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

Tressl, Marcus (2006) Pseudo completions and completions in stages of o-minimal structures. Archive for Mathematical Logic, 45 (8). pp. 983-1009. ISSN 1432-0665

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.