[gobdovan]

Unless there's some idiosyncratic meaning for the `=>`, the Antisymmetry one basically says `Orange -> Yellow => Yellow -/> Orange`. The diagram is not acurate. The prose is very imprecise. "It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me." NO. Antisymmetry doesn't exclude `x = y`. Ties are permitted in the equality case. Antisymmetry for a non-strict order says that if both directions hold, the two elements must in fact be the same element. The author is describing strict comparison or total comparability intuition, not antisymmetry.

[bubblyworld]

I don't think they are completely wrong - "=>" is just implication. A hidden assumption in their diagrams is that circles of different colours are assumed to be different elements.

A morphism from orange to yellow means "O <= Y". From this, antisymmetry (and the hidden assumption) implies that "Y not <= O".

Totality is just the other way around (all two distinct elements are comparable in one direction).

[gobdovan]

If this is meant to be an explainer, that can't be simply implicit. The text actually seems full of imprecise claims, such as:

"All diagrams that look something different than the said chain diagram represent partial orders"

"The different linear orders that make up the partial order are called chains"

The Birkhoff theorem statement, which is materially wrong. A finite distributive lattice is not isomorphic to "the inclusion order of its join-irreducible elements".

[bubblyworld]

I mean, it's a blog post, those statements are correct in spirit. A taste thing, I think. I agree about the birkhoff theorem though.

[mrkeen]

It really isn't a long enough section to get lost in.

The 'not accurate' diagram says that orange-less-than-yellow implies yellow-not-less-than-orange. Hard to find fault with.

> NO. Antisymmetry doesn't exclude `x = y`. Ties are permitted in the equality case. Antisymmetry for a non-strict order says that if both directions hold, the two elements must in fact be the same element. The author is describing strict comparison or total comparability intuition, not antisymmetry.

I like the article's "imprecise prose" better:

  You have x ≤ y and y ≤ x only if x = y

[gobdovan]

My comment is not long enough either to get lost in.

The prose "It also means that no ties are permitted - either I am better than my grandmother at soccer or she is better at it than me" is inaccurate for describing antisymmetry. In the same short section, you first state the correct condition:

You have x ≤ y and y ≤ x only if x = y

from which it doesn't follow that "It also means that no ties are permitted". The "no ties" idea belongs to a stronger notion such as a strict total order, not to antisymmetry.

[wasabi991011]

The prose is correct.

You (presumably) aren't your grandmother, so we have x=/=y. Therefore by the biimplication, (x ≤ y and y ≤ x) is false i.e. either x ≤ y (I am better than my grandmother) or y ≤ x (my grandmother is better than me). The "neither" case is excluded by the law of totality.

[gobdovan]

> The "neither" case is excluded by the law of totality.

We literally said the same thing. It doesn't follow from antisymmetry.

My point is precisely that:

(x <= y /\ y <= x) -> x = y

does not entail

x <= y \/ y <= x

The second statement is totality/comparability, not antisymmetry.