On semibounded expansions of ordered groups

Abstract

We explore "semibounded" expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal{R}=⟨R,<,+,\dots ⟩$ is a semibounded o-minimal structure and $P\subseteq R$ a set satisfying certain tameness conditions, then $⟨\mathcal{R},\mathcal{P}⟩$ remains semibounded. Examples include the cases when $\mathcal{R}=⟨\mathbb{R},<,+,(x\mapsto \lambda x{)}_{\lambda \in \mathbb{R}},{\cdot }_{[0,1{]}^2}⟩$, and $P=2^{\mathbb{Z}}$ or $P$ is an iteration sequence. As an application, we obtain that smooth functions definable in such $⟨\mathcal{R},P⟩$ are definable in $\mathcal{R}$.