We explore \emph{ semibounded} expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We introduce the notion of a \emph{ semibounded} expansion of an arbitrary ordered group, extending the usual notion from the o-minimal setting. For $\mathcal{ R}=( \mathbb{ R}, <, +, \ldots)$, a semibounded o-minimal structure and $P\subseteq \mathbb{ R}$ a set satisfying certain tameness conditions, we discuss under which conditions $(\mathcal R,P)$ defines total linear functions that are not definable in \mathcal{ R}. Examples of such structures that does define new total linear functions include the cases when $\mathcal{ R}$ is a reduct of $(\mathbb{ R},<,+,\cdot_{ \upharpoonright (0,1)^2},(x\mapsto \lambda x)_{ \lambda\in I\subseteq \mathbb{ R}})$, and $P= 2^\mathbb{ Z}$, or $P$ is an iteration sequence (for any $I$) or $P=\mathbb{ Z}$, for $I=\mathbb{ Q}$.
Categories: math.LO