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

Abstract

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

The slightly smaller object $\mathrm{id}(P)$ 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 $\mathrm{id}^n(P_n)\cong P_0$ are those having ascending chain condition. On the other hand, a wide class of cases is noted where $\mathrm{id}(P)$ is embeddable in $P.$

Counterexamples are given to many variants of the statements proved.