Definability of Completely Decomposable Torsion-Free Abelian Groups by Endomorphism Semigroups and Homomorphism Groups

Abstract

Let C be an Abelian group. A class X of Abelian groups is called a _CE^•H-class if, for every groups A, B ∈ X, the isomorphisms E^•(A) ≅ E^•(B) and Hom(C, A) ≅ Hom(C, B) imply the isomorphism A ≅ B. In the paper, necessary and sufficient conditions on a completely decomposable torsion-free Abelian group C are described under which a given class of torsion-free Abelian groupsisa _CE^•H-class.