Pseudo completions and completions in stages of o-minimal structures
Given a real-closed field \(R\) there is a largest ordered field \(\hat R\) such that \(R\) is dense in \(\hat R\). \(\hat R\) is called the completion of \(R\) and is again real-closed; if \(R=\hat R\) then \(R\) is called complete (see [\textit{S. Prieß-Crampe}, Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen. Berlin: Springer-Verlag (1983; Zbl 0558.51012)]). If \(v\) is a proper real valuation on \(R\) then \(\hat R\) is the completion of the valued field \((R,v):\hat R\) is obtained by adjoining limits of Cauchy sequences with respect to \(v\) (see [\textit{P. Ribenboim}, Théorie des valuations. Montreal: Les Presses de l'Université de Montreal (1965; Zbl 0139.26201)]). The author calls a real-closed field \(R\) complete in stages with respect to a set \(\mathcal V\) of convex valuation rings of \(R\) if the residue field of every \(V\in \mathcal V\) is complete. In the first step the author constructs for a real-closed field \(R\) and a set \(\mathcal V\) of convex valuation rings of \(R\) a smallest real-closed field extension \(R'\) of \(R\) having a limit for all sequences of \(R\) that become Cauchy sequences in the residue field of some \(V\in \mathcal V\). It is shown that up to \(R\)-isomorphisms the real-closed field \(R'\) does not depend on the chosen order of adjoining limits. \(R'\) is called the pseudo-completion of \(R\). In general the pseudo-completion is not complete in stages with respect to the set of convex hulls in \(R'\) of the elements in \(\mathcal V\). So in a second step the author iterates the construction from the first step to obtain a smallest extension \(S\) of \(R\) which is complete in stages with respect to the set of convex hulls in \(S\) of the elements in \(\mathcal V\). This extension \(S\) is called the completion in stages of \(R\) with respect to \(\mathcal V\). The value groups and residue fields of all convex valuation rings of \(S\) are computed. Step 1 can be done for o-minimal expansions of real-closed fields and a set of Th\((R)\)-convex valuation rings. Step 2 can be generalized to the setting of polynomially pounded o-minimal expansions of real-closed fields. The proofs use a new kind of dimension, called realization rank, and a model-theoretic analysis of cuts.
scientific article; zbMATH DE number 5118083