Stacked bases for a pair of homogeneous completely decomposable groups with bounded quotient

Abstract

Let A be a homogeneous completely decomposable torsion free group of infinite rank κ and let X be a torsion free abelian group containing A such that the quotient X/A is bounded. We show that there exist stacked bases for X and A, i.e. there exist b i ∈ X (i ∈ κ) and d i ∈ ℤ (i ∈ κ) such that \( X = \mathop \oplus \limits_{i \in k} \left\langle {{b_i}} \right\rangle _*^X \) and \( A = \mathop \oplus \limits_{i \in k} {d_i}\left\langle {{b_i}} \right\rangle _*^A \). This proves a stacked bases theorem for pairs of homogeneous completely decomposable torsion free abelian groups of infinite rank with bounded quotient.