Torsion Radicals and Torsion Classes of Cyclically Ordered Groups

The notion of torsion radical of cyclically ordered groups is defined analogously as in the case of lattice ordered groups. We denote by T the collection of all torsion radicals of cyclically ordered groups. For τ1, τ2 ∈ T, we put τ1 ∈ τ2 if τ1(G) □ τ2(G) for each cyclically ordered group G. We show that T is a proper class; nevertheless, we apply for T the usual terminology of the theory of partially ordered sets. We prove that T is a complete completely distributive lattice. The analogous result fails to be valid for torsion radicals of lattice ordered groups. Further, we deal with products of torsion classes of cyclically ordered groups.