Abstract

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: $\forall x \in G \forall b \in B v = v \cdot b$. 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 = v 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) endowed with its canonical valuation. We consider also the following structure: $\mathbf{M} = \langle k)_+, S, v, \times \upharpoonright_{k)\times S}\rangle,$ where k)$_+$ is the additive group of k), S is a unary predicate interpreting ${T^b | b \in B}$, and $\times \upharpoonright_{k)\times S}$ is the multiplication restricted to k)$\times$ S, structure which is a reduct of the valued field k) 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.