Long increasing chains of idempotents in $\beta G$
Volume 240 / 2018
Fundamenta Mathematicae 240 (2018), 1-13
MSC: Primary 54D35; Secondary 22A15.
DOI: 10.4064/fm363-2-2017
Published online: 23 June 2017
Abstract
We show that there is a sequence $\langle p_\alpha \rangle _{\alpha \lt \omega _1}$ of idempotents in $(\beta \mathbb Z,+)$ with the property that whenever $\alpha \lt \delta \lt \omega _1$, then $p_\alpha \lt _R p_\delta $, where $p \lt _R q$ means that $p=q+p$ and $q\not =p+q$. More generally, if $G$ is any countably infinite discrete group, $p$ is an element of $\beta G\setminus G$ which is right cancelable in $\beta G$, and $q$ is any minimal idempotent in the smallest compact subsemigroup of $\beta G$ with $p$ as a member, then there is a sequence $\langle q_\alpha \rangle _{\alpha \lt \omega _1}$ of idempotents in $\beta G$ which is $ \lt _R$-increasing with $q_0=q$.
