Selections of lattice-ordered Groups

Abstract

It is well known that the quasitorsion class of archimedean \(\ell\)-groups is the class of \(\ell\)-groups G such that every closed convex \(\ell\)-subgroup is a polar, and it is also well known that the class of \(\ell\)-groups G such that every convex \(\ell\)-subgroup is a polar is a torsion class. By defining a selection on \(\ell\)-groups, these two results are generalized to show, whenever \({\mathcal{S}}_{1}\) and \({\mathcal{S}}_{2}\) are selections on \(\ell\)-groups, the class of \(\ell\)-groups G such that \({\mathcal{S}}_{1}(G) = {\mathcal{S}}_{2}(G)\) is a radical class. Three selections in particular — all convex \(\ell\)-subgroups, all polars, and all closed convex \(\ell\)-subgroups — and the radical classes determined by them are studied in some detail.