On Finite Presentability of Subsemigroups of the Monogenic Free Inverse Semigroup

The monogenic free inverse semigroup $F{I}_1$ is not finitely presented as a semigroup due to the classic result by Schein (1975). We extend this result and prove that a finitely generated subsemigroup of $F{I}_1$ is finitely presented if and only if it contains only finitely many idempotents. As a consequence, we derive that an inverse subsemigroup of $F{I}_1$ is finitely presented as a semigroup if and only if it is a finite semilattice.