  We generalize the lexicographic product of first-order structures by
presenting a framework for constructions which, in a sense, mimic iterating the
lexicographic product infinitely and not necessarily countably many times. We
then define dense substructures in infinite products and show that any
countable product of countable transitive homogeneous structures has a unique
countable dense substructure, up to isomorphism. Furthermore, this dense
substructure is transitive, homogeneous and elementarily embeds into the
product. This result is then utilized to construct a rigid elementarily
indivisible structure.
