Truncation with a Derivation in Unions of Hahn Fields

Mourgues and Ressayre [10] showed that any real closed field is isomorphic to a truncation closed subfield of a Hahn field over R. The papers [5, 6, 7, 4] continue the study of truncation closedness. The results are typically that truncation closedness is robust in the sense of being preserved under a variety of extensions. One significance of truncation closedness is that it enables transfinite induction to be imported as a tool into valuation theory. There is increasing interest in Hahn fields with a ‘good’ derivation, and truncation closedness is potentially significant in that context for similar reasons. We show here that truncation closedness is preserved under certain extensions that involve the derivation. Our main goal is to establish results for the differential field T of transseries in the sense of [2], but initially we work in a simpler and rather general setting of Hahn fields with a ‘good’ derivation. To apply results in that setting to T we use that T is, roughly speaking, obtained by iterating a Hahn field construction: at each step one builds a Hahn field-with-derivation on top of a previously constructed Hahn field-with-derivation. For subsets of T we introduce the condition of being iteratively-logarithmically closed, IL-closed for short. We prove two preservation results for truncation closed IL-closed subsets of T , TIL-closed for short.