A tetrachotomy for expansions of the real ordered additive group

Abstract

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