Nonvaluational ordered Abelian groups of finite burden
Masato Fujita
TL;DR
This work studies expansions of ordered divisible abelian groups with finite burden, focusing on the absence of dense-codense wild sets inside definable open sets and the nonvaluational property. It proves that nonvaluational, finite-burden structures with no wild subset are $*$-locally weakly o-minimal, using two modified DG-type arguments centered at nv elements. Moreover, it provides a complete structural description for burden-two, definably complete expansions that define an infinite discrete set: such structures are interdefinable with the standard simple product of an inp-minimal discrete group and an o-minimal interval, yielding tame open cores and a precise decomposition. These results illuminate how tame topological behavior emerges in non-definably-complete and burden-two settings and clarify the landscape of finite-burden expansions of ordered groups.
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.