Embedding ω-continuous posets in function spaces of domains

We prove a number of results, including the following: IfD is an ω-algebraic poset which is not an antichain, andE is an ω-continuous poset, then there exists a Scott-continuous order-embedding fromE to D?D]; in particular, there exists a Scott-continuous order-embedding from (P(ω), ⊂-) to D?D]. The proof of this main result does not use the Axiom of Choice.