$L_{\kappa \omega }$-equivalence of abelian groups with partial decomposition bases

Abstract

We consider the class of abelian groups possessing partial decomposition bases in the language $L_{\kappa \omega }$ for uncountable cardinals $\kappa$. Jacoby, Leistner, Loth and Strüngmann developed a numerical invariant deduced from the classical global Warfield invariant and proved that if two groups have identical modified Ulm invariants and Warfield invariants up to $\omega \delta$ for some ordinal $\delta$, then they are equivalent in $L_{\infty \omega }^{\delta }$. Subsequently, Jacoby and Loth showed that the converse is true for appropriate $\delta$. In this paper we prove that the modified Warfield invariant up to $\kappa$ is expressible in $L_{\kappa \omega }$, thus a complete classification theorem in $L_{\kappa \omega }$ is obtained. This generalizes a result of Barwise and Eklof.