## On lattices and their ideal lattices, and posets and their ideal posets## George M. Bergman
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)-{Ø}.
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 Id(P).
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})≈P_{0} are those having
ascending chain condition.
On the other hand, a wide class of cases is noted
where id(P) is embeddable in
P.
Counterexamples are given to many variants of the statements proved. Tbilisi Mathematical Journal, Vol. 1(2008), pp. 89-103 |