Longer chains of idempotents in $\beta G$
Volume 220 / 2013
Abstract
Given idempotents $e$ and $f$ in a semigroup, $e \leq f$ if and only if $e = f e = e f$. We show that if $G$ is a countable discrete group, $p$ is a right cancelable element of $G^*=\beta G\setminus G$, and $\lambda $ is a countable ordinal, then there is a strictly decreasing chain $\langle q_\sigma \rangle _{\sigma <\lambda }$ of idempotents in $C_p$, the smallest compact subsemigroup of $G^*$ with $p$ as a member. We also show that if $S$ is any infinite subsemigroup of a countable group, then any nonminimal idempotent in $S^*$ is the largest element of such a strictly decreasing chain of idempotents. (It had been an open question whether there was a strictly decreasing chain $\langle q_\sigma \rangle _{\sigma <\omega +1}$ in ${\mathbb N}^*$.) As other corollaries we show that if $S$ is an infinite right cancellative and weakly left cancellative discrete semigroup, then $\beta S$ contains a decreasing chain of idempotents of reverse order type $\lambda $ for every countable ordinal $\lambda $ and that if $S$ is an infinite cancellative semigroup then the set $U(S)$ of uniform ultrafilters contains such decreasing chains.
