Pseudo completions and completions in stages of o-minimal structures

For an o-minimal expansion R of a real closed field and a set V of Th(R)-convex valuation rings, we construct a “pseudo completion” with respect to V . This is an elementary extension S of R generated by all completions of all the residue fields of the V ∈ V , when these completions are embedded into a big elementary extension of R. It is shown that S does not depend on the various embeddings up to an R-isomorphism. For polynomially bounded R we can iterate the construction of the pseudo completion in order to get a “completion in stages” S of R with respect to V . S is the “smallest” extension of R such that all residue fields of the unique extensions of all V ∈ V to S are complete. Let R be a real closed field. There is a largest ordered field R̂ such that R is dense in R̂. R̂ is again real closed and R̂ is called the completion of R (c.f. [PC]). If v is a proper real valuation on R, then R̂ is also the underlying field of the completion of the valued field (R, v) and R̂ is obtained by adjoining limits of Cauchy sequences with respect to v as explained in [Ri]. We generalize this construction as follows. Let V be a set of convex valuation rings, possibly containing R itself. We construct a “smallest” real closed field containing R which has a limit for all sequences of R that become Cauchy sequences after passing to the residue field of some V ∈ V . This can also be done for o-minimal expansions of real closed fields and Th(R)-convex valuation rings (see section 3 for the definition of the completion in this case). Our first result (4.1) basically says that we can adjoin the missing limits to R in any order and that the resulting elementary extension R′ of R does not depend on the choices, up to an R-isomorphism. We call R′ the pseudo completion of R with respect to V . If R is a pure real closed field (more generally, a polynomially bounded o-minimal expansion of a real closed field), then we can compute the value groups and the residue fields of convex valuation rings of R′. Moreover for every valuation ring V ∈ V the convex hull V ′ of V in R′ is the unique convex valuation ring of R′, lying over V . It turns out that R′ is not “complete in stages” with respect to V ′ := {V ′ | V ∈ V } in general, i.e. not all residue fields of the V ′ are complete in general (c.f. (5.7)). Therefore, in order to get a “smallest” extension of R, which is complete in stages, we have to iterate the construction of the pseudo completion. The iteration stops at an ordinal and the resulting extension S of R is called the completion in stages of R with respect to V . In (5.10), we compute the value groups and the residue fields of convex valuation rings of S. Moreover in (5.10) it is shown that every element s ∈ S \R is of the form ax+b where a, b ∈ R and x ∈ S such that for a unique convex valuation ring W of S with W ∩ R ∈ V , s/mW is the limit of a Cauchy sequence of V/mV without limits in V/mV ; here mV , mW denote the maximal ideal of V, W respectively. 2000 Mathematics Subject Classification: Primary 03C64, 12J10, 12J15; Secondary: 13B35 Partially supported by the European RTNetwork RAAG (contract no. HPRN-CT-200100271)