Talk:Distributive lattice/Proofs

1

I have reverted the edits by 132.181.160.42, which completely destroy the proofs. Here are some of the problems:

First, in lemma 1, there are not three cases, but two. y <= z is assumed, and then the last two lines constitute the two cases. The fact that in the second line x <= y implies x <= z has been lost as well.

The reformatting of lemma 2,3 and the introduction of 4 is a pretty poor idea. The distributivity requirement should be mentioned right in the statement of the lemma, not somewhere else where it may be forgotten. This is especially true if you read lemma 4 directly.

Reformating is a good idea but not if it changes the structure of the proofs. Reading them more carefully before changing stuff may be a good idea. Morana 01:38, 2 November 2007 (UTC)

2

I reverted the edit by Aleph4 because it is not at all obvious that you can deduce xvy<=xvz from y<=z without assuming distributivity. If you spell that out as I think you should, the proof becomes longer than the previous one. Morana (talk) 05:10, 17 April 2008 (UTC)

You don't need distributivity to show that the operations are monotone. If $x\leq y$, then $x\vee y=y$, so $(a\vee x)\vee (a\vee y)=a\vee x\vee y=a\vee y$, so $a\vee x\leq a\vee y$.

Or see an alternative 1-line proof, which I just added at Lattice (order). --Aleph4 (talk) 09:29, 17 April 2008 (UTC)

Birkhoff's representation theorem

I've used the second and third proofs (reworded to look more like textual explanations and less like formal proofs) in Birkhoff's representation theorem. Do they still belong here as well? Is there any point in keeping this as a separate article? —David Eppstein (talk) 22:52, 30 November 2008 (UTC)