For cycles, it looks like the projection to ¯¯¯¯¯¯W is akin to taking all the worlds that form a given cycle, and compressing them into a single world.
In your example, it’s true wi<wj and wj<wi when i≠j. That’s the condition for equivalence in the project, so you have that w1=w2=w3. If you’re thinking about the ordering as a directed graph, you can collapse those worlds to a single point and not mess up the ordering.
For cycles, it looks like the projection to ¯¯¯¯¯¯W is akin to taking all the worlds that form a given cycle, and compressing them into a single world.
In your example, it’s true wi<wj and wj<wi when i≠j. That’s the condition for equivalence in the project, so you have that w1=w2=w3. If you’re thinking about the ordering as a directed graph, you can collapse those worlds to a single point and not mess up the ordering.
Ah yes, that makes sense, thanks! I didn’t realize what ¯¯¯¯¯¯W was the set of equivalence classes of W