> Sadly, this comparator is still wrong for any sorting API that expects a general three-way comparison, because it does not handle equality as a separate case.

Let's scroll up a little bit and read from the section you're finding fault with:

  the most straightforward type of order that you think of is linear order i.e. one in which every object has its place depending on every other object

Your reply only works if the article were consistently talking about a strict order. However, it is not. It explicitly introduces linear order using reflexivity and antisymmetry, in other words, a non-strict `<=`-style relation, in which equality IS a real case.

If the author wanted to describe a 'no ties' scenario where every object has its own unique place, they should have defined a strict total order.

They may know everything about mathematics for all I care. I am critiquing what I am reading, not the author's knowledge.

Edit: for anyone wanting a basic example, ["aa", "aa", "ab"] under the usual lexicographic <=. All elements are comparable, so "every object has its place depending on every other object." It also "leaves no room for ambiguity in terms of which element comes before which": aa = aa < ab. Linear order means everything is comparable, not that there are no ties. By claiming "no ties are permitted" while defining the order as a reflexive, antisymmetric relation, the author is mixing a strict-order intuition into a non-strict-order definition.