Stone duality for Dedekind \(\sigma\)-complete \(\ell\)-groups with order-unit
If \(G\) is a lattice-ordered group (\(\ell \)-group) then \(G\) is Dedekind \(\sigma \)-complete if every non-empty countable subset of \(G\) which is bounded above in \(G\) has a supremum in \(G\). Denote by \(\mathcal{DED}\) the category whose objects are Dedekind \(\sigma \)-complete \(\ell \)-groups with a distinguished order-unit and whose morphisms are the unit preserving \(\ell \)-group homomorphisms which, moreover, preserve all denumerable infima and suprema. The authors establish a categorical duality between \(\mathcal{DED}\) and a category \(\mathcal{W}\) consisting of basically disconnected compact Hausdorff spaces equiped with a system of closed sets. Furthemore, the restiction of this duality gives analogous topological representations for certain subclasses of \(\mathcal{DED}\).
scientific article; zbMATH DE number 5078152