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1, 3Asst Prof. Department of Mathematics, Sri Vidya Mandir Arts & Science College, Uthangarai, T.N. India
2 Associate Professor of Mathematics Jamal Mohamed College, Trichy, T.N, India.
1e-mail:ragavanshana@gmail.com, 2solairama@yahoo.co.in, 3anjjukavi@gmail.com
In this paper, we introduce the notion of intuitionistic fuzzy ideal and intuitionistic fuzzy R-ideal in semi
ring and investigate some properties of intuitionistic fuzzy R-ideals of semiring.
The concept of fuzzy set μ of a set X was introduced by L. A. Zadch [9] as a function from X in [0, 1]. The concept of fuzzy ideals in a ring was introduced by W. L. Liu [8]. T. K. Dutta and B. K. Biwa’s [3, 4] studied
fuzzy ideals, fuzzy prime ideals of semi rings and they defined fuzzy R-ideals and fuzzy prime R-ideals of semi rings. Y. B. Jun, J. Naggers and M. S. Kim [5] extended the concept of an L-fuzzy left (resp. right) ideals of a ring to a semi ring. The concept of the idea of intuitionistic fuzzy set was first published by K. T. Atanassov [1, 2], as a generalization of the notion of fuzzy set. K.H. Kim and J. G. Lee [6] studied the intuitionistic Fuzzifications of the concept of several ideals in a semi groups and investigate some properties of such ideals. K. H. Kim [7] introduced the notion of intuitionistic Q-fuzzy semiprimality in a semi group and investigates some properties of intuitionistic Q-Fuzzifications of the concept of several ideals. In this paper we introduce the notion of intuitionistic fuzzy R-ideal of semi ring and investigate some properties of intuitionistic fuzzy R-ideal of semi rings. Throughout this paper R is a semiring.
Let (R, +, ·) be a smearing. By a left (right) ideal of R we mean a non-empty subset A of R such that A + A ⊆ A and RA ⊆ A (AR ⊆ A). By ideal, we mean a non-empty subset of R which both left and right ideal of R. A left ideal A of R is said to be a left R-ideal if t ∈ A, x ∈ R and if t + x ∈ A or x + t ∈ A then x ∈ A. Right R-ideal
is defined dually, and two sided R-ideal or simply a R-ideal is both a left and a right R-ideal. By a fuzzy set μ of a
non-empty set R we mean a function μ: R→ [0, 1], and the complement of μ , denoted by 𝜇 , is the fuzzy set in R
given by 𝜇 (x)=1- μ (x), for all x∈R. A fuzzy set μ in R is called fuzzy left (resp. right) ideal of R if for any x, y
∈ R, μ(x+y)≥ min{μ(x),μ(y)} and μ (xy) ≥ μ (y) ( μ (xy) ≥ μ (x)) and μ is called fuzzy ideal if μ both fuzzy left
and right ideal of R. A fuzzy ideal μ of R is called R-fuzzy ideal of R if for any x,y∈R,
μ(x)≥ min{max{μ((x+z)+(z+y)),μ((z+x)+(y+z))}, μ (y)}. A intuitionistic fuzzy set (IFS for short) A in a non-
empty set R is an object have the form: A = {(x:µA (x), λA (x)) / x ∈ R} Where the function μA : R→[0, 1] and λA
: R→[0, 1] denoted the degree of membership and the degree of non-membership, respectively, and 0 ≤ μA (x) +
λA (x) ≤1 An intuitionistic fuzzy set A = {(x : μA (x), λA (x)) / x ∈ R} in R can be identified to ordered pair
(μA,λA) in IR x IR. We shall use the symbol A = (µA, λA) for the IFS: A = {(x: μA (x), λA (x)) / x∈R}
Definition 3.1: An IFS A = (μA, λA) in R is called an intuitionistic fuzzy left (resp. right) ideal of R
1- μA (x+y) ≥ min{μA (x), μA (y)} and μA (xy) ≥ μA (y) (resp. μA (xy) ≥ μA (x)).
2- λA (x+y) ≤ max{λA (x), λA (y)} and λA (xy) ≤ λA (y) (resp. λA (xy) ≥ λA (x)). For all x, y ∈R:
Definition 3.2: An intuitionistic fuzzy ideal A= (μA, λA) of R is called an intuitionistic fuzzy R- ideal of R if
1. μA (xy) ≥ μA (y) , μA (xy) ≥ μA (x). 2 μA (x) ≥min {max {μA ((x+z) + (z+y)), μA ((z+x) +(y+z))}, μA(y)} and
3. λA (xy) ≤ λA (y), λA (xy) ≥ λA (x)). 4 λA (x) ≤max{min{λA((x+z)+(z+y)),λA((z+x)+(y+z))}, λA (y)}.∀ x,
y∈R.
Theorem 3.3: Let A = (μA, λA) an intuitionistic fuzzy set in R such that μA is fuzzy R-ideal of R then
d A = (μA, 𝜇R A ) is an intuitionistic R-ideal of R.
Proof: Let x, y∈R, since μA is a fuzzy R-ideal of R ⇒ μA is a fuzzy ideal.
So μA (x+y) ≥ min {μA (x), μA (y)} and μA (xy) ≥ μA (y), μA (xy) ≥ μA (x),
μ A (x+y) =1-μA (x+y) ≤1-min {μA (x), μA (y)} =max {1-μA (x), 1-μA (y)} = max { μ A (x), μ A (y)}
μ A (xy) = 1 – μA (xy) ≤ 1 - μA (y) = μ A (y), also μA (xy) ≤ μA (x)
Therefore dA = (μA, μ A ) is an intuitionistic fuzzy ideal of R.
Let μA (x) ≥min {max {μA ((x+z) + (z+y)), μA ((z+x) +(y+z))}, μA (y)}
𝜇 (x)=1 - μ (x) ≥min {max {μA ((x+z) + (z+y)), μA ((z+x) +(y+z))}, μA (y)},
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μ A (x) =max {1- max {μA ((x+z) + (z+y)), μA ((z+x) +(y+z))}, μA (y)}, 1-μA (y)}
= max {min {1- μA((x+z)+(z+y)),1-μA((z+x)+(y+z))},1- μA (y)}
=max {min { μ A ((x+z) + (z+y)), μ A ((z+x)+(y+z))}, μ A (y)}
so μ A (x)≤max {min { μ A ((x+z) + (z+y)), μ A ((z+x)+(y+z))}, μ A (y)}∴ dA = (μA, μ A ) is an intuitionistic R-ideal
Theorem 3.4: An IFS A = (μA, λA) is an intuitionistic fuzzy R-ideal of R if and only if the fuzzy sets μA and
Proof: Suppose that an IFS A = (μA, λA) is an intuitionistic fuzzy R-ideal of R. Clearly μA is a fuzzy R-ideal. Let x, y ∈ R. Since A = (μA, λA) is an is an intuitionistic fuzzy R-ideal.
⇒ λA(x + y) ≤ max {λA (x), λA (y)} and λA (xy) ≤ λA (y), λA (xy) ≤ λA(x)
λA (x) ≤ max {min {λA ((x+z) + (z+y)), λA ((z+x) +(y+z))}, λA (y)}
λ� RA (x + y) = 1 - λA (x + y) ≥ 1 – max {λA (x), λA (y)} = min {λ� RA (x), λ� RA (y)} So λ� RA (x + y) ≥ min {λ� RA (x), λ� RA (y)}
λ� RA (xy) = 1- λA (xy) ≥ 1 - λA (y) = λ� RA (y),
λ� RA (xy) ≥ λA (x) ≥ 1 – max λ� RA (y). Also we can get that λ� RA (xy) ≥ λ� RA (x).
λ� RA (x) = 1 − λA (x)
≥ 1 – max {min {λA ((x+z) + (z+y)), λA ((z+x) +(y+z))},λA (y)}
= max {1 – min {λA ((x+z) + (z+y)), λA ((z+x) +(y+z))}, 1 - λA (y)}
= min {max {1-λA ((x+z) + (z+y)), 1- λA ((z+x) +(y+z))}, λ� RA (y)}
= min {max {λ� RA ((x+z) + (z+y)), λ� RA ((z+x) +(y+z))},λ� RA (y)}
So λ� RA (x) ≥ min {max {λ� RA ((x+z) + (z+y)), λ� RA ((z+x) +(y+z))}, λ� RA (y)}
Therefore λ� RA fuzzy R-ideals of R. suppose that μA and λ A are fuzzy R-ideals of R
Let x, y ∈ R. Since μA is a fuzzy R-ideals of R.
μA (x +y) ≥ min {μA (x), μA(y)}
μA (x y) ≥ min {μA (x), μA(y)} and μA(x y) ≥ μA (y), μA (xy) ≥ μA (x)
μA (x) ≥ min {max {μA ((x+z)+(z+y)), μA ((z+x)+(y+z))},μA(y)}
λA(x + y) = 1 – λ� RA (x + y). Since λ� RA is a fuzzy ideal of R we get that
λA(x + y) = 1 – min { λ A (x), λ� RA (y) } = max {1-λ� RA (x), 1-λ� RA (y)} = max {λRA (x), 1-λRA (y)}
So λRA (x + y) ≤ max {λRA (x), λRA (y)} Also we get that λA (xy) ≤ λA(y), λA (xy) ≤ λA(x)
λA (x) =1− λ� RA(x) Since λA is a fuzzy R-ideals of R we get that
λA(x) ≤ 1 – min {max { λ� RA ((x+z) + (z+y)), λ� RA ((z+x) + (y+z))}, λ� RA (y)}
= max {1 - max { λ� RA ((x+z) + (z+y)), λ� RA ((z+x) + (y+z))}, 1-λ� RA (y)}
= max {min { 𝟏 − λ� RA ((x+z) + (z+y)), 1- λ� RA ((z+x) +(y+z))}, λRA (y)}
= max {min {λRA ((x+z) + (z+y)), λRA ((z+x)+(y+z))}, λRA (y)}
So λRA (x) ≤ max {min {λRA ((x+z) + (z+y)), λRA ((z+x) +(y+z))}, λRA (y)}
Therefore= (μA, λA) is an is an intuitionistic fuzzy R-ideal
Theorem 3.7: An IFS A = (μA, λA) is an intuitionistic fuzzy R-ideal of R iff for any t ∈ [a, b] such that (μA)t ≠ Φ
and (λA)t ≠ Φ. (μA)t and (λA)t are R-ideal of R, where (μA)t ={x ∈ R / μA(x) ≥ t}.
Proof: Suppose that IFS A = (μA, λA) is an intuitionistic fuzzy R-ideal of R So by theorem (2.3) μA and λ� RA are
fuzzy R-ideal of R ⇒ μA and λA are fuzzy ideal of R By [4] for any t ∈ [0, 1] such that (μA)t ≠ Φ and (λA)t ≠ Φ
(μA)t and (λA)t are ideal of R. Let x ∈ (μA)t and y ∈ R and ((x +z)+(z+ y))∈ (μA)t or ((z + x)+(y+z))∈ (μA)t .
⇒ μA(x) ≥ t and μA((x +z)+(z+ y))≥ t or μA ((z + x)+(y+z))≥t . ⇒ max {μA ((x +z)+(z+ y)), μA((z + x)+(y+z))}
≥ t. Since μA is fuzzy R-ideal of RμA (y) ≥ mini {max {μA ((x +z)+(z+ y)), μA((z + x)+(y+z))}, μA(x)} ≥ t⇒μA(y)
≥ t so y ∈ (μA )t. Therefore (μA)t is R-ideal of R.Similarly we can prove that ( λA)t is R-ideal of R.Suppose that
for any t∈[0, 1] such (μA)t ≠ Φ and (λA)t ≠ Φ, (μA)t and (λA)t are R-ideal of R.So (μA )t and (λA)t are ideal of
R.By [4] μA and λA are fuzzy ideal of R.Let x, y ∈ R and μA(y) = r 1 , μA((x +z)+(z+ y))=r 2 , μA((z +
x)+(y+z))=r 3 , (ri ∈ [0, 1])Let t= min{max{r 1 , r 3 }, r 1 }⇒ y ∈ (μA)t and ((x +z)+(z+ y)) ∈ (μA)t or ((z +
x)+(y+z))∈ (μA)t Since (μ A )t is R-ideal of R.So x ∈ (μ A )t⇒ μ A (x)≥ t, μ A (x)≥ min{max{μ A ((x +z)+(z+ y)),μ A ((z +
x)+(y+z))},μA(y)}Therefore μA is fuzzy R-ideal of R.Similarly we can prove that λA is fuzzy R-ideal of R.By
theorem (2.5) we get that A=(μA, λA) is an intuitionistic fuzzy R-ideal of R.Recall a function f from a semi-ring R
into semi-ring T homomorphism if f (x + y) = f (x) + f (y) and f (xy) = f(x) f(y) for any x,y∈ R Let f be a function
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from a set X into a set Y respectively, then the image of A under f, denoted by f(A) and the preimage of B under f, denoted by f-1(B), are IFSs in X and Y respectively and defined by f (A) = (f (μA), f(λA)), f-1(B)=(f-1 (μB ), f-1 (λB ))
Theorem 3.9: Let f: R→T be onto homomorphism of semi rings. If B= (μB , λB ) is an intuitionistic fuzzy R-ideal
of T then the preimage f-1(B) = (f-1(μB ), f-1(λB )) of B under f is an intuitionistic fuzzy R-ideal of R.
Proof:1- let x, y ∈ R. f-1(μB ) (x + y) = μB (f (x + y) = μB (f (x) + f(y))
≥ min {μB (f(x)), μB (f(y))} =min {f-1(μB ) (x), f-1(μB ) (y)}.
So f-1(μB ) (x + y) ≥ min {f-1 (μB ) (x), f-1 (μB ) (y)}
f-1 (μB ) (xy) = μB (f (xy)) = μB (f(x) f(y))
≥ μB (f(y)) = f-1 (μB ) (y).
So f-1 (μB ) (xy) ≥ f-1 (μB ) (y).Also f-1 (μB ) (xy) ≥ f-1 (μB ) (x),
f-1 (λB ) (x+y) = λB (f(x+y)) = λB (f(x) +f(y))
≤ max {λB (f(x)), λB (f(y))}
= max { f-1 (λB ) (x), f-1 (λB )(y)} So f-1(λB) (x+y) ≤ max { f-1 ( λB )(x), f-1 (λB )(y)}.
f-1 ( λB ) (xy) = λB (f(xy)) = λB (f(x) f(y))
≤ λB (f(y)) = f-1(λB ) (y).
So f-1(λB ) (xy) ≥ f-1 (λB ) (y).Also f-1 (λB ) (xy) ≥ f-1 ( λB ) (x)
2- Let x, y ∈ R⇒ f(x), f(y) ∈ T.
f-1 (μB )(x)≥ μB (f(x)) ≥ min {max {μB (f (x+z) +f (z+y)), μB (f (z+x) + f (y+z))}, μB (f(y))}
= min{max{ f-1 (μB )( (x+z)+(z+y)), f-1 (μB )((z+x)+(y+z))}, f-1 (μB )(y)}
f-1 ( λB )(x)=λB (f(x)) ≤ max {min {λB (f(x+z) +f (z+y)), λB (f (z+x) +f(y+z))}, λB (f(y))}
= max{min{ f-1 ( λB )((x+z)+(z+y)), f-1 (λB )((z+x)+(y+z))}, f-1 (λB )(y)} Therefore f-1 (B) = (f-1 (μB ), f-1 (λB )) is an intuitionistic fuzzy R-ideal of R.
Theorem 5. If μ is a intuitionistic fuzzy R- ideal in semi ring R then µ m
semi ring R.
Proof: For all x, y ∈ R.
is also intuitionistic fuzzy R- ideal in
{μ ( x )}m ≥ {min {max {μ
(( x + z ) + ( z + y )) , μ
(( z + x ) + ( y + z ))}, μ
( y )}}
A A A A
μ ( x )m ≥ min {max {μ
(( x + z ) + ( z + y )) , μ
(( z + x ) + ( y + z ))}, μ
( y )}m
A A A A
μ m ( x) ≥ min {max {μ
(( x + z ) + ( z + y )) , μ
(( z + x ) + ( y + z ))} , μ
( y )m }
A A A A
≥ min {max {μ
(( x + z ) + ( z + y ))m , μ
(( z + x ) + ( y + z ))m }, μ
( y )m }
A A A
μ m ( x ) ≥ min {max {μ m (( x + z ) + ( z + y )), μ m (( z + x ) + ( y + z ))}, μ m ( y )}
A A A A
{λA
( x )}m
≤ {max {min {λA (( x + z ) + ( z + y )), λA (( z + x ) + ( y + z ))}, λA ( y )}}
λ ( x )m ≤ max {min {λ
(( x + z ) + ( z + y )), λ
(( z + x ) + ( y + z ))}, λ
( y )}m
A A A A
λ m ( x ) ≤ max {min {λ
(( x + z ) + ( z + y )), λ
(( z + x ) + ( y + z ))} , λ
( y )m }
A A A A
≤ max {min {λ
(( x + z ) + ( z + y ))m , λ
(( z + x ) + ( y + z ))m }, λ
( y )m }
A A A
λ m ( x ) ≤ max {min {λ m (( x + z ) + ( z + y )), λ m (( z + x ) + ( y + z ))}, λ m ( y )}
Therefore µ m is a fuzzy ideal in semi ring R
Theorem 7: If μA is a fuzzy ideal in semi ring R then µA∩B is also fuzzy ideal in semi ring R. Proof: For all x, y ∈ R. μA (x+y) ≥ min {μA (x), μA (y)} and μA (xy) ≥ μA (y), μA (xy) ≥ μA (x)
μB (x+y) ≥ min{μB (x), μB (y)} and μB (xy) ≥ μB (y), μB (xy) ≥ μB (x)
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min {µ A ( x + y), µB ( x + y)} ≥ min {min {µ A ( x) + µ A ( y )} , min {µB ( x) + µB ( y )}}
min {µ A ( xy), µB ( xy)} ≥ min {µ A (y), µB ( y)}, min {µ A ( xy), µB ( xy)} ≥ min {µ A (x), µB (x)}
µ A∩ B (x + y) ≥ min {min {µ A ( x), µB ( x)} ,{min{µB (x), µB ( y)}} = min {µ A∩ B (x), µ A∩ B (y)}
µ A∩ B (xy) ≥ µ A∩ B (y), µ A∩ B (xy) ≥ µ A∩ B (x).
Theorem 8.If μA is a fuzzy R- ideal in semi ring R then µA∩B is also fuzzy R- ideal in semi ring R.
Proof: For all x, y ∈ R. ∀x, y∈R. Now,
µ A (x) ≥ min {max {µ A (( x + z ) + (z+ y)), µ A ((z+ x) + (y+ z ))}, µ A (y)}
µ B (x) ≥ min {max {µ B (( x + z ) + (z+ y)), µ B ((z+ x) + (y+ z ))}, µ B (y)}
min{µ A (x), µ B (x)} ≥ min {min {max {µ A (( x + z) + (z+ y)), µ A ((z+ x) + (y+ z))}, µ A (y)},
min {max {µ B (( x + z) + (z+ y)), µ B ((z+ x) + (y+ z))}, µ B (y)}}
If one is contained in another.
Ragavan.C1, Solairaju.A2 and Kaviyarasu.M3
µ A∩ B (x) ≥ min {max{min{µ A (( x + z ) + (z + y)), µ B (( x + z ) + (z + y))},
{µ A ((z + x) + (y+ z )), µ B ((z + x) + (y+ z ))}}, max {min {µ A (y), µ B (y)}}}
µ A∩ B (x) ≥ min {max {µ A∩ B (( x + z ) + (z + y)), µ A∩ B ((z + x) + (y+ z ))}, µ A∩ B (y)}
Theorem 9: If μA is a fuzzy ideal in semi ring R then µA∪B is also fuzzy ideal in semi ring R.
Proof: For all x, y ∈ R. μA (x+y) ≥ min {μA (x), μA (y)} and μA (xy) ≥ μA (y), μA (xy) ≥ μA (x)
μB (x+y) ≥ min{μB (x), μB (y)} and μB (xy) ≥ μB (y), μB (xy) ≥ μB (x)
max {µ A ( x + y), µB ( x + y)} ≥ max {min {µ A ( x) + µ A ( y)} , min {µB ( x) + µB ( y)}}
If one is contained in another.
max {µ A ( xy), µB ( xy)} ≥ max {µ A (y), µB ( y)} , max {µ A ( xy), µB ( xy)} ≥ max {µ A (x), µB (x)}
µ A∪ B (x + y) ≥ min {max {µ A ( x), µB ( x)} ,{max{µB (x), µB ( y)}} = min {µ A∪ B (x), µ A∪ B (y)}
µ A∪ B (xy) ≥ µ A∪ B (y), µ A∪ B (xy) ≥ µ A∪ B (x).
Theorem10.If μA is a fuzzy R- ideal in semi ring R then µA∪B is also fuzzy R- ideal in semi ring R.
Proof: For all x, y ∈ R. ∀x, y∈R. Now,
µ A (x) ≥ min {max {µ A (( x + z ) + (z+ y)), µ A ((z+ x) + (y+ z ))}, µ A (y)}
µ B (x) ≥ min {max {µ B (( x + z ) + (z+ y)), µ B ((z+ x) + (y+ z ))}, µ B (y)}
max{µ A (x), µ B (x)} ≥ max {min {max {µ A (( x + z ) + (z+ y)), µ A ((z+ x) + (y+ z ))}, µ A (y)},
min {max {µ B (( x + z ) + (z+ y)), µ B ((z+ x) + (y+ z ))}, µ B (y)}}
If one is contained in another .
µ A∪ B (x) ≥ min{max{max{µ A (( x + z ) + (z+ y)), µ B (( x + z ) + (z+ y)),
{µ A ((z+ x) + (y+ z )), µ B ((z+ x) + (y+ z ))}}, max {max {µ A (y), µ B (y)}}
µ A∪ B (x) ≥ min {max {µ A∪ B (( x + z ) + (z+ y)), {µ A∪ B ((z+ x) + (y+ z ))}}, µ A∪ B (y)}
Theorem 11.If μA is an intuitionistic fuzzy ideal of semi ring R then µA∩B is an intuitionistic fuzzy ideal of semi
ring R of one is contains another.
Proof: If μA is a fuzzy ideal in semi ring R then µA∩B is also fuzzy ideal in semi ring R.
Proof: For all x, y ∈ R. μA (x+y) ≥ min {μA (x), μA (y)} and μA (xy) ≥ μA (y), μA (xy) ≥ μA (x)
μB (x+y) ≥ min{μB (x), μB (y)} and μB (xy) ≥ μB (y), μB (xy) ≥ μB (x)
min {µA ( x + y), µB ( x + y)} ≥ min {min {µA ( x), µA ( y)}, min {µB ( x) , µB ( y)}}
µA∩ B (x + y) ≥ min {min {µA ( x), µB ( x)},{min{µB (x), µB ( y)}}
µA∩ B (x + y) ≥ min {µA∩ B (x), µA∩ B (y)}
min {µA ( xy), µB ( xy)} ≥ min {µA (y), µB ( y)}, and min {µA ( xy), µB ( xy)} ≥ min {µA (x), µB (x)}
µA∩ B (xy) ≥ µA∩ B (y), µA∩ B (xy) ≥ µA∩ B (x).
Now ∀x, y∈R, now
λA (x+y) ≥ min {λA (x), λA(y)},λA (xy) ≥ λA (y), λA (xy) ≥ λA (x)
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λB (x+y) ≥ min {λB (x), λB (y)} and λB (xy) ≥ λB (y), λB (xy) ≥ λB (x)
min {λA ( x + y), λB ( x + y)} ≥ min {max {λA ( x), λA ( y)}, max {λB ( x) + λB ( y)}}
λA∩ B (x + y) ≥ max {min {λA ( x), λB ( x)},{min{λB (x), λB ( y)}}
λA∩ B (x + y) ≥ max {min {λA ( x), λB ( x)},{min{λA (y), λB ( y)}}
λA∩ B (x + y) ≥ max {λA∩ B (x), λA∩ B (y)}
Theorem 12if μA is an intuitionistic fuzzy- R ideal of semi ring R then µA∩B is an intuitionistic fuzzy –R ideal
of
semi ring R if one is contains another. proof: for all x, y,∈ R
µ A (x) ≥ min{max{µ A (( x + z ) + (z + y)), µ A ((z + x) + (y+ z ))}, µ A (y)}
µ B (x) ≥ min{max{µ B (( x + z ) + (z + y)), µ B ((z + x) + (y+ z ))}, µ B (y)} and
µ A (xy) ≥ µ A (y), µB (xy) ≥ µB (y)
min{µ A (x), µ B (x)} ≥ min{min{max{µ A (( x + z) + (z+ y)), µ A ((z+ x) + (y+ z))}, µ A (y)},
min{max{µ B (( x + z) + (z+ y)), µ B ((z+ x) + (y+ z))}, µ B (y)}}and,
min{µ A (xy), µ A (xy) } ≥ min{µ A (y), µB (y)}
µ A∩ B (x) ≥ min{max{min{µ A (( x + z ) + (z+ y)), µ B (( x + z ) + (z+ y)), µ A ((z+ x) + (y+ z )),
µ B ((z+ x) + (y+ z ))}}, min{µ A (y), µ B (y)}} and µ A∩ B (xy) ≥ µ A∩ B (y)
µ A∩ B (x) ≥ min{max{µ A∩ B (( x + z ) + (z+ y)), µ A∩ B (( x + z ) + (z+ y))}, µ A∩ B (y)},
and µ A∩ B (xy) ≥ µ A∩ B (y)
λ A (x) ≤ max{min{λ A (( x + z ) + (z+ y)), λ A ((z+ x) + (y+ z))}, λ A (y)}
λ B (x) ≤ max{min{λ B (( x + z ) + (z+ y)), λ B ((z+ x) + (y+ z))}, λB (y)} and
λ A (xy) ≤ λ A (y), λB (xy) ≤ λB (y)
max{λ A (x), λ B (x)} ≤ max{max{min{λ A (( x + z) + (z+ y)), λ A ((z+ x) + (y+ z))}, λ A (y)},
max{min{λ B (( x + z) + (z+ y)), λ B ((z+ x) + (y+ z))}, λ B (y)}} and,
max{λ A (xy), λ B (xy) } ≤ max{ λ A (y), λ B (y)}
λA∩ B (x) ≤ max{min{max{λ A (( x + z) + (z+ y)), λ B (( x + z) + (z+ y)), λ A ((z+ x) + (y+ z)),
λ B ((z+ x) + (y+ z))}}, max{λ A (y), λ B (y)}} and λ A∩ B (xy) ≤ λ A∩ B (y)
λ A∩ B (x) ≤ max{min{λ A∩ B (( x + z ) + (z+ y)), λ A∩ B (( x + z ) + (z+ y))}, λ A∩ B (y)},
and λ A∩ B (xy) ≤ λ A∩ B (y)
13 if μA is an intuitionistic fuzzy- R ideal of semi ring R then µA∪B is an intuitionistic fuzzy –R ideal of semi ring
R if one is contains another.
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ISSN 2229-5518
µ A (x) ≥ min{max{µ A (( x + z ) + (z+ y)), µ A ((z+ x) + (y+ z ))}, µ A (y)}
µ B (x) ≥ min{max{µ B (( x + z ) + (z+ y)), µ B ((z+ x) + (y+ z ))}, µ B (y)} and
µ A (xy) ≥ µ A (y), µB (xy) ≥ µB (y)
max{µ A (x), µ B (x)} ≥ max{min{max{µ A (( x + z ) + (z+ y)), µ A ((z+ x) + (y+ z ))}, µ A (y)},
min{max{µ B (( x + z ) + (z+ y)), µ B ((z+ x) + (y+ z ))}, µ B (y)}} and,
max{µ A (xy), µ A (xy) } ≥ max{µ A (y), µB (y)}
µ A∪ B (x) ≥ max{min{max{µ A (( x + z ) + (z+ y)), µ B (( x + z ) + (z+ y)), µ A ((z+ x) + (y+ z )),
µ B ((z+ x) + (y+ z ))}}, max{µ A (y), µ B (y)}} and µ A∪ B (xy) ≥ µ A∪ B (y)
µ A∪ B (x) ≥ min{max{µ A∪ B (( x + z ) + (z+ y)), µ A∪ B (( x + z ) + (z+ y))}, µ A∪ B (y)},
and µ A∪ B (xy) ≥ µ A∪ B (y)
λ A (x) ≤ max{min{λ A (( x + z ) + (z+ y)), λ A ((z + x) + (y+ z ))}, λ A (y)}
λ B (x) ≤ max{min{λ B (( x + z ) + (z+ y)), λ B ((z + x) + (y+ z ))}, λB (y)} and
λ A (xy) ≤ λ A (y), λB (xy) ≤ λB (y)
max{λ A (x), λ B (x)} ≤ max{max{min{λ A (( x + z ) + (z+ y)), λ A ((z + x) + (y+ z ))}, λ A (y)},
max{min{λ B (( x + z ) + (z + y)), λ B ((z + x) + (y+ z ))}, λ B (y)}} and,
max{λ A (xy), λ B (xy) } ≤ max{ λ A (y), λ B (y)}
λA∩ B (x) ≤ max{min{max{λ A (( x + z ) + (z + y)), λ B (( x + z ) + (z + y)), λ A ((z + x) + (y+ z )),
λ B ((z + x) + (y+ z ))}}, max{λ A (y), λ B (y)}} and λ A∩ B (xy) ≤ λ A∩ B (y)
λ A∩ B (x) ≤ max{min{λ A∩ B (( x + z ) + (z+ y)), λ A∩ B (( x + z ) + (z+ y))}, λ A∩ B (y)},
and λ A∩ B (xy) ≤ λ A∩ B (y)
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