  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)-\{\emptyset\}. 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 S admits no semilattice homomorphism from any
subsemilattice of itself onto Id(S).
  The slightly smaller object id(P) is known to be isomorphic to P if and only
if P has ascending chain condition. This result is strengthened to say that the
only posets P_0 such that for every natural number n there exists a poset P_n
with id^n(P_n)\cong P_0 are those having ascending chain condition. On the
other hand, a wide class of cases is noted here where id(P) is embeddable in P.
  Counterexamples are given to many variants of the results proved.
