  A characterization is given of the subsets of a group that extend to the
positive cone of a right order on the group and used to relate validity of
equations in lattice-ordered groups (l-groups) to subsets of free groups that
extend to positive cones of right orders. This correspondence is used to obtain
new proofs of the decidability of the word problem for free l-groups and
generation of the variety of l-groups by the l-group of automorphisms of the
real number line. A characterization of the subsets of a group that extend to
the positive cone of an order on the group is also used to establish a
correspondence between the validity of equations in varieties of representable
l-groups (equivalently, classes of ordered groups) and subsets of relatively
free groups that extend to positive cones of orders.
