We consider the class of mixed Zp-modules with partial decomposition bases. This class includes those modules classified by Ulm and Warfield and is closed under L∞ω-equivalence. In the context of L∞ω- equivalence, Jacoby defined invariants for this class and proved a classification theorem. Here we examine this class relative to Lδ∞ω, those formulas of quantifier rank ≤ some ordinal δ, defining invariants and proving a classification theorem. This generalizes a result of Barwise and Eklof.
Jacoby, C., & Loth, P. (2014). Zp-modules with partial decomposition bases in L δ ∞ω. Houston Journal of Mathematics, 40(4), 1007-1019.