$T$-convexity, Weakly Immediate Types, and $T$-$\lambda$-Spherical Completions of o-minimal Structures

It is well known that ordered exponential fields with a compatible non-trivial valuation cannot be spherically complete, but there are some that are ``complete enough''. This paper gives analogues of Kaplansky's theorem on maximally valued fields that hold for a suitable class of elementary extensions of some ordered exponential fields with a compatible valuation. More precisely it does so for models of any theory $T_{\text{convex}}$ given by the expansion of a fixed complete o-minimal theory of ordered fields $T$, by a predicate $\mathcal{O}$ for a non-trivial $T$-convex valuation ring. For $\lambda$ an uncountable cardinal, say that a unary type $p(x)$ over a model of $T_{\text{convex}}$ is \emph{$\lambda$-bounded weakly immediate} if its cut is defined by an empty intersection of fewer than $\lambda$ many nested valuation balls. Call an elementary extension \emph{$\lambda$-bounded wim-constructible} if it is obtained as a transfinite composition of extensions each generated by one element whose type is $\lambda$-bounded weakly immediate. I show that $\lambda$-bounded wim-constructible extensions do not extend the residue-field sort and that any two wim-constructible extensions can be amalgamated in an extension which is again $\lambda$-bounded wim-constructible over both. A consequence of this is that given an uncountable cardinal $\lambda$, every model of $T_{\text{convex}}$ has a unique-up-to-isomorphism $\lambda$-spherically complete $\lambda$-bounded wim-constructible extension providing an analogue of Kaplansky's theorem. I call this extension the $T$-$\lambda$-spherical completion. Another consequence is that $T_{\mathrm{convex}}$ is \emph{definably spherically complete}. When $T$ is power bounded wim-constructible extensions are just the immediate extensions. I discuss the example of power bounded theories expanded by $\exp$ (\emph{simply exponential} theories).