Characterization of irreducible lower Eulerian posets

Characterize all lower Eulerian posets X for which every strong formal subdivision from X to a lower Eulerian poset Y with fewer elements than X is impossible, equivalently, for which every strong formal subdivision with source X is an isomorphism of posets.

Background

The paper observes that the one- and two-element lower Eulerian posets B_0 and B_1 admit no strong formal subdivision to a lower Eulerian poset with strictly fewer elements. It also develops parity and boundary constraints on morphisms in the category of lower Eulerian posets.

The unresolved question asks for a complete characterization of all lower Eulerian posets exhibiting this rigidity and for structural information about the resulting category of strong formal subdivisions.

References

Which lower Eulerian posets $X$ have the property that they do not admit a strong formal subdivision $\sigma: X \to Y$ for some $Y$ with $|Y| < |X|$ (equivalently, any strong formal subdivision $\sigma: X \to Y$ is an isomorphism of posets)? What can be said about the structure of the category $\EulPos$?

— Subdivisions of lower Eulerian posets  (2511.16608 - Stapledon, 20 Nov 2025) in Section 10, Section 'Generalizations and further questions'