Nonvaluational ordered Abelian groups of finite burden

TL;DR

This paper studies certain ordered Abelian groups with finite burden, showing they are locally weakly o-minimal under specific conditions, and characterizes definable sets in expansions with burden two that define infinite discrete sets.

Contribution

It establishes that nonvaluational ordered Abelian groups of finite burden are $*$-locally weakly o-minimal and provides a full description of definable sets in expansions with burden two that define infinite discrete sets.

Findings

Nonvaluational ordered Abelian groups of finite burden are $*$-locally weakly o-minimal.

Complete characterization of definable sets in burden two expansions with infinite discrete sets.

Abstract

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