Timeline for Poset of antichains of given cardinality
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 19, 2020 at 5:40 | comment | added | Tri | I threw in the vague phrase "in some sense," but maybe one way to make it concrete is this: I was wondering if, as you go from $A_{m_P}$ (for all $P$) to $A_{m_P-1}$ (for all $P$) to $A_{m_P-2}$ (for all $P$)..., you get less and less "distributive lattice-like"---maybe one interpretation of this is that you don't get all distributive lattices, but you don't get the class of all finite posets and the class of posets you get grows larger and its members less and less like distributive lattices; and maybe something similar happens in the opposite direction although it won't be exactly analogous | |
| Sep 18, 2020 at 13:25 | comment | added | Sam Hopkins | Ah, I realized another possible interpretation of what you're asking as, if $\mathcal{A}_k(P)$ is always a lattice, then is it always distributive. But note the case of $P$ an antichain shows that $\mathcal{A}_k(P)$ is always a lattice only for $k=0,m_P$. | |
| Sep 18, 2020 at 3:31 | comment | added | Sam Hopkins | But maybe I'm misunderstanding your query, because of course $P$ itself could be a non-distributive lattice... | |
| Sep 18, 2020 at 3:30 | comment | added | Sam Hopkins | @Tri: Consider the graded poset $P$ with ranks of sizes 2, 3, and 2, and all relations between adjacent ranks. The size 2 antichains of $P$ form a $5$-element non-distributive lattice consisting of a minimum, maximum, and $3$ incomparable elements. | |
| Sep 17, 2020 at 23:33 | comment | added | Tri | I find this interesting. We go from all posets to all distributive lattices... I have not thought about it much, but I wonder if the classes of posets you get are weakly increasing (in some sense I'm not pinning down, since $m_P$ of course depends on $P$). I wonder if, when you have joins and meets existing, you have the distributive law. P.S. Trotter's Axiom: "All posets are finite." | |
| Sep 15, 2020 at 19:04 | comment | added | Sam Hopkins | @Mare: it should be completely straightforward to code, but I don't think there's an existing command like that. But also, it's not so obvious to me that looking at small examples would tell you much: e.g., to get a given distributive lattice as the maximum sized antichains of a poset requires a poset of a very particular form. | |
| Sep 15, 2020 at 19:02 | comment | added | Mare | Good question. Is there a command to get $A_{{m_P} -1}(P)$ or more generally $A_k(P)$ for a given poset $P$ using Sage (Im not too experienced yet with Sage)? Looking at some examples might reveal some properties. | |
| Sep 15, 2020 at 18:17 | history | asked | Sam Hopkins | CC BY-SA 4.0 |