We consider the class of abelian groups with partial decomposition bases, which includes groups classified by Ulm, Warfield, Stanton and others. We define an invariant and classify these groups in the language L∞ω, or equivalently, up to partial isomorphism. This generalizes a result of Barwise and Eklof and builds on Jacoby's classification of local groups with partial decomposition bases in L∞ω.
Jacoby, C. & Loth, P. (2017). The classification of infinite abelian groups with partial decomposition bases in L∞ω. Rocky Mountain Journal of Mathematics, 47(2), 463--477. doi:10.1216/RMJ-2017-47-2-463.