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Un principe d'ax-kochen-ershov pour des structures intermédiates entre groupes et corps valués

Published online by Cambridge University Press:  12 March 2014

Françoise Delon
Affiliation:
Équipe de Logique Mathématique de Paris VII, 2, Place Jussieu - Case 012, 75251 Paris Cedex 05, France E-mail: delon@logique.jussieu.fr
Patrick Simonetta
Affiliation:
Équipe de Logique Mathématique de Paris VII, 2, Place Jussieu - Case 012, 75251 Paris Cedex 05, France E-mail: simbaud@logique.jussieu.fr

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 〈G, B, *, υ〉 where

G is an abelian group,

B is an ordered group,

– υ is a valuation denned on G taking its values in B,

– * is an action of B on G satisfying: ∀x ϵ GbB υ(x * b) = ν(x) · 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 υ(x) = υ(nx) for every integer n ≠ 0 and x ϵ G, B is solvable and acts on G in such a way that, for the induced action, Z[B] ∖ {0} embeds in the automorphism group of G. Then 〈G, B, *, υ〉 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:

where k((B))+ is the additive group of k((B)), S is a unary predicate interpreting {Tb ∣ b ϵB}, and ×↾k((B))×S is the multiplication restricted to k((B)) × 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 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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1] Bourbaki, N., Algèbre commutative, ch. VI: Valuations, Hermann, Paris, 1964.Google Scholar
[2] Delon, F., ℚ muni de l'arithmétique faible de Penzin est décidable, Proceedings of the American Mathematical Society, vol. 125 (1997), pp. 27112717.Google Scholar
[3] Delon, F. and Simonetta, P., Undecidable wreath products and skew power series fields, this Journal, vol. 63 (1998), pp. 237245.Google Scholar
[4] Kaplanski, I., Maximal fields with valuations, Duke Mathematics Journal, vol. 9 (1942), pp. 303321.Google Scholar
[5] Neumann, B. H., On ordered division rings, Transactions of the American Mathematical Society, vol. 66 (1949), pp. 202252.Google Scholar
[6] Passman, D. S., The algebraic structure of group rings, Krieger, Malabar, Florida, 1985.Google Scholar
[7] Ribenboim, P., Théorie des valuations, Les Presses de l'Université de Montréal, Montréal, 1964.Google Scholar
[8] Schilling, O. F. G., The theory of valuations, Mathematical Surveys, vol. IV, American Mathematical Society, New York, 1950.CrossRefGoogle Scholar
[9] Shoenfield, J. R., A theorem on quantifier elimination, Symposia Mathematica, vol. 5 (1971), pp. 173176.Google Scholar
[10] Simonetta, P., Equivalence élémentaire et décidabilité pour des structures du type groupe agissant sur un groupe abélien, to appear in this Journal.Google Scholar
[11] Simonetta, P., Décidabilité et interprétabilité dans les corps et les groupes non commutatifs, Thèse de l'Université Paris 7 , 1994.Google Scholar
[12] Simonetta, P., Une correspondance entre anneaux partiels et groupes, this Journal, vol. 62 (1997), pp. 6078.Google Scholar
[13] Wehrfritz, B. A. F., Infinite linear groups, Springer-Verlag, Berlin, Heidelberg, New York, 1973.Google Scholar