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Long increasing chains of idempotents in $\beta G$

Volume 240 / 2018

Neil Hindman, Dona Strauss Fundamenta Mathematicae 240 (2018), 1-13 MSC: Primary 54D35; Secondary 22A15. DOI: 10.4064/fm363-2-2017 Published online: 23 June 2017


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 $q_0=q$.


  • Neil HindmanDepartment of Mathematics
    Howard University
    Washington, DC 20059, U.S.A.
  • Dona StraussDepartment of Pure Mathematics
    University of Leeds
    Leeds LS2 9J2, UK

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