Every Hausdorff locally compact semitopological semigroup IPF(N^n) with an adjoined zero is either discrete or topologically isomorphic to the Alexandroff one-point compactification of the discrete semigroup.
Andersen, Ein Bericht ¨ uber die Struktur abstrakter Halbgruppen, PhD Thesis, Hamburg, 1952
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On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of $\mathbb{N}^n$ with adjoined zero
Every Hausdorff locally compact semitopological semigroup IPF(N^n) with an adjoined zero is either discrete or topologically isomorphic to the Alexandroff one-point compactification of the discrete semigroup.