Relatively thin and sparse subsets of groups. (English. Russian original) Zbl 1258.20027

Lutsenko, Ie.; Protasov, I. V.

Ukr. Math. J. 63, No. 2, 254-265 (2011); translation from Ukr. Mat. Zh. 63, No. 2, 216-225 (2011).

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.

Reviewer: Mihail I. Ursul (Oradea)

MSC:

20F05	Generators, relations, and presentations of groups
20A05	Axiomatics and elementary properties of groups
22A05	Structure of general topological groups
54H11	Topological groups (topological aspects)
43A07	Means on groups, semigroups, etc.; amenable groups
06E15	Stone spaces (Boolean spaces) and related structures
20F69	Asymptotic properties of groups

Keywords:

left invariant families of subsets; downward closed families of subsets; additive families of subsets; ideals; large subsets; small subsets; thick subsets; sparse subsets; Stone-Čech compactifications of discrete groups; compact right topological semigroups

References:

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