Definability of Completely Decomposable Torsion-Free Abelian Groups by Semigroups of Endomorphisms and Groups of Homomorphisms

Abstract

Let C be an Abelian group. A class X of Abelian groups is called a CE• H-class if for any groups A, B ∈ X, it follows from the existence of isomorphisms E• (A) ≅ E• (B) and Hom(C,A) ≅ Hom(C,B) that there is an isomorphism A ≅ B. In this paper, conditions are studied under which the class \( {\mathfrak{I}}_{\mathrm{cd}}^{\mathrm{ad}} \) of completely decomposable almost divisible Abelian groups and class \( {\mathfrak{I}}_{\mathrm{cd}}^{\ast } \) of completely decomposable torsion-free Abelian groups A where Ω(A) contains only incomparable types are CE• H-classes, where C is a completely decomposable torsion-free Abelian group.