Structural Investigations in some Classes of o-minimal Fields

This thesis contributes to the study of generalized power series and their relation to o-minimal fields.
In the first part I provide a categorical framework for strong linearity in the form of reasonable categories of strong vector spaces (r.c.s.v.s.).
I show that, in a precise sense, for any fixed base field k, the largest r.c.s.v.s. is the category ΣVect of small k-additive endofunctors X of Vect such that for every cardinal λ, the only natural transformation from the quotient of the λ-fold power of the identity functor by the λ-fold copower of the identity functor to X is the zero one. I compare such ΣVect to the category of “naive” strong vector spaces BΣVect constructed in a specific way and to the r.c.s.v.s. KTVect_{s} of separated linearly topologized spaces that are generated by linearly compact spaces. I study the reflector of Ind(Vect^{op}) onto ΣVect as a transfinite iteration of a natural approximate reflector and show that the approximate reflector agrees with the reflector on ℵ_1-ary objects. This entails that, provided X ∈ ΣVect is ℵ_1-ary, the closure under infinite sums of a linear subspace H of X is given by the set of infinite sums of summable families in H. Finally I describe a natural monoidal closed structure on ΣVect_k, induced by the one on Ind(Vect^{op}). Most of the technical results apply to a more general class of reflective subcategories of Ind(Vect^{op}).
In the second part I study analogues, in the context of T-convexly valued o-minimal fields, of Kaplansky’s Theorem on maximally valued fields.
I show that for each uncountable cardinal λ, every T-convexly valued o-minimal field (E, O) has unique T-λ-spherical completion (up to a non unique isomorphism): that is, an elementary extension which is λ-spherically complete and prime among all λ-spherically complete elementary extensions. I also show that such a completion does not properly enlarge the residue-field sort. When T is power bounded and λ is large enough, the RCVF-reduct of the T-λ-spherical completion is the usual spherical completion.
Afterwards, I show that certain expansions of Hahn fields by generalized power series interpreted as functions on the positive infinitesimal elements, have the property that truncation-closed subsets generate truncation-closed
substructures. This allows me to adapt Mourgues’ and Ressayre’s constructions to deduce structural results for T_0-reducts of T-λ-spherical completion of models of T_{convex}, where T_0 is a power-bounded reduct of T satisfying a certain property I call seriality. The results entail that whenever R_L is a reduct of R_{an,exp} defining the exponential, every elementary extension of R_L has an elementary truncation-closed embedding in the field of surreal numbers, thus partially answering a question of Ehrlich and Kaplan (2021).