[brynbryn]

That article is misinterpreting the meaning of the symbol. 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.

The author of the original article uses it correctly - think about it more in regards to importance for their example.

The business is no more or less important than the developer, but they are NOT equal.

It doesn't have to mean importance though, just the method by which you are comparing things.

Monday ≹ Wednesday

Come to think of it, it should be called the 'No better than' operator.

[p4bl0]

> if "neither of the two compared entities is greater or lesser than the other" then they are equal.

Not in a partial order.

For example in this simple lattice structure, where lines mark that their top end in greater than their bottom end:

      11
     /  \
    01  10
     \  /
      00

11 is > to all other (by transitivity for 00), 00 is < to all other (by transitivity for 11), but 01 is not comparable with 10, it is neither lesser nor greater given the described partial order.

You can actually see this kind of structure everyday: unix file permissions for example. Given a user and a file, the permissions of the user are is an element of a lattice where the top element is rwx (or 111 in binary, or 7 in decimal, which means the user has all three permissions to read, write, and execute) and the bottom element is --- (or 000, in binary, or 0 in decimal, which means the user has no permissions). All other combination of r, w, and x are possible, but not always comparable: r-x is not greater nor lesser than rw- in the permissions lattice, it's just different.

[astrobe_]

Yes, or for more familiar examples: coordinates and complex numbers. the "default" less-than and greater-than don't have any meaning for them; you have to define one, which may be "imperfect" (because one can't do better), hence the concept of partial order.

[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.

[Chirono]

That’s only true for linearly ordered structures, but isn’t true for partially ordered ones.

For example, set inclusion. Two different sets can be neither greater than not smaller than each other. Sets ordered by inclusion form a partially ordered lattice.

[jcparkyn]

Is that really a contradiction? What about complex numbers?