Normal subgroups and elementary theories of lattice-ordered groups

We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups H _i which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups H _i can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically nice properties of their normal subgroup lattices. Moreover, they can be taken to be 2-transitive automorphism groups A(S _i ) of infinite linearly ordered sets (S _i , ) such that each group A(S _i ) has only inner automorphisms. We also show that any countable l-group G can be l-embedded into a countable l-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B.