A lattice-ordered group (“l-group”) G will be called
a P-group if G = g″ ⊕ g′ for each g ∈ G (projectable)
an SP-group if G = C ⊕ C′ for each polar C of G (strongly projectable)
an L-group if each disjoint subset has a 1. u. b. (laterally complete)
an O group if it is both an L-group and a P-group (orthocomplete).
G is representable if it is an l-subgroup of a cardinal product of totally ordered groups. It follows that a P-group must be representable and hence SP-groups and O-groups are also representable.