Complete embeddings of linear orderings and embeddings of lattice-ordered groups

An infinite linearly ordered set (S,≦) is called doubly homogeneous, if its automorphism group Aut(S,≦) acts 2-transitively on it. We study embeddings of linearly ordered sets into Dedekind-completions of doubly homogeneous chains which preserve all suprema and infima, and obtain necessary and sufficient conditions for the existence of such embeddings. As one of several consequences, for each lattice-ordered groupG and each regular uncountable cardinalκ≧|G | there are 2⋉ non-isomorphic simple divisible lattice-ordered groupsH of cardinalityκ all containingG as anl-subgroup.