THE DP-RANK OF ABELIAN GROUPS
THE DP-RANK OF ABELIAN GROUPS
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阿贝尔群的 DP 秩
DOI：
10.1017/jsl.2018.89
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
D. Palacín
中科院分区：
文献类型：
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作者：
Yatir Halevi;D. Palacín
摘要给出了计算任意交换群的DP-秩的一个方程。我们还证明了它的DP-秩或者说任何一基群的DP-秩都与它的Vapnik-Chervonenkis密度一致。进一步，我们刻画了强交换群A的特征，使得只有有限多个素数p使得群A/PA是无穷的，并且对于每个素数p，只有有限多个自然数n使得$\Left({p^n A}\right)[p]/\Left({p^{n+1}A}\right)[p]$是无穷的。最后证明了有限DP-秩无穷稳定域是代数闭的。
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.