On the dichotomy of a locally compact semitopological bicyclic monoid with adjoined zero

Abstract

We prove that a Hausdorff locally compact semitopological bicyclic semigroup with adjoined zero ${\mathcal{C}}^0$ is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological bicyclic semigroup with an adjoined compact ideal and construct an example which witnesses that a counterpart of the statements does not hold when ${\mathcal{C}}^0$ is a Čech-complete metrizable topological inverse semigroup.

7 pages. Submitted to Visnyk Lviv Univ. Ser. Mech. Math

Cited by in corpus (11)

• Classifying locally compact semitopological polycyclic monoids
• Topological properties of Taimanov semigroups
• On semitopological interassociates of the bicyclic monoid
• On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers
• On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of ${\mathbb{N}}^n$ with adjoined zero
• On the lattice of weak topologies on the bicyclic monoid with adjoined zero
• On a semitopological polycyclic monoid
• The monoid of monotone injective partial selfmaps of the poset $({\mathbb{N}}^3,⩽)$ with cofinite domains and images
• On variants of the extended bicyclic semigroup
• The monoid of order isomorphisms of principal filters of a power of the positive integers
• On the semigroup ${\text{ID}}_{\infty }$