Regular idempotents in $\beta S$

Abstract:

Let $S$ be a discrete semigroup and let $\beta S$ be the Stone-Čech compactification of $S$. We take the points of $\beta S$ to be the ultrafilters on $S$. Being a compact Hausdorff right topological semigroup, $\beta S$ has idempotents. Every idempotent $p\in \beta S$ determines a left invariant topology $\mathcal {T}_p$ on $S$ with a neighborhood base at $a\in S$ consisting of subsets $aB\cup \{a\}$, where $B\in p$. If $S$ is a group and $p$ is an idempotent in $S^*=\beta S\setminus S$, $(S,\mathcal {T}_p)$ is a homogeneous Hausdorff maximal space. An idempotent $p\in \beta S$ is regular if $p$ is uniform and the topology $\mathcal {T}_p$ is regular. We show that for every infinite cancellative semigroup $S$, there exists a regular idempotent in $\beta S$. As a consequence, we obtain that for every infinite cardinal $\kappa$, there exists a homogeneous regular maximal space of dispersion character $\kappa$. Another consequence says that there exists a translation invariant regular maximal topology on the real line of dispersion character $\mathfrak {c}$ stronger than the natural topology.