Let A be a torsion free abelian group. We say that a group K is a finite essential extension of A if K contains an essential subgroup of finite index which is isomorphic to A. Such K admits a representation as (A ℤ xkx)/ℤky where y = Nx + a for some k x k matrix N over Z and α ∈ Ak satisfying certain conditions of relative primeness and ℤk = {(α1,..., αk) : αi, ∈ ℤ}. The concept of absolute width of an f.e.e. K of A is defined and it is shown to be strictly smaller than the rank of A. A kind of basis substitution with respect to Smith diagonal matrices is shown to hold for homogeneous completely decomposable groups. This result together with general properties of our representations are used to provide a self contained proof that acd groups with two critical types are direct sum of groups of rank one and two.
Let A X be two homogeneous completely decomposable torsion free abelian groups of the same type t and of countable rank such that the quotient X =A is a direct sum of torsion cyclic groups and a homogeneous completely decomposable summand of type t. Furthermore assume that A and X have a common direct summand of countable rank. We show that there exist stacked bases for A and X , i.e. there existThis proves a stacked bases theorem for pairs of homogeneous completely decomposable torsion free abelian groups of countable rank and the same type with a large common summand.
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