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

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

2008-01-04

arxiv.org/abs/0801.0751v2

Mathematics

Rings and Algebras

8 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy. After publication,

Scientific paper

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.

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