Truncations in languages of generalized power series and the structure of $T$-$\lambda$-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 $T$ is the theory of a reduct of $\mathbb{R}_{an,\exp}$ defining the exponentiation (e.g.\ $T=T_{\exp}$, the theory of the field of reals expanded by the exponential function), every model of $T$ has an initial elementary embedding in the field $\mathbf{No}$ of surreal numbers. This answers positively an open question in (arXiv:2002.07739). The main technical result is that expanding an integral domain of generalized series in the sense of Hahn-Higman-Ribenboim (such as a Hahn field) by a family of generalized power series interpreted as functions defined on certain infinitesimal elements, has the property that truncation closed subsets generate truncation closed substructures, provided that the family of generalized power series is itself closed under truncations and partial derivatives. It is also shown that the further closure of the generated set under solutions to certain equations is as well closed under truncations. The formal results on power series leave room for possible generalizations to the case in which $T_0$ is power bounded but not necessarily a reduct of $T_{an}$.