Ordered Algebraic Structures : Hahn Fields, Convex Valuations, and Exponentials

Abstract

This habilitation thesis contains seven articles dedicated to the valuation and model theoretic study of ordered algebraic structures. It is divided into three parts, each consisting of those articles focussing on a particular class of structures: Hahn fields, definable convex valuations in ordered fields, and ordered exponential fields. In the first part, substructures of fields of generalised power series are investigated. These substructures, mostly subfields, are either induced by families of prescribed supports or solution sets of generalised linear recurrence equations. The second part undertakes a systematic study of first-order definable convex valuation rings in ordered fields, mostly dealing with the henselian case. Several definability results are established that rely on (topological) conditions on the residue field and on the value group of the valuations. The third part is concerned with models of real exponentiation, that is, ordered exponential fields that are elementarily equivalent to the real exponential field. The two main threads in this part are integer parts of such models as well as prime models of real exponentiation.