Complete distributivity in lattice-ordered groups
Richard Byrd · Pacific Journal of Mathematics · 1967
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 ("o-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 1-subgroup of all o-permutations of a totally ordered set and that this 1isomorphism preserves all joins and meets if and only if G possesses a collection of closed prime subgroups whose intersection contains no nonzero 1-ideal.Both theorems lead to results concerning complete distributivity.