Linear O-minimal structures

A linearly ordered structure {Mathematical expression} is called o-minimal if every definable subset of M is a finite union of points and intervals. Such an {Mathematical expression} is a CF structure if, roughly said, every definable family of curves is locally a one-parameter family. We prove that if {Mathematical expression} is a CF 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.