Factoring a minimal ultrafilter into a thick part and a syndetic part

Abstract

Let $S$ be an infinite discrete semigroup. The operation on $S$ extends uniquely to the Stone-\v{C}ech 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 \emph{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 $\mathcal F \cup \mathcal G$. Letting $L = \widehat{\mathcal F}$ and $C = \widehat{\mathcal G}$, the sets of ultrafilters containing $\mathcal F$ and $\mathcal G$ respectively, we have that $L$ 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 $S$, one has also 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.