Linear O-minimal structures

Abstract

A linearly ordered structure $$\mathcal{M} = (M,< , \cdot \cdot \cdot )$$ is called o-minimal if every definable subset ofM is a finite union of points and intervals. Such an $$\mathcal{M}$$ is aCF structure if, roughly said, every definable family of curves is locally a one-parameter family. We prove that if $$\mathcal{M}$$ is aCF structure which expands an (interval in an) ordered group, then it is elementary equivalent to a reduct of an (interval in an) ordered vector space. Along the way we prove several quantifier-elimination results for expansions and reducts of ordered vector spaces.