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

Abstract

Let $n$ be any positive integer and $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$ 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 $\mathcal{I}\mathcal{P}\mathcal{F}({\mathbb{N}}^n)$ with an adjoined zero is either compact or discrete.