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

Let T be the theory of an o-minimal field, and T₂₎₍ₕ₄ₗ the theory of its expansion by a predicate O for a non-trival T-convex valuation ring. For an uncountable cardinal, say that a unary type p (x) over a model of T₂₎₍ₕ₄ₗ is -bounded weakly immediate if its cut is defined by an empty intersection of fewer than many nested valuation balls. Call an elementary extension -bounded wim-constructible if it is obtained as a transfinite composition of extensions each generated by one element whose type is -bounded weakly immediate. I show that -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 -bounded wim-constructible over both. A consequence is that given a cardinal, every model of T₂₎₍ₕ₄ₗ has a unique-up-to-non-unique-isomorphism -spherically complete -bounded wim-constructible extension. We call this extension the T--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.