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

On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of $\mathbb{N}^n$ with adjoined zero