Very large subgroups of a lattice-ordered group

This article introduces the concept of a very large subgroup in the theory of lattice-ordered groups. The existence of a minimal very large subgroup is connected to some previously known structure theory, but it is also linked to conditions not studied before. Very large subgroups are useful in studying torsion and radical classes, and among other things, extension of lattice-ordered groups using very large kernels yields an intriguing completion operation for torsion classes. In the final section there is a new contruction which produces a lattice-ordered group in which every value is essential, having no special values.