Pseudo Completions and Completion in

For an o-minimal expansion R of a real closed eld 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 elds of the V 2 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 elds of the unique extensions of all V 2 V to S are complete. Let R be a real closed eld. There is a largest ordered eld ^ R such that R is dense in ^ R. ^ R is again real closed and ^ R is called the completion of R (c.f. PC]). If v is a proper real valuation on R, then ^ R is also the underlying eld of the completion of the valued eld (R; v) and ^ R 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 eld containing R which has a limit for all sequences of R that become Cauchy sequences after passing to the residue eld of some V 2 V. This can also be done for o-minimal expansions of real closed elds and Th(R)-convex valuation rings (see section 3 for the deenition of the completion in this case). Our rst result (4.1) basically says that we can adjoin the missing limits to R in any order and that the resulting elementary extension R 0 of R does not depend on the choices, up to an R-isomorphism. We call R 0 the pseudo completion of R with respect to V. If R is a pure real closed eld (more generally, a polynomially bounded o-minimal expansion of a real closed eld), then we can compute the value groups and the residue elds of convex valuation rings of R 0. Moreover for every valuation ring V 2 V the convex hull V 0 of V …