Journal of Symbolic Logic

Un Principe D'Ax-Kochen-Ershov Pour des Structures Intermediares Entre Groupes et Corps Values

Francoise Delon and Patrick Simonetta

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An Ax-Kochen-Ershov principle for intermediate structures between valued groups and valued fields. We will consider structures that we call valued B-groups and which are of the form $\langle G, B, *, v\rangle$ where - G is an abelian group, - B is an ordered group, - v is a valuation defined on G taking its values in B, - * is an action of B on G satisfying: $\forall x \in G \forall b \in B v(x * b) = v(x) \cdot b$. The analysis of Kaplanski for valued fields can be adapted to our context and allows us to formulate an Ax-Kochen-Ershov principle for valued B-groups: we axiomatise those which are in some sense existentially closed and also obtain many of their model-theoretical properties. Let us mention some applications: 1. Assume that v(x) = v(nx) for every integer n $\neq$ 0 and x $\in$ G, B is solvable and acts on G in such a way that, for the induced action, $\mathbb{Z}[B] \setminus \{0\}$ embeds in the automorphism group of G. Then $\langle G, B, *, v\rangle$ is decidable if and only if B is decidable as an ordered group. 2. Given a field k and an ordered group B, we consider the generalised power series field k((B)) endowed with its canonical valuation. We consider also the following structure: $\mathbf{M} = \langle k((B))_+, S, v, \times \upharpoonright_{k((B))\times S}\rangle,$ where k((B))$_+$ is the additive group of k((B)), S is a unary predicate interpreting ${T^b | b \in B}$, and $\times \upharpoonright_{k((B))\times S}$ is the multiplication restricted to k((B))$\times$ S, structure which is a reduct of the valued field k((B)) with its canonical cross section. Then our result implies that if B is solvable and decidable as an ordered group, then $\mathbf{M}$ is decidable. 3. A valued B-group has a residual group and our Ax-Kochen-Ershov principle remains valid in the context of expansions of residual group and value group. In particular, by adding a residual order we obtain new examples of solvable ordered groups having a decidable theory.

Article information

J. Symbolic Logic, Volume 64, Issue 3 (1999), 991-1027.

First available in Project Euclid: 6 July 2007

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Zentralblatt MATH identifier


Primary: 03C60: Model-theoretic algebra [See also 08C10, 12Lxx, 13L05]

Decidability of Groups Decidability of Ordered Groups Decidability of Valued Groups Ax-Kochen-Ershov Groups Acting on Groups


Delon, Francoise; Simonetta, Patrick. Un Principe D'Ax-Kochen-Ershov Pour des Structures Intermediares Entre Groupes et Corps Values. J. Symbolic Logic 64 (1999), no. 3, 991--1027.

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