Relatively thin and sparse subsets of groups.
type: Item
instance of: scholarly article
title: Relatively thin and sparse subsets of groups.
zbMATH Open document ID: 1258.20027
author: Ie. Lutsenko
author: Igor V. Protasov
published in: Ukrainian Mathematical Journal
publication date: 2012-03-19T00:00:00Z
full work available at URL: http://dspace.nbuv.gov.ua/handle/123456789/166348
review text: Let \(G\) be a group, \(\mathcal P_G\) the Boolean algebra of all subsets of \(G\), and \(\mathcal I\) a left invariant ideal of \(\mathcal P_G\). A subset \(A\subset G\) is called `\(\mathcal I\)-thin' if \(gA\cap A\in\mathcal I\) for all \(e_G\neq g\in G\). A subset \(B\) of \(G\) is called `\(\mathcal I\)-sparse' if for every infinite subset \(S\) of \(G\) there exists a finite subset \(F\subset S\) such that \(\bigcap_{g\in F}gB\in\mathcal I\). In the paper are studied properties of \(\mathcal I\)-thin and \(\mathcal I\)-sparse subsets of an arbitrary group \(G\). The authors consider the Stone-Čech compactification \(\beta G\) for every discrete group \(G\) and the structure of a compact right topological semigroup on \(\beta G\) induced by the multiplication in \(G\). This compact right topological semigroup is used to prove some properties of \(\mathcal I\)-thin sets.
reviewed by: Mihail I. Ursul
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20A05
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 06E15
Mathematics Subject Classification ID: 20F69
zbMATH DE Number: 6015579
zbMATH Keywords: left invariant families of subsets
zbMATH Keywords: downward closed families of subsets
zbMATH Keywords: additive families of subsets
zbMATH Keywords: ideals
zbMATH Keywords: large subsets
zbMATH Keywords: small subsets
zbMATH Keywords: thick subsets
zbMATH Keywords: sparse subsets
zbMATH Keywords: Stone-Čech compactifications of discrete groups
zbMATH Keywords: compact right topological semigroups
MaRDI profile type: Publication
OpenAlex ID: W1986973211
cites work: SPARSE, THIN AND OTHER SUBSETS OF GROUPS
cites work: Completeness of invariant ideals in groups
cites work: On thin-complete ideals of subsets of groups.
cites work: Algebra in the Stone-Čech compactification: theory and applications
cites work: Q4940870
cites work: Boolean group ideals and the ideal structure of \(\beta G\).
DOI: 10.1007/S11253-011-0502-3
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description: scientific article; zbMATH DE number 6015579