  This paper considers the word problem for free inverse monoids of finite rank
from a language theory perspective. It is shown that no free inverse monoid has
context-free word problem; that the word problem of the free inverse monoid of
rank $1$ is both $2$-context-free (an intersection of two context-free
languages) and ET0L; that the co-word problem of the free inverse monoid of
rank $1$ is context-free; and that the word problem of a free inverse monoid of
rank greater than $1$ is not poly-context-free.
