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 IPF (N n ) be the semigroup of all order isomorphisms between principal filters of the n-th power of the set of positive integers N with the product order. We prove that a Hausdorff locally compact semitopological semigroup IPF (N n ) with an adjoined zero is either compact or discrete.

On the lattice of weak topologies on the bicyclic monoid with adjoined zero

“…In [6] and [8] this result was extended for polycyclic monoids and graph inverse semigroups over strongly connected graphs with finitely many vertices, respectively. Similar dichotomy also holds for other generalizations of the bicyclic monoid (see [9,24,30]).…”