Longer chains of idempotents in $\beta G$

Neil Hindman ¹   ; Dona Strauss ²   ; Yevhen Zelenyuk ³

¹ Department of Mathematics Howard University Washington, DC 20059, U.S.A.
² Department of Pure Mathematics University of Leeds Leeds LS2 9J2, UK
³ School of Mathematics University of the Witwatersrand Private Bag 3 Wits 2050, South Africa

Fundamenta Mathematicae, Tome 220 (2013) no. 3, pp. 243-261

Résumé

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.

Détail

DOI : 10.4064/fm220-3-5

Keywords: given idempotents semigroup leq only countable discrete group right cancelable element * beta setminus lambda countable ordinal there strictly decreasing chain langle sigma rangle sigma lambda idempotents smallest compact subsemigroup * member infinite subsemigroup countable group nonminimal idempotent * largest element strictly decreasing chain idempotents had question whether there strictly decreasing chain langle sigma rangle sigma omega mathbb * other corollaries infinite right cancellative weakly cancellative discrete semigroup beta contains decreasing chain idempotents reverse order type lambda every countable ordinal lambda infinite cancellative semigroup set uniform ultrafilters contains decreasing chains