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

TL;DR

This paper demonstrates that minimal ultrafilters on an infinite discrete semigroup can be uniquely decomposed into a thick part and a syndetic part, revealing structural insights into the minimal ideal and ultrafilter behavior.

Contribution

It introduces a novel factorization of minimal ultrafilters into thick and syndetic components, linking ultrafilter structure to the topology of the Stone-ish compactification.

Findings

Minimal ultrafilters factor into thick and syndetic parts.

The minimal ideal can be partitioned into relatively closed sets with specific intersection properties.

Under weak cancellation, certain ultrafilters are shown to have non-normal complements.

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.