We consider the class of abelian groups possessing partial decomposition bases in Lδ∞ω for δ an ordinal. This class contains the class of Warfield groups which are direct summands of simply presented groups or, alternatively, are abelian groups possessing a nice decomposition basis with simply presented cokernel. We prove a classification theorem using numerical invariants that are deduced from the classical Ulm-Kaplansky and Warfield invariants. This extends earlier work by Barwise-Eklof, Göbel and the authors.
Jacoby, C., Leistner, K., Loth, P., & Strüngmann, L. (2012). Abelian groups with partial decomposition bases in Lδ∞ω, Part I", in groups and model theory, Contemporary Mathematics, 576: 163-175. doi: 10.1090/conm/576/11349