# Talk:Antichain

WikiProject Mathematics (Rated Start-class, Mid-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 Start Class
 Mid Importance
Field: Discrete mathematics

## Correction

[...]the non-existence of an antichain A of size n+1 in S is a necessary and sufficient condition for S to be the union of n total orders. That doesn't sound quite right. Surely, if S has no antichain of size n+1, then it doesn't have one of size n+2 either, and by this equivalence it would then be the union of n and n+1 total orders at the same time; in fact we could continue this reasoning to conclude that S is the union of n, n+1, n+2, ... total orders. Presumably the equivalence holds only for the minimal such n? --Eriatarka 13:29, 15 January 2006 (UTC)

Some combinatorics articles, some mathematical logic articles, and some algebra articles probably ought to link to this page. Michael Hardy 00:16, 25 Oct 2003 (UTC)

Agreed. Don't think Dilworth's theorem is here? Or Sperner's lemma?

Charles Matthews 09:50, 25 Oct 2003 (UTC)

I think the definition of join and meet are both broken, and should be defined as follows:

$A \vee B = \{ x \in A\cup B \mid \not\exists y\in A\cup B\mbox{ s.t. }x < y\}.$
$A \wedge B = \{ x\in L_A\cap L_B\mid \not\exists y\in L_A\cap L_B\mbox{ s.t. }x < y\}.$

--Iwehrman (talk) 10:38, 13 August 2009 (UTC)