[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.