Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms

Abstract

Let C be an Abelian group. An Abelian group A in some class $\mathcal{X}$ of Abelian groups is said to be _C H-definable in the class $\mathcal{X}$ if, for any group B\in $\mathcal{X}$, it follows from the existence of an isomorphism Hom(C,A) ≅ Hom(C,B) that there is an isomorphism A ≅ B. If every group in $\mathcal{X}$ is _C H-definable in $\mathcal{X}$, then the class $\mathcal{X}$ is called an _C H-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a _C H-class, where C is a completely decomposable torsion-free Abelian group.