Torsion Classes of Vector Lattices

Let Vl be the class of all vector lattices, and let S and T be torsion classes of ℓ-groups. T ∩ Vl 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 T ∩ Vl is a torsion class, so is S ∩ T ∩ Vl. The following classes of vector lattices form torsion classes: the hyperarchimedean vector lattices; the finite-valued vector lattices; the class of all vector lattices of the form Σ(Δ,R). In particular, the principal torsion class $$ \tilde \sum (\Delta, R) $$ determined by Σ(Δ,R) consists of vector lattices; it consists of all cardinal sums of ℓ-groups Σ(Λ, R) where Λ is a direct limit of connected, convex subsets of Λ. 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, then 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.