Strongly complete almost maximal left invariant topologies on groups (Q390427)

In this paper \(G\) is an arbitrary countable discrete group. Every left invariant topology \(\mathcal{T}\) on \(G\) determines a closed subsemigroup \(\mathrm{Ult}(\mathcal{T})\subseteq \beta G\). It is called the ultrafilter semigroup of \(\mathcal{T}\). An object \(P\) is an absolute coretract if for every surjective morphism \(g:R\rightarrow P\) there exists a morphism \(h:P\rightarrow R\) such that \(g\circ h=id_P\). Let \(\mathfrak{C}\) denote the category of compact Hausdorff right topological semigroups. In this paper two important Theorems are proved.NEWLINENEWLINE Theorem 1: Assume MA (Martin's Axiom). Then for every finite absolute coretract \(S\) in \(\mathfrak{C}\), there is a strongly complete left invariant topology \(\mathcal{T}\) on \(G\) such that \(\mathrm{Ult}(\mathcal{T})\) is isomorphic to \(S\), and in the case \(G=\bigoplus_\omega \mathbb{Z}_2\), \(\mathcal{T}\) can be chosen to be a group topology.NEWLINENEWLINE Theorem 2: It is consistent with \(ZFC\) that if \(G\) can be embedded into a compact group, then there is no \(d-\)complete almost maximal left invariant topology on \(G\).NEWLINENEWLINE As a consequence, they obtain from Theorem 1 the followingNEWLINENEWLINE Corollary: Assume MA. Then there is an absolutely maximal idempotent \(p\in G^*\) such that \((\beta G)p\in \mathcal{I}(G)\), where \(\mathcal{I}(G)\) denotes the finest decomposition of \(G^*\) into closed left ideals of \(\beta G\) with the property that the corresponding quotient space of \(G^*\) is Hausdorff.