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

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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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Authors and Affiliations

  1. Jacoby Consulting, Long Beach, CA, USA

    Carol Jacoby

  2. Department of Mathematics, Sacred Heart University, Fairfield, CT, 06825, USA

    Peter Loth

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  1. Carol Jacoby
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  2. Peter Loth
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Correspondence to Peter Loth.

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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

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