Intuitionistic Fuzzy Ideals

International Journal of Mathematics and Computer Application Research (IJMCAR) ISSN (P): 2250-2378; ISSN (E): 2319-4510 Vol. 9, Issue 2, Dec 2019, 63–72 © TJPRC Pvt. Ltd.

INTUITIONISTIC FUZZY IDEALS B. ANITHA Annamalai University, Annamalainagar, Tamil Nadu, India ABSTRACT In this paper, we introduce the concept of

-intuitionistic fuzzy ideal and

-intuitionistic fuzzy ideal and study their

properties.

KEYWORDS:

-Intuitionistic Fuzzy Set,

-Intuitionistic Fuzzy Subsemiring,

-Intuitionistic Fuzzy Ideal &

-Intuitionistic Fuzzy Ideal.

Received: Jun 13, 2019; Accepted: Jul 03, 2019; Published: Oct 25, 2019; Paper ID.: IJMCARDEC20196

INTRODUCTION Zadeh [13] in 1965 introduced fuzzy sets once that many researchers explored on the generalizations of the notion of

represent a genralization of the notion of fuzzy sets [1, 2]. A Solairaju and R. Nagarajan [8, 9, 10] have introduced and outlined a brand new algebraical structure known as -fuzzy subgroups. S. Hemalatha, et. al. [3] introduced the thought of

-fuzzy subring of a ring and established some results. A study on anti

has been introduced by Vanathi et. al. [11]. Some theorems in

-fuzzy subsemiring of a semiring

-intuitionistic fuzzy subsemiring of a semiring has

been introduced by Vanathi et. al. [12]. O. Ratnabala Devi [7] introduced the thought of intuitionistic of Near-rings. In this paper we introduce the thought of

-intuitionistic fuzzy ideals and

ideals. So far all Q-fuzzy subsets of rings, semirings, near-rings are studied with introduce the thought of

-fuzzy ideals

Original Article

fuzzy sets and its application to many mathematical branches. Atanassov introduced intuitionistic fuzzy set that

-intuitionistic fuzzy

as a set only. In this paper, we

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

2. PRELIMINARIES Definition 2.1 Let (R, +,⋅) be a semiring. Let (Q,⋅) be a semigroup. A map A: R × Q → [0,1] is said to be a Q-fuzzy subset of R. Definition 2.2 Let (R, +, ,⋅) be a semiring. Let (Q,⋅) be a semigroup. An Q -intuitionistic fuzzy set is defined as A = {< (x, q), μ (x, q), ν (x, q) >/x ∈ R, q ∈ Q} where the function μ : R × Q → [0,1] and ν : R × Q → [0,1] denote the degree of membership and degree of non-membership for each element x ∈ R, q ∈ Q to the set A respectively and 0 ≤ μ (x, q) + ν (x, q) ≤ 1, for each x ∈ R, q ∈ Q.

Definition 2.3 Let (R, +,⋅) be a semiring. A Q-fuzzy subset A of R is said to be a Q-fuzzy subsemiring of R if it satisfies the following conditions: [(i)]

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B. Anitha

• • •

%& (' + (, )) ≥ min{%& (', )), %& ((, ))}

%& ('(, )) ≥ min{%& (', )), %& ((, ))}

%& (', ), ) ) ≥ min{%& (', ) ), %& (', ) )}, ∀', ( ∈ / and ), ) , ) ∈ .

Definition 2.4 Let (R, +,⋅) be a semiring. A Q-fuzzy subset A of R is said to be an anti-Q-fuzzy subsemiring of R if it satisfies the following conditions: • • •

%& (' + (, )) ≤ max{%& (', )), %& ((, ))} %& ('(, )) ≤ max{%& (', )), %& ((, ))}

%& (', ), ) ) ≤ max{%& (', ) ), %& (', ) )}

Definition 2.5 Let (R, +,⋅) be a semiring. A Q-fuzzy subset A of R is said to be a Q-intuitionistic fuzzy subsemiring of R if it satisfies the following conditions: • • • • • •

3.

%& (' + (, )) ≥ min{%& (', )), %& ((, ))} %& ('(, )) ≥ min{%& (', )), %& ((, ))}

%& (', ). ) ) ≥ min{%& (', ) ), %& (', ) )}

1& (' + (, )) ≤ max{1& (', )), 1& ((, ))}

1& ('(, )) ≤ max{1& (', )), 1& ((, ))}

1& (', ). ) ) ≤ max{1& (', ) ), 1& (', ) )} for all ', ( ∈ / and ), ) , ) ∈ .

-INTUITIONISTIC FUZZY IDEAL: (

-INTUITIONISTIC FUZZY IDEAL)

Definition 3.1 Let (R, +,⋅) be a ring. A Q -intuitionistic fuzzy subset A of R is said to be Q -intuitionistic fuzzy ideal of R (Q -intuitionistic fuzzy ideal of R) for every q ∈ Q if it satisfies the following conditions.

• • • • • •

%& (2 + 3, )) ≥ min{%& (2, )), %& (3, ))} %& (23, )) ≥ max{%& (2, )), %& (3, ))}

%& (2, ) . ) ) ≥ min{%& (2, ) ), %& (2, ) )}[%& (', ) . ) ) ≥ max{%& (', ) ), %& (', ) )}]

1& (2 + 3, )) ≤ max{1& (2, )), 1& (3, ))} 1& (23, )) ≤ min{1& (2, )), 1& (3, ))}

1& (2, ) . ) ) ≤ max{1& (2, ) ), 1& (2, ) )}[1& (2, ) . ) ) ≤ min{1& (2, ) ), 1& (2, ) )}]

for all 2, 3 ∈ / and ) , ) ∈ .

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Definition 3.2 Q -intuitionistic fuzzy ideal ⇒ Q -intuitionistic fuzzy ideal. But the converse is not true. It is shown by the following

example. Let R be the set of real numbers and Q be the set of rational numbers. Then (R, +,⋅) is a semiring and (Q,⋅) is a semigroup. We define Q-intuitionistic fuzzy set A =< μ , ν > in R as follows: 1 0

if 2 ∈ / and ) ∈ ? otherwise

0 1

if 2 ∈ / and ) ∈ ? otherwise

%& (2, )) = 5 and 1& (2, )) = 5

Clearly, @ =< %& , 1& > is a

-intuitionistic fuzzy ideal, but it is not a

-intuitionistic fuzzy ideal since

%& (1,2√2) ≱ D2'{%& (1,2), %& (1, √2).

Definition 3.3 Let θ be a mapping from semiring X to semiring Y. For any intuitionistic fuzzy set B =< μI , νI > in Y, we define a new

intuitionistic fuzzy set denoted as θJ (B) =< μKLM(I) , νKLM(I) > in X where μKLMI (a, q) = μI (θ(a), q) and νKLM (I) (a, q) = νI (θ(a), q) for all a ∈ X, q ∈ Q. For any intuitionistic fuzzy set A =< μ , ν > in X, we define θ(A) denoted as θ(A) =< μK( ) , νK( %N(&) (3, )) =

)

> in Y by

sup %& (2, )), if T J (3) ≠ V

OR∈NLM(S) 0,

inf 1& (2, )), 1N(&) (3, )) = OR∈NLM (W) 1,

otherwise

if T J (3) ≠ V otherwise

?

?

Theorem 3.4 Let f: R → R′ be surjective ring homomorphism. Then if A is Q -intuitionistic fuzzy ideal in R(Q -intuitionistic fuzzy

ideal in R) then f(A) is Q -intuitionistic fuzzy ideal in R′. (Q -intuitionistic fuzzy ideal in R′.)

Proof

Let A be Q -intuitionistic fuzzy ideal in R. Let a , a ∈ R, b , b ∈ R′ and q, q , q ∈ Q.

Now %Y(&) (3 + 3 , )) = sup{%& (2 − 2 , ))/2 − 2 ∈ [ J (3 − 3 )}

≥ sup{min{%& (2 , )), %& (2 , ))}/2 ∈ [ J (3 ), 2 ∈ [ J (3 )}

= min{sup{%& (2 , ))/2 ∈ [ J (3 )}, sup{%& (2 , ))/2 ∈ [ J (3 )}} = min{%Y(&) (3 , )), %Y(&) (3 , ))}

Also %Y(&) (3 3 , )) = sup{%& (2 2 , ))/2 2 ∈ [ J (3 3 )}

≥ sup{max{%& (2 , )), %& (2 , ))}/2 ∈ [ J (3 ), 2 ∈ [ J (3 )}

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B. Anitha

= max{sup{%& (2 , ))/2 ∈ [ J (3 )}, sup{%& (2 , ))/2 ∈ [ J (3 )}}

= max{%Y(&) (3 , )), %Y(&) (3 , ))}

and %Y(&) (3 , ) . ) ) = sup{%& (2 , ) . ) )/2 ∈ [ J (3 )}

≥ sup{min{%& (2 , ) ), %& (2 , ) )}/2 ∈ [ J (3 )}

= min{sup{%& (2 , ) )/2 ∈ [ J (3 )}, sup{%& (2 , ) )/2 ∈ [ J (3 )}}

= min{%Y(&) (3 , ) ), %Y(&) (3 , ) )}

Now 1Y(&) (3 + 3 , )) = inf{1& (2 − 2 , ))/2 − 2 ∈ [ J (3 − 3 )}

≤ inf{max{1& (2 , )), 1& (2 , ))}/2 ∈ [ J (3 ), 2 ∈ [ J (3 )}

= max {inf{1& (2 , ))/2 ∈ [ J (3 )}, inf{1& (2 , ))/2 ∈ [ J (3 )}} = max{1Y(&) (3 , )), 1Y(&) (3 , ))}

Also 1Y(&) (3 3 , )) = inf{1& (2 2 , ))/2 2 ∈ [ J (3 3 )}

≤ inf{min{1& (2 , )), 1& (2 , ))}/2 ∈ [ J (3 ), 2 ∈ [ J (3 )}

= min{inf{1& (2 , ))/2 ∈ [ J (3 )}, inf{1& (2 , ))/2 ∈ [ J (3 )}}

= min{1Y(&) (3 , )), 1Y(&) (3 , ))}

and 1Y(&) (3 , ) . ) ) = inf{1& (2 , ) . ) )/2 ∈ [ J (3 )}

≤ inf{max{1& (2 , ) ), 1& (2 , ) )}/2 ∈ [ J (3 )}

= max{inf{1& (2 , ) )/2 ∈ [ J (3 )}, inf{1& (2 , ) )/2 ∈ [ J (3 )}}

= max{1Y(&) (3 , ) ), 1Y(&) (3 , ) )} Hence [(@) is

-intuitionistic fuzzy ideal of semiring /.

Theorem 3.5 Let f: R → R′ be surjective ring homomorphism. If B is a Q -intuitionistic fuzzy (Q -intuitionistic fuzzy ideal) in R′ then

f J (B) is Q -intuitionistic fuzzy ideal (Q -intuitionistic fuzzy ideal) in R. Proof Let \ be a

-intuitionistic fuzzy ideal in /′. Let 2 , 2 ∈ / and ), ) , ) ∈ .

Now %YLM(]) (2 + 2 , )) = %] ([(2 + 2 ), )) = %] ([(2 ) + [(2 ), ))

≥ min{%] ([(2 ), )), %] ([(2 ), )) = min{%YLM(]) (2 , )), %YLM(]) (2 , ))}

Also %YLM(]) (2 2 , )) = %] ([(2 2 ), )) = %] ([(2 )[(2 ), ))

≥ min{%] ([(2 ), )), %] ([(2 ), )) = min{%YLM(]) (2 , )), %YLM(]) (2 , ))}

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and %YLM(]) (2 , ) . ) ) = %] ([(2 ), ) . ) ) ≥ min{%] ([(2 ), ) ), %] ([(2 ), ) ) = min{%YLM(]) (2 , ) ), %YLM(]) (2 , ) )}

Now 1YLM(]) (2 + 2 , )) = 1] ([(2 + 2 ), )) = 1] ([(2 ) + [(2 ), ))

≤ max{1] ([(2 ), )), 1] ([(2 ), )) = max{1YLM (]) (2 , )), 1YLM(]) (2 , ))}

1YLM(]) (2 2 , )) = 1] ([(2 2 ), )) = 1] ([(2 )[(2 ), ))

≤ max{1] ([(2 ), )), 1] ([(2 ), )) = max{1YLM(]) (2 , )), 1YLM(]) (2 , ))}

and 1YLM(]) (2 , ) . ) ) = 1] ([(2 ), ) . ) ) ≤ max{1] ([(2 ), ) ), 1] ([(2 ), ) ) = max{1YLM(]) (2 , ) ), 1YLM(]) (2 , ) )}

Hence [ J (\) is a

-intuitionistic fuzzy ideal of /.

Definition 3.6 Let (R, +,⋅) be a semiring, (Q,⋅) be a semigroup and μ: R × Q → [0,1] be a Q-fuzzy set. An upper level set U_ (μ; t) of a

Q-fuzzy set μ is defined as for q ∈ Q, U_ (μ; t) = {a ∈ R/μ(a, q) ≥ t} and a lower level set L_ (μ; t) of a Q-fuzzy set μ is defined as for q ∈ Q, L_ (μ; t) = {a ∈ R/μ(a, q) ≤ t}, for all t ∈ [0,1].

Definition 3.7 The intersection of all upper level sets U_ (μ; t) is denoted by U(μ; t)i. e. , U(μ; t) = ⋂_∈c U_ (μ; t)

The intersection of all lower level sets de (%; f) is denoted by d(%; f)g. h. , d(%; f) = ⋂e∈i de (%; f)

Definition 3.8 Let (R, +,⋅) be a semiring, (Q,⋅) be a semigroup and μ: R × Q → [0,1] be a Q-fuzzy set. An upper level set Uj (μ; t) for a

given a ∈ R of a Q-fuzzy set μ is defined as Uj (μ; t) = {q ∈ Q/μ(a, q) ≥ t} and a lower level set Lj (μ; t) for a given a ∈ R of a Q-fuzzy set μ is defined asLj (μ; t) = {q ∈ Q/μ(a, q) ≤ t}.

Definition 3.9 The intersection of all upper level sets Uj (μ; t) is denoted by U(μ; t)i. e. , U(μ; t) = ⋂j∈k Uj (μ; t)

The intersection of all lower level sets dR (%; f) is denoted by d(%; f)g. h. , d(%; f) = ⋂R∈l dR (%; f) Theorem 3.10 If A is a Q -intuitionistic fuzzy ideal (Q -intuitionistic fuzzy ideal) of a semiring (R, +,⋅), then U_ (μ ; t) and L_ (ν ; t)

are ideals of R for every q ∈ Q. Proof Given @ is a

are ideals of /. www.tjprc.org

-intuitionistic fuzzy ideal of a semiring (/, +,⋅). We have to prove that me (%& ; f) and de (1& ; f)

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B. Anitha

Let ) ∈ , 2, 3 ∈ me (%& ; f) and f ∈ [0,1] then %& (2, )) ≥ f and %& (3, )) ≥ f

Now %& (2 + 3, )) ≥ min{%& (2, )), %& (3, ))} ≥ min{f, f} = f ⇒ 2 + 3 ∈ me (%& ; f) Let ) ∈ , 2 ∈ me (%& ; f) and b∈ /.

%& (ab, )) ≥ max{%& (2, )), %& (3, ))} ≥ min{2, )} ≥ f ⇒ 23 ∈ me (%& ; f)

Thus me (%& ; f) is an ideal of /.

|||opq We can prove that de (1& ; f) is an ideal of /.

Corollary 3.11 If A is Q -intuitionistic fuzzy ideal(Q -intuitionistic fuzzy ideal) of a semiring (R, +,⋅) then U(μ ; t) = ⋂_∈c U_ (μ ; t) and L(ν ; t) = ⋂_∈c L_ (μ ; t) are ideals of R.

Theorem 3.12 If A is a Q -intuitionistic fuzzy ideal (Q -intuitionistic fuzzy ideal) of a semiring R, then Uj (μ ; t) and Lj (ν ; t) are subsemigroup (ideals) of Q for every a ∈ R. Proof Given @ is a subsemigroup of

.

-intuitionistic fuzzy ideal of a semiring /. We have to prove that mR (%& ; f) and dR (1& ; f) are

Let 2 ∈ /, ) , ) ∈ mR (%& ; f) and f ∈ [0,1] then %& (2, ) ) ≥ f and %& (2, ) ) ≥ f.

Now %& (2, ) ) ) ≥ min{%& (2, ) ), %& (2, ) )} ≥ min{f, f} = f ⇒ ) ) ∈ mR (%& ; f)

Let 2 ∈ /, ) , ) ∈ dR (1& ; f) and f ∈ [0,1] then 1& (2, ) ) ≤ f, 1& (2, ) ) ≤ f.

Now 1& (2, ) ) ) ≤ max{1& (2, ) ), 1& (2, ) )} ≤ max{f, f} = f ⇒ ) ) ∈ dR (1& ; f)

Hence mR (%& ; f) and dR (1& ; f) are subsemigroup of

.

Corollary 3.13 If A is a Q -intuitionistic fuzzy ideal of a semiring R (Q -intuitionistic fuzzy ideal) then U(μ ; t) = ⋂j∈k Uj (μ ; t) and

L(ν ; t) = ⋂j∈k Lj (ν; t) are subsemigroup (ideals) of Q. Theorem 3.14

If A is a Q -intuitionistic fuzzy ideal of a semiring R if and only if for given q ∈ Q and a ∈ R, U_ (μ ; t), L_ (ν ; t) are

ideals of R and Uj (μ ; t), Lj (ν ; t) are subsemigroup of Q. Proof

Consider A is a Q -intuitionistic fuzzy ideal of R. Then for q ∈ Q, by Theorem 3.10, U_ (μ ; t) and L_ (ν; t) are

ideals of R. For x ∈ R, by Theorem 3.12, Uj (μ ; t) and Lj (ν ; t) are subsemigroup of Q. Impact Factor (JCC): 6.9195

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Conversely, let U_ (μ ; t) and L_ (ν ; t) be ideals of R and Uj (μ ; t) and Lj (ν ; t) be subsemigroup of Q. We

have to prove A is a Q -intuitionistic fuzzy ideal of R. Let a, b ∈ R, q ∈ Q and t = min{μ (a, q), μ (b, q)} and

t = max{ν (a, q), ν (b, q)}. Then μ (a, q) ≥ t , μ (b, q) ≥ t and a, b ∈ L_ (ν ; t )

and ν (a, q) ≤ t , ν (b, q) ≤ t ⇒ a, b ∈ U_ (μ ; t )

⇒ a + b ∈ U_ (μ ; t ) and a + b ∈ L_ (ν ; t )[∵ U_ (μ ; t ) and L_ (ν ; t ) are ideals of R].

So %& (2 + 3, )) ≥ f = min{%& (2, )), %& (3, ))}

and 1& (2 + 3, )) ≤ f = max{1& (2, )), 1& (3, ))}

Now let fs = max{%& (2, )), %& (3, ))}, say fs = %& (2, ))

Since %& (2, )) + 1& (2, )) ≤ 1, 1& (2, )) ≤ 1 − fs = ft Thus 2 ∈ me (%& ; fs )

Since me (%& ; fs ) is an ideal, for 3 ∈ /, we get 23 ∈ me (%& ; fs )

∴ %& (23, )) ≥ fs = max{%& (2, )), %& (3, ))}

|||opq we can show that 1& (23, )) ≤ min{1& (2, )), 1& (3, ))}

Now let us show that %& (2, ) ) ) ≥ min{%& (2, ) ), %& (a, ) )} for any 2 ∈ /, ) , ) ∈ Let fv = min{%& (2, ) ), %& (2, ) )} then %& (2, ) ) ≥ fv , %& (2, ) ) ≥ fv ⇒ ) , ) ∈ mR (%& ; fv )

⇒ ) ) ∈ mR (%& ; fv )[∵ mR (%& ; fv ) is a subsemigroup of ∴ %& (2, ) . ) ) ≥ fv = min{%& (2, ) ), %& (2, ) )}

]

|||opq we can show that 1& (2, ) . ) ) ≤ max{1& (2, ) ), 1& (2, ) )}.

Hence @ is a

-intuitionistic fuzzy ideal of /.

Theorem 3.15 A is a Q -intuitionistic fuzzy ideal of a semiring (R, +,⋅), if and only if for given q ∈ Q, x ∈ R, U_ (μ ; t) and L_ (ν ; t) are

ideals of R and Uj (μ ; t), Lj (ν ; t) are ideals of Q. Proof Consider @ is a

-intuitionistic fuzzy ideal of /. Then for ) ∈ , by Theorem 3.10, me (%& ; f) and de (1; f) are

ideals of /. For 2 ∈ /, by Theorem 3.12, mR (%& ; f) and dR (1& ; f) are ideals of

.

Conversely, let me (%& ; f) and de (1& ; f) be ideals of / and mR (%& ; f) and dR (1& ; f) be ideals of

prove @ is a

-intuitionistic fuzzy ideal of /.

Let 2, 3 ∈ /, ) ∈

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. We have to

and f = min{%& (2, )), %& (3, ))} and f = max{1& (2, )), 1& (3, ))} editor@tjprc.org

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B. Anitha

Then %& (2, )) ≥ f , %& (3, )) ≥ f and 1& (2, )) ≤ f , 1& (3, )) ≤ f

⇒ 2, 3 ∈ me (%& ; f ) and 2, 3 ∈ de (1& ; f )

⇒ 2 + 3 ∈ me (%& ; f ) and 2 + 3 ∈ de (1& ; f )[∵ me (%& ; f ) and de (1& ; f ) are ideals of /].

So %& (2 + 3, )) ≥ f = min{%& (2, )), %& (3, ))}

and 1& (2 + 3, )) ≤ f = max{1& (2, )), 1& (3, ))}

Now let fs = max{%& (2, )), %& (3, ))}, say fs = %& (2, ))

Since %& (2, )) + 1& (2, )) ≤ 1, 1& (2, )) ≤ 1 − fs = ft Thus 2 ∈ me (%& ; fs )

Since me (%& ; fs ) is an ideal of /, for 3 ∈ /, we get 23 ∈ me (%& ; fs )

∴ %& (23, )) ≥ fs = max{%& (2, )), %& (3, ))}

|||opq we can show that, 1& (23, )) ≤ min{1& (2, )), 1& (3, ))}

Now let us show that %& (2, ) . ) ) ≥ max{%& (2, ) ), %& (a, ) )} for any ' ∈ /, ) , ) ∈ .

Let fv = max{%& (2, ) ), %& (2, ) )}, say fv = %& (2, ) ) Thus ) ∈ me (%& ; fv ). Since me (%& ; fv ) is an ideal of

∴ %& (2, ) . ) ) ≥ fv = max{%& (2, ) ), %& (2, ) )}

, for ) ∈ , we get ) . ) ∈ me (%& ; fv )

|||opq we can show that 1& (2, ) . ) ) ≤ min{1& (2, ) ), 1& (2, ) )}.

4. CONCLUSIONS In this paper Q -intuitionistic fuzzy ideal and Q -intuitionistic fuzzy ideal were introduced and their basic algebraic

properties were studied. The upper level sets and lower level sets of a Q- fuzzy setswere defined and their basic structures

were explored. A similar approach can be explained for manyrings.

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