Young Bae Jun, Madad Khan, Florentin Smarandache, Saima Anis
Proof. Let x ∈ S. Then δ ∗ (λ ◦ µ)(x) = sup (λ ◦ µ)(y) y∈[x]δ
µ
¶ sup min{λ(a), µ(b)}
= sup y∈[x]δ
y=ab
µ
≤ sup
¶
sup min{λ(ab), µ(ab)} y=ab
y∈[x]δ
= sup min{λ(y), µ(y)} y∈[x]δ
≤
sup
min{λ(a), µ(b)}
a∈[x]δ , b∈[x]δ
(
= min
)
sup λ(a), sup µ(b) a∈[x]δ
b∈[x]δ
= min{δ ∗ (λ)(x), δ ∗ (µ)(x)} = (δ ∗ (λ) ∧ δ ∗ (µ)) (x), and δ∗ (λ ◦ µ)(x) = inf (λ ◦ µ)(y) y∈[x]δ µ ¶ = inf sup min{λ(a), µ(b)} y∈[x]δ y=ab µ ¶ ≤ inf sup min{λ(ab), µ(ab)} y∈[x]δ
y=ab
= inf min{λ(y), µ(y)} y∈[x]δ ½ ¾ = min inf λ(a), inf µ(b) a∈[x]δ
b∈[x]δ
= min{δ∗ (λ)(x), δ∗ (µ)(x)} = (δ∗ (λ) ∧ δ∗ (µ)) (x). Therefore δ ∗ (λ ◦ µ) ≤ δ ∗ (λ) ∧ δ ∗ (µ) and δ∗ (λ ◦ µ) ≤ δ∗ (λ) ∧ δ∗ (µ). Theorem 3.17. Let δ be a congruence on S and let λ and µ be a fuzzy right ideal and a fuzzy left ideal, respectively, of S. If S is regular, then δ ∗ (λ ◦ µ) = δ ∗ (λ) ∧ δ ∗ (µ) and δ∗ (λ ◦ µ) = δ∗ (λ) ∧ δ∗ (µ). Proof. Let a be any element of S. Then a = aca for some c ∈ S since S is regular. Hence
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