Ultrafilters on \(G\)-spaces

• Datum: 2015
• Hauptverfasser: Petrenko, O. V., Protasov, I. V.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Lugansk National Taras Shevchenko University 2015
• Schlagworte: \(G\)-space; ultrafilters; ultracompanion; \(G\)-selective ultrafilter; \(G\)-Ramsey ultrafilter; \(T\)-point; ballean; asymorphism; 05D10; 22A15; 54H20
• Online Zugang: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/69
• Назва журналу: Algebra and Discrete Mathematics

For a discrete group \(G\) and a discrete \(G\)-space \(X\), we identify the Stone-Cech compactifications \(\beta G\) and \(\beta X\) with the sets of all ultrafilters on \(G\) and \(X\), and apply the natural action of \(\beta G\) on \(\beta X\) to characterize large, thick, thin, sparse and scattered subsets of \(X\). We use \(G\)-invariant partitions and colorings to define \(G\)-selective and \(G\)-Ramsey ultrafilters on \(X\). We show that, in contrast to the set-theoretical case, these two classes of ultrafilters are distinct. We consider also universally thin ultrafilters on \(\omega\), the \(T\)-points, and study interrelations between these ultrafilters and some classical ultrafilters on \(\omega\).