Journal of Symbolic Logic

Some Model Theory for Almost Real Closed Fields

Francoise Delon and Rafel Farre

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We study the model theory of fields $k$ carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the definable real valuation rings of $k$ are in correspondence with the definable convex subgroups of the value group of a certain real valuation of $k$.

Article information

J. Symbolic Logic, Volume 61, Issue 4 (1996), 1121-1152.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03C30: Other model constructions
Secondary: 12D15: Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) [See also 11Exx] 12J10: Valued fields 60F20: Zero-one laws 03B25: Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10]

Henselian fields real closed fields ordered abelian groups decidability


Delon, Francoise; Farre, Rafel. Some Model Theory for Almost Real Closed Fields. J. Symbolic Logic 61 (1996), no. 4, 1121--1152.

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