Relatively thin and sparse subsets of groups.
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.
Stone-Čech compactifications of discrete groups
scientific article; zbMATH DE number 6015579
scientific article