On the atomic conditions of lattice-ordered groups

We introduce large convex l-subgroups to study the structure of the lattice-ordered groups (l-groups) G whose C(G), P(G) and Gamma(G) satisfy atomic conditions, where C(G), P(G) and Gamma(G) denote respectively the lattice of all convex l-subgroups, the lattice of all polar subgroups and the root system of all regular subgroups of G. In particular, we construct a new torsion class S defined as the class of l-groups G for which all large prime subgroups are maximal. We prove that the class of hyperarchimedean l-groups is properly contained within epsilon, and that any l-group within epsilon has the property that any chain of prime subgroups has length at most 2.