  Let $n$ be any positive integer and $\mathscr{I\!P\!F}(\mathbb{N}^n)$ be the
semigroup of all order isomorphisms between principal filters of the $n$-th
power of the set of positive integers $\mathbb{N}$ with the product order. We
prove that a Hausdorff locally compact semitopological semigroup
${\mathscr{I\!P\!F}(\mathbb{N}^n)}$ with an adjoined zero is either compact or
discrete.
