$T$-convexity, Weakly Immediate Types, and $T$-$λ$-Spherical Completions of o-minimal Structures
Pietro Freni
TL;DR
The work develops a Kaplansky-type framework for o-minimal structures expanded by a $T$-convex valuation, introducing $oldsymbol{\lambda}$-bounded weakly immediate (wim) types and Wim-constructible extensions. It proves a fivefold classification of unary extensions, an amalgamation lemma for $oldsymbol{\lambda}$-bounded Wim-constructible extensions, and establishes the existence and uniqueness of a $oldsymbol{\lambda}$-spherically complete, $oldsymbol{\lambda}$-bounded Wim-constructible extension that does not enlarge the residue-field sort; this yields definable spherical completeness for $T_{ ext{convex}}$. In the power-bounded setting Wim-constructible extensions coincide with immediate ones, enabling clean amalgamation and spherical completion results. The paper also initiates the exponential case, outlining the extension framework for simply exponential theories and providing foundational results on weakly immediate types in that context. Collectively, the results offer a robust structural analog of Kaplansky’s theorem for a broad class of o-minimal expansions, with implications for definable spherical completeness and residue-field stability across models.
Abstract
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 $λ$ an uncountable cardinal, say that a unary type $p(x)$ over a model of $T_{\text{convex}}$ is \emph{$λ$-bounded weakly immediate} if its cut is defined by an empty intersection of fewer than $λ$ many nested valuation balls. Call an elementary extension \emph{$λ$-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 of this is that given an uncountable cardinal $λ$, every model of $T_{\text{convex}}$ has a unique-up-to-isomorphism $λ$-spherically complete $λ$-bounded wim-constructible extension providing an analogue of Kaplansky's theorem. I call this extension the $T$-$λ$-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).