Nonvaluational ordered Abelian groups of finite burden

Abstract

Consider an expansion $\mathcal G=(G,<,+,\ldots)$ of an ordered divisible Abelian group of finite burden defining no nonempty subset $X$ of $G$ which is dense and codense in a definable open subset $U$ of $G$ with $X \subseteq U$. We further assume that $\mathcal G$ is nonvaluational, that is, for every nonempty definable subsets $A,B$ of $G$ with $A<B$ and $A \cup B=G$, $\mathrm{inf}{b-a;|;a \in A,, b \in B}=0$. Then, $\mathcal G$ is $*$‑locally weakly o‑minimal. We also give a complete description of sets definable in a definably complete expansion of an ordered group of burden two if it defines an infinite discrete set.