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The classification of \({\mathbb {Z}}_p\)-modules with partial decomposition bases in \(L_{\infty \omega }\)

Abstract

Ulm’s Theorem presents invariants that classify countable abelian torsion groups up to isomorphism. Barwise and Eklof extended this result to the classification of arbitrary abelian torsion groups up to \(L_{\infty \omega }\)-equivalence. In this paper, we extend this classification to a class of mixed \({\mathbb {Z}}_p\)-modules which includes all Warfield modules and is closed under \(L_{\infty \omega }\)-equivalence. The defining property of these modules is the existence of what we call a partial decomposition basis, a generalization of the concept of decomposition basis. We prove a complete classification theorem in \(L_{\infty \omega }\) using invariants deduced from the classical Ulm and Warfield invariants.

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Jacoby, C., Loth, P. The classification of \({\mathbb {Z}}_p\)-modules with partial decomposition bases in \(L_{\infty \omega }\) . Arch. Math. Logic 55, 939–954 (2016). https://doi.org/10.1007/s00153-016-0506-7

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  • DOI: https://doi.org/10.1007/s00153-016-0506-7

Keywords

  • Partial decomposition basis
  • Partial isomorphism
  • Infinitary equivalence
  • Ulm-Kaplansky invariants
  • Warfield invariants

Mathematics Subject Classification

  • Primary 03C52
  • 13C05
  • 20K21
  • Secondary 03E10
  • 20K25
  • 20K35