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THE DP-RANK OF ABELIAN GROUPS

Published online by Cambridge University Press:  01 February 2019

YATIR HALEVI  and
DANIEL PALACÍN
YATIR HALEVI
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM GIVAT RAM9190401, JERUSALEM, ISRAELE-mail: yatir.halevi@mail.huji.ac.il
DANIEL PALACÍN
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM GIVAT RAM9190401, JERUSALEM, ISRAELE-mail: daniel.palacin@mail.huji.ac.il
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Abstract

An equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik–Chervonenkis density. Furthermore, strong abelian groups are characterised to be precisely those abelian groups A such that there are only finitely many primes p such that the group A / pA is infinite and for every prime p, there are only finitely many natural numbers n such that $\left( {p^n A} \right)[p]/\left( {p^{n + 1} A} \right)[p]$ is infinite.

Finally, it is shown that an infinite stable field of finite dp-rank is algebraically closed.

Keywords

03C6003C4520K99abelian groupone-basedvc-densitydp-rank
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 3 , September 2019 , pp. 957 - 986
Copyright
Copyright © The Association for Symbolic Logic 2019 

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