2017•arXiv (Cornell University)Open access

Strong ordered Abelian groups and dp-rank

Rafel Farré

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Abstract

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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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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Available abstract

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.

Key concepts: Abelian group, Rank of an abelian group, Elementary abelian group, Mathematics, Rank (graph theory), Bounded function, Combinatorics, Free abelian group

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