We study completely decomposable torsion-free abelian groups of the form $\mathcal{G}_S := \oplus_{n \in S} \mathbb{Q}_{p_n}$ for sets $S \subseteq \omega$. We show that $\mathcal{G}_S$has a decidable copy if and only if S is $\Sigma^0_2$and has a computable copy if and only if S is $\Sigma^0_3$.

We prove that for any k≥3 each element of the h-quasiorder of finite k-labeled forests is definable in the ordinary first order language and, respectively, each element of the h-quasiorder of countable k-labeled forests is definable in the language Lω1ω, in both cases provided that the minimal non-smallest elements are allowed as parameters. As corollaries, we characterize the automorphism groups of both structures and show that the structure of finite k-forests is atomic. Similar results hold true for two other relevant (...) structures: the h-quasiorder of finite k-labeled trees and of finite k-labeled trees with a fixed label of the root element. (shrink)