  Ordering theorems, characterizing when partial orders of a group extend to
total orders, are used to generate hypersequent calculi for varieties of
lattice-ordered groups (l-groups). These calculi are then used to provide new
proofs of theorems arising in the theory of ordered groups. More precisely: an
analytic calculus for abelian l-groups is generated using an ordering theorem
for abelian groups; a calculus is generated for l-groups and new decidability
proofs are obtained for the equational theory of this variety and extending
finite subsets of free groups to right orders; and a calculus for representable
l-groups is generated and a new proof is obtained that free groups are
orderable.
