Throughout this note let G be a
lattice-ordered group (1-group”). G is said to be representable if there exists an
1-isomorphism of G into a cardinal sum of totally ordered groups ( 0-groups”).
The main result of §3 establishes five conditions in terms of certain convex
1-subgroups each of which is equivalent to representability. In §4 it is shown that
there is an 1-isomorphism of G onto a subdirect product of 1-groups where
each 1-group is a transitive l-subgroup of all o-permutations of a totally
ordered set and that this 1-isomorphism preserves all joins and meets if and
only if G possesses a collection of closed prime subgroups whose intersection
contains no nonzero l-ideal. Both theorems lead to results concerning complete
distributivity.
