The Classification of $\mathbb{Z}_p$-Modules with Partial Decomposition Bases in $L_{\infty\omega}$

TL;DR

This paper extends the classification of certain mixed $Z_p$-modules using invariants derived from Ulm and Warfield invariants, generalizing the concept of decomposition basis within $L_{ ext{ } extomega}$ logic.

Contribution

It introduces the concept of partial decomposition bases for mixed $Z_p$-modules and provides a complete classification theorem in $L_{ ext{ } extomega}$.

Findings

Classifies a broad class of mixed $Z_p$-modules using invariants.

Includes all Warfield modules and is closed under $L_{ ext{ } extomega}$-equivalence.

Extends Ulm's and Warfield's classification results.

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.