On semitopological interassociates of the bicyclic monoid

Abstract

Semitopological interassociates ${\mathcal{C}}_{m,n}$ of the bicyclic semigroup $\mathcal{C}(p,q)$ are studied. In particular, we show that for arbitrary non-negative integers $m$, $n$ and every Hausdorff topology $\tau$ on ${\mathcal{C}}_{m,n}$ such that $\left({\mathcal{C}}_{m,n},\tau \right)$ is a semitopological semigroup, is discrete. Also, we prove that if an interassociate of the bicyclic monoid ${\mathcal{C}}_{m,n}$ is a dense subsemigroup of a Hausdorff semitopological semigroup $(S,\cdot )$ and $I=S\setminus {\mathcal{C}}_{m,n}\ne \varnothing$ then $I$ is a two-sided ideal of the semigroup $S$ and show that for arbitrary non-negative integers $m$, $n$, any Hausdorff locally compact semitopological semigroup ${\mathcal{C}}_{m,n}^0={\mathcal{C}}_{m,n}\bigsqcup \{0\}$ is either discrete or compact.

8 pages

References in corpus (1)

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

Cited by in corpus (6)

• 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 semigroup ${\text{ID}}_{\infty }$
• 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