On Boolean algebras and integrally closed commutative regular rings

Journal of Symbolic Logic 57 (4):1305-1318 (1992)
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Abstract

In this paper we consider properties, related to model-completeness, of the theory of integrally closed commutative regular rings. We obtain the main theorem claiming that in a Boolean algebra B, the truth of a prenex Σn-formula whose parameters ai partition B, can be determined by finitely many conditions built from the first entry of Tarski invariant T(ai)'s, n-characteristic D(n, ai)'s and the quantities S(ai, l) and S'(ai, l) for $l < n$. Then we derive two important theorems. One claims that for any Boolean algebras A and B, an embedding of A into B preserving D(n, a) for all a ∈ A is a Σn-extension. The other claims that the theory of n-separable Boolean algebras admits elimination of quantifiers in a simple definitional extension of the language of Boolean algebras. Finally we translate these results into the language of commutative regular rings

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References found in this work

Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
Comparing The Expressive Power of Some Languages for Boolean Algebras.Lutz Heindorf - 1981 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 27 (25-30):419-434.
Comparing The Expressive Power of Some Languages for Boolean Algebras.Lutz Heindorf - 1981 - Mathematical Logic Quarterly 27 (25‐30):419-434.
Artin-Schreier Theory for Commutative Regular Rings.L. van den Dries - 1977 - Annals of Mathematical Logic 12 (2):113.

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