Increasing sequences of principal left ideals of $\beta \mathbb{Z}$ are finite
Volume 258 / 2022
Fundamenta Mathematicae 258 (2022), 225-235
MSC: Primary 22A15, 54H20; Secondary 05D10, 54D80.
DOI: 10.4064/fm17-8-2021
Published online: 10 January 2022
Abstract
We show that increasing sequences of principal left ideals of $\beta \mathbb {Z}$ are finite. As a consequence, $\beta \mathbb {Z}\setminus \mathbb {Z}$ is a disjoint union of maximal principal left ideals of $\beta \mathbb {Z}$. Another consequence is that increasing chains of idempotents ($p\le q\Leftrightarrow p+q=q+p=p$) in $\beta \mathbb {Z}$ are finite. All these are answers to long-standing open questions.
