> That is simply not the theorem.

> The article is plain wrong.

> This does not excuse the article from reversing the meaning of the theorem.

What's with this hyperbole? Even the best math books have loads of errors (typographical, factual, missing conditions, insufficient reasoning, incorrect reasoning, ...). Just look at any errata list published by any university for their set books! Nobody does this kind of hyperbole for errors in math books. Only on HN do you see this kind of takedown, which is frankly very annoying. In universities, professors and students just publish errata and focus on understanding the material, not tearing it down with such dismissive tone. It's totally unnecessary.

I don't know if you've got an axe to grind here or if you're generally this dismissive but calling it "simply not the theorem" or "plain wrong" is a very annoying kind of exaggeration that misses all nuance and human fallibility.

Yes, the precise statement of Birkhoff's representation theorem involves down-sets of the poset of join-irreducibles. Yes, the article omits that. I agree that it is imprecise.

But it's not "reversing the meaning". It still correctly points to reconstructing the lattice via an inclusion order built from join-irreducibles. What's missing is a condition. It is sloppy wording but not a fundamental error like you so want us to believe.

Feels like the productive move here is just to suggest the missing wording to the author. I'm sure they'll appreciate it. I don't really get the impulse to frame it as a takedown and be so dismissive when it's a small fix.