On the structure of $T$-$λ$-spherical completions of o-minimal fields

Abstract: Let $T$ be the theory of an o-minimal field and $T_0$ a common reduct of $T$ and $T_{an}$. I adapt Mourgues' and Ressayre's constructions to deduce structure results for $T_0$-reducts of $T$-$\lambda$-spherical completion of models of $T_{\mathrm{convex}}$. These in particular entail that whenever $\mathbb{R}L$ is a reduct of $\mathbb{R}{an,\exp}$ defining the exponential, every elementary extension of $\mathbb{R}L$ has an elementary truncation-closed embedding in the class-sized field $\mathbf{No}$ of surreal numbers. This partially answers a question in 3. The main technical result is that certain expansion of Hahn fields by generalized power series interpreted as functions defined on the positive infinitesimal elements, have the property that truncation closed subsets generate truncation closed substructures. This leaves room for possible generalizations to the case in which $T_0$ is power bounded but not necessarily a reduct of $T{an}$.