T-convexity, Weakly Immediate Types and T-λ-Spherical Completions of o-minimal Structures

Let T be the theory of an o-minimal field, and Tconvex the theory of its expansion by a predicate O for a non-trival T-convex valuation ring. For λ an uncountable cardinal, say that a unary type $p(x)$ over a model of Tconvex is λ-bounded weakly immediate if its cut is defined by an empty intersection of fewer than λ many nested valuation balls. Call an elementary extension λ-bounded wim-constructible if it is obtained as a transfinite composition of extensions each generated by one element whose type is λ-bounded weakly immediate. I show that λ-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 λ-bounded wim-constructible over both. A consequence is that given a cardinal λ, every model of Tconvex has a unique-up-to-non-unique-isomorphism λ-spherically complete λ-bounded wim-constructible extension. We call this extension the T-λ-spherical completion. In the case T is power bounded, wim-constructible extensions are just the immediate extensions. I discuss the example of power bounded theories expanded by exp.