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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2015, Volume 19, Issue 2, Pages 254–269 (Mi adm521)

RESEARCH ARTICLE

Ultrafilters on $G$-spaces

O. V. Petrenko, I. V. Protasov

Department of Cybernetics, Taras Shevchenko National University

Abstract: For a discrete group $G$ and a discrete $G$-space $X$, we identify the Stone-Čech 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 $\omega$.

Keywords: $G$-space, ultrafilters, ultracompanion, $G$-selective ultrafilter, $G$-Ramsey ultrafilter, $T$-point, ballean, asymorphism.

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Bibliographic databases:
MSC: 05D10, 22A15, 54H20
Revised: 26.06.2015
Language:

Citation: O. V. Petrenko, I. V. Protasov, “Ultrafilters on $G$-spaces”, Algebra Discrete Math., 19:2 (2015), 254–269

Citation in format AMSBIB
\Bibitem{PetPro15}
\by O.~V.~Petrenko, I.~V.~Protasov
\paper Ultrafilters on $G$-spaces
\jour Algebra Discrete Math.
\yr 2015
\vol 19
\issue 2
\pages 254--269
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3376354}