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Fuzzy Join Prime Semi L-Ideals B. Chellappa Associate Professor, Department of Mathematics, Alagappa Govt. Arts College, Karaikudi-3, chellappab_1958@yahoo.com B. Anandh Head, Department of Mathematics, Sudharsan College of Arts and Science, Perumanadu, Pudukkottai-1 maharishibalaanandh@gmail.com Abstract The concept of fuzzy join prime semi L-ideals in fuzzy join subsemilattices is introduced. The properties of fuzzy join prime semi

Keywords:

L-ideals are discussed.

Fuzzy join prime semi L-ideals, f-invariant fuzzy join prime semi L-ideals, fuzzy join prime

semi L-ideal homomorphism.

Introduction Zhang Yue[2] defined Prime L-Fuzzy Ideals in Fuzzy Sets. Rajesh Kumar[4] derived the concepts of Fuzzy Semiprime Ideals in Ring. Swami & Swamy[3] generalized the concepts of Fuzzy Prime Ideals of Rings. Definition: 1 A fuzzy join semi L-ideal S(µ) of a fuzzy join semilattice A is said to be a fuzzy join prime semi L-ideal of A if (i) S(µ) is not a constant function and (ii) For any two fuzzy join semi L-ideals S(σ) and S(θ) in A if S(σ) ∨ S(θ) ⊆ S(µ), then either S(σ) ⊆ S(µ) or S(θ) ⊆ S(µ). Example: 1 Let A = { 0, a, b, 1 } be a fuzzy join semilattice. Consider S(µ) is a fuzzy join prime semi L-ideal of A. Then S[ µ(0) ] = 0.6, S[ µ(a) ] = 0.5, S[ µ(b) ] = 0.4, S[ µ(1) ] = 0.7 Let S(σ) and S(θ) be any fuzzy join prime semi L-ideals of A. Then S[ σ(0) ] = 0.4, S[ σ(a) ] = 0.2, S[ σ(b) ] = 0.3, S[ σ(1) ] = 0.8 and S[ θ(0) ] = 0.5, S[ θ(a) ] = 0.3, S[ θ(b) ] = 0.4, S[ θ(1) ] = 0.7 Here S(σ) ∨ S(θ) ⊆ S(µ), S(σ) ⊄ S(µ) or S(θ) ⊄ S(µ).

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Hence S(µ) is a fuzzy join prime semi L-ideal of A. Note: 1 S(σ) ⊆ S(µ) means S[ σ(x) ] ≤ S[ µ(x) ] for all x ∈ A. Definition: 2 A fuzzy join prime semi L-ideal S(µ) of a fuzzy join semilattice A is called fuzzy level join semi L-prime if the fuzzy level join semi

L-ideal S( µt ), where t = S[ µ(0) ] is a prime semi L-ideal of A.

Proposition: 1 Let S(µ) be any fuzzy join prime semi L-ideal of a fuzzy join semilattice A such that each fuzzy level join prime semi L-ideal S( µt ), t ∈ Im S(µ) is prime. If

S[ µ(x) ] < S[ µ(y) ] for some x, y ∈ A then

S[ µ( x∨y ) ] = S[ µ(y) ]. Proof: Let S[ µ(x) ] = t; S[ µ(y) ] = t′; and S[ µ(x∨y) ] = s Given S[ µ(x) ] < S[ µ(y) ] (i.e) t < t′ Now, s = S[ µ( x∨y ) ] ≥ max { S[ µ(x) ], S[ µ(y) ] } = max { t, t′} = t′ Therefore t < t′ ≤ s Suppose that t′ < s. If x∨y ∈ S( µs ) then either x ∈ S( µs )

or y ∈ S( µs ), Since S( µs ) is a fuzzy level join prime semi

L-ideal of A. Now, x ∈ S( µs ) ⇒ S[ µ(x) ] = s or y ∈ S( µs ) ⇒ S[ µ(y) ] = s Hence, t = S[ µ(x) ] ≥ s or t′ = S[ µ(y) ] ≥ s, which is not possible. Therefore, t′ = s (i.e) S[ µ( x∨y ) ] = S[ µ(y) ] Corollary: 1 If S(µ) is any fuzzy join prime semi L-ideal of a fuzzy join semilattice A then S[ µ(x∨y) ] = max { S[ µ(x) ], S[ µ(y) ] }, for all x, y ∈ A. Proof: Let x, y ∈ A ⇒ x∨y ∈ A S(µ) is a fuzzy join prime semi L-ideal. ⇒S[ µ( x∨y ) ] ≥ max { S[ µ(x) ], S[ µ(y) ] }. ⇒ If S[ µ(x) ] < S[ µ(y) ], then S[ µ( x∨y ) ] = S[ µ(y) ] Similarly, if S[ µ(x) ] > S[ µ(y) ], then S[ µ( x∨y ) ] = S[ µ(x) ], by theorem 5.1.5 Therefore, S[ µ( x∨y ) ] = max { S[ µ(x) ], S[ µ(y) ] }. Theorem: 1 Let S(µ) be a fuzzy join prime semi L-ideal of a fuzzy join semilattice A then Card Im S(µ) = 2. Proof: Since S(µ) is non constant, card Im S(µ) ≥ 2. Suppose that card Im S(µ) ≥ 3.

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Let S[ µ(0) ] = s and k = Sup { S[ µ(x) ] / x ∈ A }. Then there exists t, m ∈ Im S(µ) such that t < m < s and t ≤ k. Let S(σ) and S(θ) be two fuzzy join prime semi L-ideals of A such that S[ σ(x) ] = ½ ( t + m ), for all x ∈ A and k, if x ∉ S( µm ) = { x ∈ A / S[ µ(x) ] ≥ m} S[ θ(x) ] = s, if x ∈ S( µm ) Clearly, S(σ) is a fuzzy join prime semi L-ideal of A. To show that S(θ) is a fuzzy join prime semi L-ideal of A. Let x, y ∈ A. Case (i): If x, y ∈ S( µm ), then S[ θ(x) ] = s, S[ θ(y) ] = s,

x∨y ∈ S( µm )

Also, S[ θ( x∨y ) ] = s = max { S[ θ(x) ], S[ θ(y) ] } ⇒ S[ θ( x∨y ) ] ≥ max{ S[ θ(x) ], S[ θ(y) ] } Therefore S(θ) is s fuzzy join prime semi L-ideal of A. Case (ii): If x ∈ S( µm ) and y ∉ S( µm ), then S[ θ(x) ] = s, S[ θ(y) ] = k, x∨y ∉ S( µm ) Also, S[ θ( x∨y ) ] = k = max { S[ θ(x) ], S[ θ(y) ] } = max { s, k } ⇒ S[ θ( x∨y ) ] ≥ max { S[ θ(x)], S[ θ(y) ] } Therefore, S(θ) is a fuzzy join prime semi L-ideal of A. Case(iii): If x ∉ S( µm ) and y ∉ S( µm ), then S[ θ(x) ] = S[ θ(y) ] = k, x∨y ∉ S( µm ) Also, S[ θ( x∨y ) ] = k = max { S[ θ(x) ], S[ θ(y) ] } = max { k, k } ⇒ S[ θ( x∨y ) ] ≥ max { S[ θ(x) ], S[ θ(y) ] } Therefore, S(θ) is a fuzzy join prime semi L-ideal of A. Claim: S(σ) ∨ S(θ) ⊆ S(µ) Let x ∈ A. Consider the following cases: (i)

Let x = 0.

Then [ S(σ) ∨ S(θ) ] (x) = max { max { S[ σ(y) ], S[ θ(z) ] } } x = y∨z

≤½(t+m) <s = S[ µ(0) ] (ii)

Let x ≠ 0, x ∈ S( µm ). Then S[ µ(x) ] ≥ m and [ S(σ) ∨ S(θ) ] (x) = max { max { S[ σ(y) ], S[ θ(z) ] } }

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x = y∨z

≤½(t+m) <m = S[ µ(x) ], Since max { S[ σ(y) ], S[ θ(z) ] } ≤ S[ σ(y) ]. (iii) Let x ≠ 0, x ∉ S( µm ). Then for any y, z ∈ A, such that x = y∨z, y ∉ S( µm ) and z ∉ S( µm ). Thus, S[ θ(y) ] = k and S[ θ(z) ] = k Hence, [ S(σ) ∨ S(θ) ] (x) = max { max { S[ σ(y) ], S[ θ(z) ] } } x = y∨z

= max { max ( k, k ) } =k ≤ S[ µ(x) ]. Thus in any case, [ S(σ) ∨ S(θ) ] (x) ≤ S[ µ(x) ] Hence S(σ) ∨ S(θ) ≤ S(µ) Now, there exists y ∈ A such that S[ µ(y) ] = t Then S[ σ(y) ] = ½ ( t + m ) > S[ µ(y) ] ⇒ S[ σ(y) ] > S[ µ(y) ] Hence, S(σ) ⊆ S(µ) Also there exists x ∈ A such that S[ µ(x) ] = t. Then, x ∈ S( µm ) and thus S[ θ(x) ] = s > m = S[ µ(x) ] ⇒ S[ θ(x) ] > S[ µ(x) ] Hence S(θ) ⊆ S(µ) This shows that S(µ) is not a fuzzy join prime semi L-ideal of A, which is a contradiction to the hypothesis. Hence, card Im S(µ) = 2. Theorem: 2 Let A be a fuzzy join semilattice and let S(µ) be a fuzzy join semi L-ideal of A such that Card Im

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S(µ) = 2, S[ µ(0) ] = 1 and the set S( µ0 ) = { x ∈ A / S[ µ(x) ] = S[ µ(0) ] } is a fuzzy level join prime semi L-ideal of A. Proof:

Then S(µ) is a fuzzy join prime semi L-ideal of A.

Let Im S(µ) = { t, 1}, t < 1.

Then S[ µ(0) ] = 1. Let x, y ∈ A. Case (i): If x, y ∈ S( µ0 ) then x∨y ∈ S( µ0 ) and S[ µ( x∨y ) ] = 1 = max { S[ µ(x) ], S[ µ(y) ] }. Case (ii): If

x ∈ S( µ0 ) and y ∉ S( µ0 ), then x∨y ∉ S( µ0 ) and S[ µ( x∨y ) ] = t = max { S[ µ(x) ], S[ µ(y) ] },

Since S[ µ(x) ] = 1 > t = S[ µ(y) ]. Case (iii): If x, y ∈ S( µ0 ) and y ∉ S( µ0 ), then S[ µ(x) ] = S[ µ(y) ] = t Thus, S[ µ( x∨y) ] ≥ t = max { S[ µ(x) ], S[ µ(y) ] }. Hence, S[ µ( x∨y ) ] ≥ max { S[ µ(x) ], S[ µ(y) ] }, for all x, y ∈ A. Now, if

x ∈ S( µ0 ), then x∨y, y∨x ∈ S( µ0 )

Therefore, S[ µ( x∨y ) ] = S[ µ( y∨x ) ] = 1 = S[ µ(x) ]. If x ∉ S( µ0 ), then S[ µ( x∨y ) ] ≥ t = S[ µ(x) ] and S[ µ( y∨x ) ] ≥ t = S[ µ(x) ]. Hence, S(µ) is a fuzzy join prime semi L-ideal of A. Let S(σ) and S(θ) be fuzzy join prime semi L-ideals of A such that S(σ) ∨ S(θ) ⊆ S(µ). Suppose that S(σ) ⊆ S(µ) and S(θ) ⊆ S(µ). Τhen there exists x, y ∈ A such that S[ σ(x) ] > S[ µ(x) ] and S[ θ(x) ] > S[ µ(y) ]. Since for all a ∈ S( µ0 ), S[ µ(a) ] = 1 = S[ µ(0) ], x ∉ S( µ0 ) and y ∉ S( µ0 ). Now, since S(µ0) is a fuzzy join prime semi L-ideal of A, there Let a = x∨z∨y. Then, S[ µ(a) ] = S[ µ(x) ] = S[ µ(y) ] = t. Now, [ S(σ) ∨ S(θ) ] (x) = max { max { S[ σ(u) ] ∨ S[ θ(v) ] } x = u∨v

≥ max { S[ σ(x) ], S[ θ( z∨y ) ] }

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exists

z ∈ A such that x∨z∨y ∉ S( µ0 ).

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.4, 2012

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> t = S[ µ(a) ], Since S[ σ(x) ] > S[ µ(y) ] = t and S[ θ( z∨y ) ] ≥ S[ θ(y) ] > S[ µ(y) ] = t That is , S(σ) ∨ S(θ) ⊆ S(µ). This contradicts the assumption that S(σ) ∨ S(θ) ⊆ S(µ)

Thus, S(µ) is a fuzzy join prime semi L-ideal of A.

Theorem: 3

If f is a fuzzy join prime semi

L-ideal homomorphism from a

fuzzy join semi L-ideal of A onto

a fuzzy join semi L-ideal of A′ and S(µ′) is any fuzzy join prime semi L-ideal of A′, then f -1[ S( µ′ ) ] is a fuzzy join prime semi L-ideal of A.

Proof: Let S(µ) and S(σ) be any two fuzzy join prime semi L-ideals of A such that S(µ) ∨ S(σ) -1

[ S( µ′ ) ].

⇒ f [ S(µ) ∨ S(σ) ] ⊆ ff -1[ S(µ) ] = S( µ′ ) ⇒ f [ S(µ) ] ∨ f [ S(σ) ] ⊆ S( µ′ ), Since f is a fuzzy join prime semi L-ideal homomorphism. ⇒ Either f [ S(µ) ] ⊆ S( µ′ ) or f [ S(σ) ] ⊆ S( µ′ ), Since S(µ′) is fuzzy join prime semi L-ideal of A′. ⇒ Either f -1f [ S(µ) ] ⊆ f -1[ S( µ′ ) ] or f -1f [ S(σ) ] ⊆ f -1[ S( µ′ ) ]. ⇒ Either S(µ) ⊆ f -1[ S( µ′ ) ] or S(σ) ⊆ f -1[ S( µ′ ) ] Hence f -1[ S( µ′ ) ] is a fuzzy join prime semi L-ideal of A.

Reference:

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f

Mathematical Theory and Modeling ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.2, No.4, 2012

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Zhang Yue., and Peng Xingtu(1984), “ The Maximum Fuzzy Ideals and Prime Fuzzy Ideals on Rings”, Fuzzy Mathematics 1, 115 – 116.

Zhang Yue.,(1988), “Prime L- Fuzzy Ideals and Primary L-Fuzzy Ideals”, Fuzzy Sets and Systems 27, 345-350.

Swami, U.M., and Swami, K.L.N.,(1998), “Fuzzy Prime Ideals of Rings”, J. Math Anal.Appl.134, 94–103.

Rajesh Kumar.,(1992), “Fuzzy Ideals and Fuzzy Semiprime Ideals:Some Ring Theoretic analogues”, Inform. Sci. 46 , 43 – 52.

Mukherjee, T.K., and Sen, M.K.,(1989),”Prime Fuzzy Ideals in Rings”, Fuzzy Sets and Systems 32, 337 – 341.

Malik., and John, N.Mordesen.,(1990), “Fuzzy Prime Ideals of a Ring”, Fuzzy Sets and Systems 37 , 93 98.

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