Right coherence of free left h-Ehresmann monoids

Determine whether free left h-Ehresmann monoids are right coherent.

Background

The paper establishes that free left h-Ehresmann monoids are left and right ideal Howson, finitely left and right equated, and weakly left and right coherent. It also proves that, in ranks at least two, they are not left coherent. Full right coherence remains unresolved: free left Ehresmann monoids are known not to be left coherent in rank at least two, while their right coherence is unknown, and free left h-Ehresmann monoids lie between the relevant Ehresmann and ample classes.

The question is motivated by the contrast that free left ample monoids are right coherent in all ranks, whereas right coherence has not been established for free left Ehresmann monoids or for the intermediate free left h-Ehresmann monoids. In the monogenic case, both left and right coherence remain open.

References

As previously mentioned, it is open whether free left Ehresmann monoids are right coherent. On the other hand, free left ample monoids in all ranks are right coherent Theorem 5.7. With left $h$-Ehresmann monoids sitting between these classes, a natural question arises.

Are free left $h$-Ehresmann monoids right coherent?

— Free left $h$-Ehresmann semigroups  (2608.24657 - Heath, 25 Aug 2026) in Section 5, Section “Open Questions”