A lattice of homomorphs (Q5906465)

If \(\mathcal H\) is a homomorph, we denote by \(D{\mathcal H}\) the class of all those groups that have \(\mathcal H\)-covering subgroups. \(D{\mathcal H}\) is also a homomorph. Let \(\mathbb{H}({\mathcal U}) = \{{\mathcal H} \mid D{\mathcal H} = {\mathcal U}\}\), where \(\mathcal U\) is a homomorph. The author studies the relation of ``strong containment'' \(\ll\) in \(\mathbb{H}({\mathcal U})\), defined by: \({\mathcal X} \ll {\mathcal Y}\) if, for each \(G \in {\mathcal U}\), an \(\mathcal X\)-covering subgroup of \(G\) is contained in some \(\mathcal Y\)-covering subgroup of \(G\). The main result gives a necessary and sufficient condition for \((\mathbb{H}({\mathcal U}),\ll)\) to be a lattice, where \(\mathcal U\) is a totally unsaturated homomorph.