤ max{max{" ( , ), " ( , )}, max{"1 ( , ), "1 ( , )}} = max{max{" ( , ), "1 ( , )}, max{" ( , ), "1 ( , )}} = max{"4 ( , ), "4 ( , )} ∴ "4 (

, ) ≤ max{"4 ( , ), "4 ( , )}∀ ,

and "4 ( ,

,. +)

= max{" ( ,

, . + ), "1 (

∈ ,

and

, . + )}

∈

76

B. Anitha

≤ max{max{" ( ,

, ), "

( ,

= max{max{" ( ,

, ), "1 (

+ )}, max{"1 (

,

, )}, max{"

= max{"4 ( , ∴ "4 ( ,

,. +)

≤ max{"4 ( ,

, ), "4 (

,

Theorem 4.2 The intersection of a family of subsemiring of a semiring

+ ), "1 (

,

+ )}}

∈ is a -intuitionistic fuzzy

∈

Then

( + , ) = inf

and ,

,, +

, ) = inf 8∈=

( ,

,. +)

8∈=

?@ (

,

,. +) ?@ (

8∈=

,. +)

≥ min{

B@ (

, ), inf 8∈=

?@ (

( ,

, ),

( ,

+ )}

, )} ( , ),

∈

, )}

, ),

+ )}

for all

∈

( , ),

( , )}

∈ .

and

,

( , )}

∈ .

, )C = min{

,

, ), inf ?@ ( 8∈=

?@ (

and

?@ (

?@ (

8∈=

,

∈

, ),

≥ infmin{

, ),

, )} = min{

( , )} for all ,

= min{inf ( ,

?@ (

?@ (

8∈=

( , ),

= inf

8∈=

, ) ≥ infmin{ 8∈=

, ) ≥ min{

, ), inf

( , )} for all ,

= min Ainf (

?@ (

( , ),

?@ (

?@ (

8∈=

8∈=

( + , ) ≥ min{

and let = ⋂8∈= 78 .

-intuitionistic fuzzy subsemiring of a ring

+ , ) ≥ infmin{

= min{inf

(

-intuitionistic fuzzy

∈ . ?@ (

8∈=

Now

is a

.

Let ,

∴

( ,

-intuitionistic fuzzy subsemiring of a semiring

Proof. Let {78 ; : ∈ ;} be a family of

and

+ )}}

+ )}

,, +

and

,

.

subsemiring of

∴

∈

,

, ), "1 (

-intuitionistic fuzzy subsemiring of a semiring

Hence the intersection of any two

∴

+ )}∀

, ), "4 (

,

?@ (

,

+ )}

= min{ and

( , ,, +

, ),

( ,

+ )}

∈ .

Now " ( + , ) = sup"?@ ( + , ) ≤ supmax{"?@ ( , ), "?@ ( , )} 8∈=

8∈=

= max{sup"?@ ( , ), sup"?@ ( , )} = max{" ( , ), " ( , )}∀ , 8∈=

8∈=

" (

, ) = sup"?@ ( 8∈=

= max{sup"?@ ( , 8∈=

That is

is a

,. +)

, ), sup"?@ ( 8∈=

,

= sup"?@ ( , 8∈=

+ )}

,. +)

= max{" ( ,

Impact Factor (JCC): 6.9195

is a

∈

≥ supmax{"?@ ( , 8∈=

, ), "

-intuitionistic fuzzy subsemiring of a ring

fuzzy subsemiring of a semiring

∈ .

8∈=

8∈=

" ( ,

and

, ) ≤ supmax{"?@ ( , ), "?@ ( , )}

= max{sup"?@ ( , ), sup"?@ ( , )} = max{" ( , ), " ( , )}∀ , 8∈=

∈

( ,

+ )}∀

∈

and , ), "?@ (

and

∈ . ,

+ )}

,, +

∈ .

. Hence the intersection of a family of

-intuitionistic fuzzy subsemiring of a semiring

-intuitionistic

.

NAAS Rating: 3.76

Intuitionistic Fuzzy Subsemiring

and - are any two

Theorem 4.3 If × - is a

77

-intuitionistic fuzzy subsemirings of the semirings

-intuitionistic fuzzy subsemiring of and - be two

Proof. Let Let

,, +

Now

∈

,, ,, +

×1 [( , , , )

∈

×

Then ( , ,

+ ),

]=

,)

≥ min{min{

Also

+ ( +,

+ ),

] ≥ min{

×1 [( , , , )( + , + ),

]=

),

( , , ),

Also

×1 [( , , , ), , . + ]

= min{

∴

×1 [( , , , ), , . + ]

Now "

×1 [( , , , )

= min{min{

( ,,

, ),

]="

),

( + , ),

( ,,

1( +, 1( +,

)}} )}}

)}

)}

1( , +,

)}, min{ ),

),

)}

1( ,,

),

( + , ),

×1 (( + , + ),

1( +, 1( +,

)}} )}}

)}

)}

1( ,

+ )}, min},

1 ( , , , )}, min{

×1 (( , , , ), , ),

×1 (( , , , ), , ), + ),

),

+,

, ),

+ ( +,

1( ,,

(

( ,,

≥ min{

)}

1 ( , , , . + )}

≥ min{min{

= min{

+,

, . + ),

+

+

+,

)}

1 ( , , , ),

( +,

+ ),

1 ( + , + )}} 1 ( + , + )}}

×1 (( , , , ), + )}

×1 (( , , , ), + )}

×1 (( ,

= max{" (

+ ,

+, ,

+

+,

+

+ ),

), "1 (

,

) +

+,

)}

≤ max{max{" ( , , ), " ( + , )}, max{"1 ( , , ), "1 ( + , )}} = max{max{" ( , , ), "1 ( , , )}, max{" ( + , ), "1 ( + , )}} = max{" ∴

×1 [( , , , )

Also "

+ ( +,

+ ),

] ≤ max{"

×1 [( , , , )( + , + ),

×1 (( , , , ),

×1 (( , , , ),

]="

), "

), "

×1 (( + , + ),

×1 (( + , + ),

×1 (( , + , , + ),

)

= max{" (

), "1 (

, +,

and

+.

×1 (( + , + ),

×1 (( + , + ),

,

×

,

)

, +,

1( ,,

),

= min{

+

),

×1 (( , , , ),

( ,,

1( ,

( + , )}, min},

( , ),

×1 (( , , , ),

)

×1 (( + , + ),

(

,

+ ),

)}, min{

×1 (( , , , ),

= min{

+

),

×1 (( , + , , + ),

= min{min{

] ≥ min{

+,

1( ,,

×1 (( , , , ),

≥ min{min{

×1 [( , , , )( + , + ),

+

are in

( + , )}, min},

= min{

∴

+, ,

,

( , ),

= min{ ×1 [( , , , )

(

+)

+

( , , ),

= min{min{

∴

and ( + ,

×1 (( ,

= min{

and

+

respectively, then

+.

-intuitionistic fuzzy subsemirings of the semirings

+.

+ ( +,

,

,

)}

, +,

)}

)}

+

respectively.

78

B. Anitha

≤ max{max{" ( , , ), " ( + , )}, max{"1 ( , , ), "1 ( + , )}} = max{max{" ( , , ), "1 ( , , )}, max{" ( + , ), "1 ( + , )}} = max{" ∴

×1 [( , , , )( + , + ),

Now "

] ≤ max{"

×1 [( , , , ), , . + ]

×1 (( , , , ),

×1 (( , , , ),

= max{" ( , ,

≤ max{max{" ( , ,

= max{" ×1 [( , , , ), , . + ]

× - is a

Hence Theorem 4.4 Let G. Then

≤ max{"

Proof. Let

×1 (( + , + ),

)}

)}

( ,,

+ )}, max{"1 ( , , , ), "1 ( , , + )}}

, ), "1 ( , , , )}, max{"

( + , ), "1 ( + , )}}

×1 (( , , , ), , ), " ×1 (( , , , ), + )}

×1 (( , , , ), , ), " ×1 (( , , , ), + )}

-intuitionistic fuzzy subsemiring of

,

×

+.

-intuitionistic fuzzy subsemiring of a semiring G and H is an isomorphism from a ring

be a

∘ H is a

×1 (( + , + ),

, . + ), "1 ( , , , . + )}

, ), "

= max{max{" ( , ,

∴

), "

), "

-intuitionistic fuzzy subsemiring of and

. and ,

be arbitrary elements in

on to

,, +

∈

and

be a

-intuitionistic fuzzy subsemiring

of G. Then we have (

∘ H)( + , ) =

(H( + ), ) =

(H( ) + H( ), )

≥ min{

(H( ), ),

≥ min{( ⇒(

∘ H)( + , ) ≥ min{( Also (

∘ H)(

∘ H)( , ), (

, )=

(H(

∘ H)( , ), (

), )

≥ min{ ≥ min{( ∘ H)( and (

, ) ≥ min{( ∘ H)( ,

∘ H)( , ), (

,. +)

=

(H( ),

,. +)

≥ min{(

∘ H)( ,

(H( ), ), ∘ H)( , ), (

(H( ), )} ∘ H)( , )}

,. +)

≥ min{( ∘ H)( ,

(H( )H( ), )

∘ H)( , )}

≥ min{

⇒(

∘ H)( , )}

∘ H)( , )}

=

⇒(

(H( ), )}

, ), (

(H( ), ∘ H)( , ,

∘ H)( ,

, ), , ), (

(H( ),

+ )}

∘ H)( ,

+ )}

+ )}

Then we have (" ∘ H)( + , ) = " (H( + ), )

Impact Factor (JCC): 6.9195

NAAS Rating: 3.76

Intuitionistic Fuzzy Subsemiring

79

= " (H( ) + H( ), ) ≤ max{" (H( ), ), " (H( ), )} ≤ max{(" ∘ H)( , ), (" ∘ H)( , )} ⇒ (" ∘ H)( + , ) ≤ max{(" ∘ H)( , ), (" ∘ H)( , )} Also (" ∘ H)(

, ) = " (H(

), ) = " (H( )H( ), ) ≤ max{" (H( ), ), " (H( ), )} ≤ max{(" ∘ H)( , ), (" ∘ H)( , )}

⇒ (" ∘ H)(

, ) ≤ max{(" ∘ H)( , ), (" ∘ H)( , )}

and (" ∘ H)( ,

,. +)

= " (H( ),

,. +)

≤ max{" (H( ), ≤ max{(" ∘ H)( , ⇒ (" ∘ H)( , ∴

,. +)

∘ H is a

Theorem 4.5 Let

≤ max{(" ∘ H)( ,

∘ H)( ,

(H( ),

, ), ("

+ )}

+ )}

.

-intuitionistic fuzzy subsemiring of a semiring G and H is an anti-isomorphism from a semiring

∘ H is a

Proof. Let

+ )}

∘ H)( ,

-intuitionistic fuzzy subsemiring of a semiring be a

onto G. Then

, ), ("

, ), "

and

.

-intuitionistic fuzzy subsemiring of and ,

be arbitrary elements in

,, +

∈

and

of G. Then we have (

∘ H)( + , ) =

(H( + ), ) = ≥ min{ ≥ min{(

⇒(

∘ H)( + , ) ≥ min{( Also (

∘ H)(

∘ H)( , ), (

, )=

(H(

(H( ) + H( ), ) (H( ), ), ∘ H)( , ), (

), )

≥ min{

(H( )H( ), ) (H( ), ),

= min{ ( H( ), ), ≥ min{( ∘ H)( and (

, ) ≥ min{( ∘ H)( ,

∘ H)( , ), (

,. +)

=

∘ H)( , )}

∘ H)( , )}

=

⇒(

(H( ), )}

(H( ),

∘ H)( , ), (

∘ H)( , )} ,. +)

(H( ), )} (H( ), )} ∘ H)( , )}

be a

-intuitionistic fuzzy subsemiring

80

B. Anitha

≥ min{

(H( ),

≥ min{( ⇒(

∘ H)( ,

,. +)

≥ min{(

∘ H)( ,

, ), (

∘ H)( ,

∘ H)( ,

, ),

(H( ),

, ), (

+ )}

∘ H)( ,

+ )}

+ )}

Now (" ∘ H)( + , ) = " (H( + ), ) = " (H( ) + H( ), ) ≤ max{" (H( ), ), " (H( ), )} ≤ max{(" ∘ H)( , ), (" ∘ H)( , )} ⇒ (" ∘ H)( + , ) ≤ max{(" ∘ H)( , ), (" ∘ H)( , )} Also (" ∘ H)(

, ) = " (H(

), ) = " (H( )H( ), ) ≤ max{" (H( ), ), " (H( ), )} = max{" (H( ), ), " (H( ), )} ≤ max{(" ∘ H)( , ), (" ∘ H)( , )}

⇒ (" ∘ H)(

, ) ≤ max{(" ∘ H)( , ), (" ∘ H)( , )}

and (" ∘ H)( ,

,. +)

= " (H( ),

,. +)

≤ max{" (H( ), ≤ max{(" ∘ H)( , ⇒ (" ∘ H)( , ∴

,. +)

∘ H is a

≤ max{(" ∘ H)( ,

, ), ("

∘ H)( ,

, ), "

(H( ),

, ), ("

+ )}

∘ H)( ,

+ )}

+ )}

-intuitionistic fuzzy subsemiring of a ring

.

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2.

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3.

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8.

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Impact Factor (JCC): 6.9195

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81

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