> Generally speaking the "rating structure" is a lattice where you can, for any two players A and B, tell whether A is a better player than B or the other way around.

Not really. You tend to have cycles, where person A can beat person B who beats person C who beats person A.

There's a guy who was even with the guys playing Go who were 2 or 3 stones weaker than me that would tend to beat me because of some of the unorthodox things he did. (Eventually I strengthened my game against these things).

Considering ratings to be a total ordering is a useful approximation.