  We prove that the equality problem is decidable for rational subsets of the
monogenic free inverse monoid $F$. It is also decidable whether or not a
rational subset of $F$ is recognizable. We prove that a submonoid of $F$ is
rational if and only if it is finitely generated. We also prove that the
membership problem for rational subsets of a finite $\mathcal{J}$-above monoid
is decidable, covering the case of free inverse monoids.
