Factoring a minimal ultrafilter into a thick part and a syndetic part
Volume 252 / 2021
Abstract
Let $S$ be an infinite discrete semigroup. The operation on $S$ extends uniquely to its Stone–Čech compactification $\beta S$, making $\beta S$ a compact right topological semigroup with $S$ contained in its topological center. As such, $\beta S$ has a smallest two-sided ideal, $K(\beta S)$. An ultrafilter $p$ on $S$ is minimal if and only if $p\in K(\beta S)$.
We show that any minimal ultrafilter $p$ factors into a thick part and a syndetic part. That is, there exist filters ${\mathcal F} $ and ${\mathcal G} $ such that ${\mathcal F} $ consists only of thick sets, ${\mathcal G} $ consists only of syndetic sets, and $p$ is the unique ultrafilter containing ${\mathcal F} \cup {\mathcal G} $.
If $L=\widehat {\mathcal F} $ and $C=\widehat {\mathcal G} $ are the sets of ultrafilters containing ${\mathcal F} $ and ${\mathcal G} $ respectively, then $L$ is a minimal left ideal of $\beta S$, $C$ meets every minimal left ideal of $\beta S$ in exactly one point, and $L\cap C=\{p\}$. We show further that $K(\beta S)$ can be partitioned into relatively closed sets, each of which meets each minimal left ideal in exactly one point.
With some weak cancellation assumptions on $S$, we also prove that for each minimal ultrafilter $p$, $S^*\setminus \{p\}$ is not normal. In particular, if $p$ is a member of either of the disjoint sets $K(\beta \mathbb{N} ,+)$ or $K(\beta \mathbb{N} ,\cdot )$, then $\mathbb{N} ^*\setminus \{p\}$ is not normal.