On definability of completely decomposable torsion-free Abelian groups by certain groups of homomorphisms

Let C be an Abelian group. An Abelian group A from a class X of Abelian groups is said to be _C H-definable in X if, for any group B ∈ X, the isomorphism Hom(C,A) ≅ Hom(C,B) implies that A ≅ B. If every group from X is _C H-definable in X, then X is called an _C H-class. In this paper, we study conditions under which a class of completely decomposable torsion-free Abelian groups is an _C H-class, where C is a vector group.