Let V be the class of all vector lattices, and let S and T be torsion classes of l-groups. TÇ V is a torsion class if and only if each divisible abelian l-group in T contains a largest l-ideal that is a vector lattice. Moreover, if TÇ V is a torsion class, so is SÇ TÇ V. The following classes of vector lattices form torsion classes: the hyperarchimedean vector lattices; the finite-valued vector lattices. In particular, the principal torsion class S (D , R) determined by S (D ,R); it consists of all cardinal sums of l-groups S (L ,R) where L is a direct limit of connected, convex subsets of D .

The following classes of vector lattices form pseudo torsion classes: the archimedean l-groups; the special-valued and conditionally laterally complete l-groups. Underlying this theory is the fact that if K is a finite-valued l-group or a conditionally laterally complete l-group, the K is a vector lattice if and only if each K(k) is a vector lattice, which is true if and only if each K(k), with k a special element, is a vector lattice.