Presentations for wreath products involving symmetric inverse monoids and categories

Wreath products involving symmetric inverse monoids/semigroups/categories arise in many areas of algebra and science, and presentations by generators and relations are crucial tools in such studies. The current paper finds such presentations for $M\wr\mathcal I_n$, $M\wr\operatorname{Sing}(\mathcal I_n)$ and $M\wr\mathcal I$. Here $M$ is an arbitrary monoid, $\mathcal I_n$ is the symmetric inverse monoid, $\operatorname{Sing}(\mathcal I_n)$ its singular ideal, and $\mathcal I$ is the symmetric inverse category.