On homological finiteness properties and free inverse monoids

We construct a simple and useful sufficient condition, based on actions on a lattice of idempotents, for monoids admitting homomorphisms to the monogenic free inverse monoid $\mathrm{FIM}(1)$ to not be of type $\mathrm{FP}_2$. This recovers a result of Gray and Steinberg that free inverse monoids are not of type $\mathrm{FP}_2$. The same technique is then used to show that a finitely generated submonoid of $\mathrm{FIM}(1)$ is of type $\mathrm{FP}_2$ if and only if it is finitely presented, answering a question of Cho & Ru\v{s}kuc.