Oracle Database Enterprise User Administrators Guide 10g

Read or Download Oracle Database Enterprise User Administrators Guide 10g Release 2 (10.2) b14269-01 PDF

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Let A = SymA (Wp , RadA). Then by Lemma 31 N (Wp ) = RadA. The other implication is trivial. A Now we show (i) ⇔ (ii). 18] RadA ∈ V (Sp ) A a )= and, by Theorem 33, RadA = Sym A (Wp , {RadA}). Let a ∈ A. Wp ( RadA Wp (a) RadA RadA = RadA. Thus Wp (a) ∈ RadA and A = SymA (Wp , RadA). Assume Wp (a) a ) = RadA = RadA. now A = SymA (Wp , RadA) and a ∈ A. 18] A ∈ V (Spω ). By the above theorem immediately we have: Corollary 36. Let A be an MV-algebra. If A ∈ V (Spω ), then RadA is an hyperarchimedean ideal.

Let M = Sp and Wp (a) a a ∈ A. By Proposition 21(i), Wp ( M ) = M = 0. Thus Wp (a) ∈ M and NA (Wp ) ⊆ M = A. So we proved (i) ⇔ (ii). (ii) ⇒ (iii). Corollary 29 allows us to consider the subalgebra SymA (Wp , NA (Wp )) of A. As a matter of fact A = SymA (Wp , NA (Wp )). Indeed for a ∈ A, Wp (a) ∈ NA (Wp ). (iii) ⇒ (iv) is trivial. (iv) ⇒ (ii). Assume A = SymA (Wp , I), for some semisimple proper ideal I of A. Then Wp (a) ∈ I, for every a ∈ A. Thus NA (Wp ) ⊆ I = A. Proposition 38. Let A be an MV-algebra and p ∈ P.

By arbitrariety of J the thesis follows. Let I be an ideal of an MV-algebra A and W an MV- polynomial. Set SymA (W, I) = {a ∈ A : W (a) ∈ I}. Then, with the above notations, we have: Proposition 10. Let A be an MV-algebra, W a symmetric and stable (resp. strongly stable) MV-polynomial and I a semisimple ideal (resp. ideal) of A. Then SymA (W, I) is a subalgebra of A and I is an ideal of SymA (W, I). Proof. It is immediate to see that SymA (W, I) is a subalgebra of A. Let us now show that I ⊆ SymA (W, I).

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