[bah_humbug]

The definition of antisymmetric relations is an unusual one. As given, it's incompatible with the definition of reflexivity (presumably there's an implicit assumption on `a ≠ b`).

The usual definition is `x ≤ y AND y ≤ x → x = y`.

[gugagore]

Yeah. That's unfortunate. The discussion on totality makes clear that it subsumes reflexivity when a = b.

The diagram, which indicates implication, is also not consistent with the logical statement, which indicates iff.

[danbruc]

There is also a mistake in the discussion of totality and reflexivity.

By the way, this law makes the reflexivity law redundant, as it is just a special case of reflexivity when a and b are one and the same object, but I still want to present it for reasons that will become apparent soon.

This should be as follows.

[...] a special case of totality when a and b are one and the same [...]

[scubbo]

Is there also a mistake in the discussion of joins? At the start, it says:

> The least upper bound of two elements that are connected as part of an order is called the join of these elements...

but then at the end of that section it says

> Like with the maximum element, if two elements have several upper bounds that are equally big, then none of them is a join (a join must be unique). If, however, one of those elements is established as bigger than another, it immediately qualifies.

If the join is the least-upper-bound, shouldn't the final sentence read "...is established as smaller than another"? Or, I guess, the "it" could be referring to "another" rather than "one of". Maybe it's simply unclear rather than incorrect.

[Sharlin]

Yeah, antisymmetry would be correct if the order was strict, which the article does mention but only in passing.

[boris_m]

Thanks everyone for pointing that and other problems, just made some corrections.

[13415]

Yes, the property defined on the page is more commonly called asymmetry.