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

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 ) − {∅}. 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 Pn with id n (Pn) ∼ = 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.2000 Mathematics Subject Classification. Primary: 06A06, 06A12, 06B10. Key words and phrases. lattice of ideals of a lattice or semilattice; poset of ideals (upward directed downsets) of a poset; ideals generated by chains.The author is indebted to the referee for many helpful corrections and references. After publication of this note, updates, further references, errata, etc., if found, will be recorded at

Representation of artinian partially ordered sets over semiartinian von Neuman regular algebras

“…The problem of determining which algebraic and distributive lattices are isomorphic to L 2 (R) for some von Neumann regular ring R is one among the so called Realization (or Representation) Problems for Von Neumann regular rings. A first answer to this problem was given by Bergman in his famous 1986 unpublished note (see [12]):…”