2024/07/10 by Pietro Freni, Freni, Pietro
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals #math.LO
paper · pdf · doi:10.48550/arxiv.2407.07442
openalex publication_date 2024/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let T be the theory of an o-minimal field and T0 a common reduct of T and Tan. I adapt Mourgues' and Ressayre's constructions to deduce structure results for T0-reducts of T-λ-spherical completion of models of Tconvex. These in particular entail that whenever T is the theory of a reduct of ℝan,exp defining the exponentiation (e.g. T=Texp, the theory of the field of reals expanded by the exponential function), every model of T has an initial elementary embedding in the field 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 T0 is power bounded but not necessarily a reduct of Tan.