Abstract
We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups Hiwhich are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups Hican 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(Si) of infinite linearly ordered sets (Si, ≤) such that each group A(Si) 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.
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