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

Let T be the theory of an o-minimal field and T₀ a common reduct of T and T₀₍. I adapt Mourgues' and Ressayre's constructions to deduce structure results for T₀-reducts of T--spherical completion of models of T₂₎₍ₕ₄ₗ. These in particular entail that whenever RL is a reduct of R₀₍, defining the exponential, every elementary extension of RL has an elementary truncation-closed embedding in No. This partially answers a question in 3 (arXiv: 2002. 07739). 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 T₀ is power bounded but not necessarily a reduct of T₀₍.