Completely decomposable groups with prescribed invariants (Q1271928)

Let \(L\) be an arbitrary finite lattice of types. A function \(f\colon L\to\mathbb{N}\) is called admissible if there exists a completely decomposable torsion free abelian group such that \(\text{typeset}(G)\subseteq L\) and \(\text{rank}(G(\tau))=f(\tau)\) for all \(\tau\in L\), where \(G(\tau)=\{x\in G:\text{type}(x)\geq\tau\}\) is the \(\tau\)-socle of \(G\). In this case the group \(G\) admits \(f\). The authors give a technical necessary and sufficient condition for a function \(f\colon L\to\mathbb{N}\) to be admissible and derive a recursive algorithm which produces the completely decomposable group \(G\) that admits \(f\). Moreover the authors apply their methods to certain classes of Butler-groups called \(L\)-bracket-groups and \(L\)-Butler-groups. Recall that a Butler-group is a pure subgroup of a finite rank completely decomposable (c.d.) group or equivalently a torsion free homomorphic image of a c.d. group. Characterizations for \(L\)-bracket-groups and \(L\)-Butler-groups to be almost completely decomposable are given.