On a locally compact monoid of cofinite partial isometries of $\mathbb{N}$ with adjoined zero

Abstract

Let ${\mathcal{C}}_{\mathbb{N}}$ be a monoid which is generated by the partial shift $\alpha :n\mapsto n+1$ of the set of positive integers $\mathbb{N}$ and its inverse partial shift $\beta :n+1\mapsto n$. In this paper we prove that if $S$ is a submonoid of the monoid $\mathbf{I}{\mathbb{N}}_{\infty }$ of all partial cofinite isometries of positive integers which contains ${\mathcal{C}}_{\mathbb{N}}$ as a submonoid then every Hausdorff locally compact shift-continuous topology on $S$ with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup $S$ with an adjoined compact ideal.

arXiv comment15 pages. arXiv admin note: text overlap with arXiv:2104.14149, arXiv:2008.03159

References in corpus (4)

• On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers
• Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image
• On the monoid of cofinite partial isometries of $\mathbb{N}$ with a bounded finite noise
• The semigroup of partial co-finite isometries of positive integers

Cited by in corpus (1)

• On a semitopological semigroup ${\mathbf{B}}_{\omega }^{\mathcal{F}}$ when a family $\mathcal{F}$ consists of inductive non-empty subsets of $\omega$