Further obstructions to non-minimal join-admissible elements

Identify obstructions, beyond the parity obstruction established in Corollary~\ref{cor:oddequaleven}, to a lower Eulerian poset containing a non-minimal join-admissible element, equivalently to being the non-Hausdorff mapping cylinder of a strong formal subdivision between lower Eulerian posets.

Background

The main bijection shows that a lower Eulerian poset is the non-Hausdorff mapping cylinder of a strong formal subdivision precisely when it contains a non-minimal join-admissible element. The paper proves one obstruction: such a poset must contain equally many elements of even and odd rank and therefore have even cardinality.

The authors ask whether additional structural or numerical obstructions exist that would further characterize which lower Eulerian posets arise as mapping cylinders.