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2008 On lattices and their ideal lattices, and posets and their ideal posets
George M. Bergman
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Tbilisi Math. J. 1: 89-103 (2008). DOI: 10.32513/tbilisi/1528768825

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.

Acknowledgment

The author is indebted to the referees for many helpful corrections and references. Any updates, errata, etc. found after publication, will be noted at the author's webpage.

Citation

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George M. Bergman. "On lattices and their ideal lattices, and posets and their ideal posets." Tbilisi Math. J. 1 89 - 103, 2008. https://doi.org/10.32513/tbilisi/1528768825

Information

Received: 5 January 2008; Accepted: 23 June 2008; Published: 2008
First available in Project Euclid: 12 June 2018

zbMATH: 1171.06001
MathSciNet: MR2563808
Digital Object Identifier: 10.32513/tbilisi/1528768825

Subjects:
Primary: 06A06, 06A12, 06B10

Rights: Copyright © 2008 Tbilisi Centre for Mathematical Sciences

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