On lattices and their ideal lattices, and posets and their ideal posets

For P a poset or lattice, let Id(P ) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P ) = Id(P )−{∅}. This note obtains various results to the effect that Id(P ) is always, and id(P ) often, “essentially larger” than P. In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone map) from Id(P ) into P, and, going the other way, that an upper semilattice P admits no semilattice homomorphism from any subsemilattice of itself onto