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## Fundamenta Mathematicae

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

### Tom 252 / 2021

Fundamenta Mathematicae 252 (2021), 121-145 MSC: Primary 54D35, 54D80, 22A15; Secondary 06E15, 03E05. DOI: 10.4064/fm886-4-2020 Opublikowany online: 23 July 2020

#### Streszczenie

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.

#### Autorzy

• Will BrianDepartment of Mathematics and Statistics
University of North Carolina at Charlotte
9201 University City Blvd.
Charlotte, NC 28223, U.S.A.
e-mail
• Neil HindmanDepartment of Mathematics
Howard University
Washington, DC 20059, U.S.A.
e-mail

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