[denotational]

> It isn't useful in mathematics because it is a contradiction in terms: if "neither of the two compared entities is greater or lesser than the other" then they are equal.

That’s only true for a total order; there are many interesting orders that do not have this property.

It holds for the usual ordering on N, Z, Q and R, but it doesn’t hold for more general partially ordered sets.

In general one has to prove that an order is total, and this is frequently non-trivial: Cantor-Schröder-Bernstein can be seen as a proof that the cardinal numbers have a total order.

[adastra22]

Example: alphabetic ordering in most languages with diacritics. For example, "ea" < "éz", but also "éa" < "ez". That's because e and é are treated the same as far as the ordering function is concerned, but they are obviously also NOT the same glyph.