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, Carol; Leistner, Katrin; Loth, Peter; and Strungmann, Lutz, "Abelian Groups with Partial Decomposition Bases in Lδ∞ω, Part I" (2012). Mathematics Faculty Publications. 13.