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

TL;DR

This paper explores the relationship between posets and lattices and their ideal structures, revealing that ideal lattices are generally larger and more complex than the original posets, with various embedding and homomorphism limitations.

Contribution

It establishes new results on the size and embeddability of ideal lattices and posets, including conditions for isomorphism and the non-existence of certain maps.

Findings

Id(P) is always larger than P in a structural sense.

Posets P with no ascending chain condition cannot be reconstructed from their ideal structures.

Id(P) can often be embedded into P, but not always via certain homomorphisms.

Abstract

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.