Unavoidable subprojections in union-closed set systems of infinite breadth

We consider union-closed set systems with infinite breadth, focusing on three particular configurations ${\mathcal{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}(E)$, ${\mathcal T}{\rm min}(E)$and${\mathcal T}{\rm ort}(E)$. We show that these three configurations are not isolated examples; in any given union-closed set system of infinite breadth, at least one of these three configurations will occur as a subprojection. This characterizes those union-closed set systems which have infinite breadth, and is the first general structural result for such set systems.