Pseudo completions and completions in stages of o-minimal structures

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.