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3 results

  • THE DP-RANK OF ABELIAN GROUPS

  • YATIR HALEVI, DANIEL PALACÍN
  • Journal:
    The Journal of Symbolic Logic / Volume 84 / Issue 3 / September 2019
    Published online by Cambridge University Press:
    01 February 2019, pp. 957-986
    Print publication:
    September 2019

  • 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.

  • Ample thoughts

  • Daniel Palacín, Frank O. Wagner
  • Journal:
    The Journal of Symbolic Logic / Volume 78 / Issue 2 / June 2013
    Published online by Cambridge University Press:
    12 March 2014, pp. 489-510
    Print publication:
    June 2013

  • Non-n-ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.