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arxiv: 1706.05471 · v1 · submitted 2017-06-17 · 🧮 math.LO

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Strong ordered Abelian groups and dp-rank

Rafel Farr\'e
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classification 🧮 math.LO
keywords abelianordereddp-rankstronggroupgroupsfinitealgebraic
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We provide an algebraic characterization of strong ordered Abelian groups: An ordered Abelian group is strong iff it has bounded regular rank and almost finite dimension. Moreover, we show that any strong ordered Abelian group has finite Dp-rank. We also provide a formula that computes the exact valued of the Dp-rank of any ordered Abelian group. In particular characterizing those ordered Abelian groups with Dp-rank equal to $n$. We also show the Dp-rank coincides with the Vapnik-Chervonenkis density.

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    A weakly o-minimal structure without infinite groups has a Shelah completion that interprets an infinite field, with a new local trace definability notion and a linearity-field dichotomy for certain ordered groups.