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

 Algebra Discrete Math., 2014, Volume 17, Issue 1, Pages 98–109 (Mi adm461)

RESEARCH ARTICLE

On the subset combinatorics of $G$-spaces

Igor Protasova, Sergii Slobodianiukb

a Department of Cybernetics, Kyiv National University, Volodymyrska 64, 01033, Kyiv, Ukraine
b Department of Mechanics and Mathematics, Kyiv National University, Volodymyrska 64, 01033, Kyiv, Ukraine

Abstract: Let $G$ be a group and let $X$ be a transitive $G$-space. We classify the subsets of $X$ with respect to a translation invariant ideal $J$ in the Boolean algebra of all subsets of $X$, introduce and apply the relative combinatorical derivations of subsets of $X$. Using the standard action of $G$ on the Stone-Čech compactification $\beta X$ of the discrete space $X$, we characterize the points $p\in\beta X$ isolated in $Gp$ and describe a size of a subset of $X$ in terms of its ultracompanions in $\beta X$. We introduce and characterize scattered and sparse subsets of $X$ from different points of view.

Keywords: $G$-space, relative combinatorial derivation, Stone-Čech compactification, ultracompanion, sparse and scattered subsets.

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Bibliographic databases:
MSC: 20F69, 22A15, 54D35
Revised: 15.01.2014
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Citation: Igor Protasov, Sergii Slobodianiuk, “On the subset combinatorics of $G$-spaces”, Algebra Discrete Math., 17:1 (2014), 98–109

Citation in format AMSBIB
\Bibitem{ProSlo14}
\by Igor~Protasov, Sergii~Slobodianiuk
\paper On the subset combinatorics of $G$-spaces
\jour Algebra Discrete Math.
\yr 2014
\vol 17
\issue 1
\pages 98--109
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3288187}

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This publication is cited in the following articles:
1. Protasov I. Slobodianiuk S., “Partitions of Groups”, Adv. Appl. Discret. Math., 15:1 (2015), 33–60
2. O. V. Petrenko, I. V. Protasov, “Ultrafilters on $G$-spaces”, Algebra Discrete Math., 19:2 (2015), 254–269
3. I. V. Protasov, S. V. Slobodianiuk, “Ultrafilters on balleans”, Ukr. Math. J., 67:12 (2016), 1922–1931
4. I. Protasov, K. Protasova, “Ideals in $\mathcal{P}_G$ and $\beta G$”, Topology Appl., 238 (2018), 24–31
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