# Pseudo completions and completions in stages of o-minimal structures

Research paper by Marcus Tressl

Indexed on: 14 Jul '06Published on: 14 Jul '06Published in: Archive for Mathematical Logic

#### Abstract

For an o-minimal expansion R of a real closed field and a set $$\fancyscript{V}$$ of Th(R)-convex valuation rings, we construct a “pseudo completion” with respect to $$\fancyscript{V}$$. This is an elementary extension S of R generated by all completions of all the residue fields of the $$V \in \fancyscript{V}$$, 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 $$\fancyscript{V}$$. S is the “smallest” extension of R such that all residue fields of the unique extensions of all $$V \in \fancyscript{V}$$ to S are complete.