A tetrachotomy for expansions of the real ordered additive group

Let $$\mathcal {R}$$ be an expansion of the ordered real additive group. When is o-minimal, it known that either defines field isomorphic to $$(\mathbb {R},<,+,\cdot )$$ on some open subinterval $$I\subseteq \mathbb , or a reduct vector space. We say field-type if satisfies former condition. In this paper, we prove more general result for arbitrary expansions {R},<,+)$$ . particular, show do not define dense $$\omega $$ -orders (we call these type A expansions), appropriate version Zilber’s principle holds. Among other things conclude in field-type, every continuous definable function $$[0,1]^m \rightarrow {R}^n$$ locally affine outside nowhere set.