This a bonus question for HW 2 Basic Modal Logic course 2016/2017.
Show that transitive fitration on is a finite list of clusters, maybe interspersed by some irreflexive singletons, no two of which may be adjacent.
What I did is to consider it in four steps.
Finiteness and transitivity
This follows from filtration property (upon finite ) and the property that has as a transitive filtration.
For all , we have .
Proof. WLOG, . Then, we show that for all , we have that if , then . If , then because , we have . If , then there exists some , such that , and thus becaue , .
Moreover, we basically have .
Properties of irreflexive points
If , then there must be some , such that is the greatest number satisfying .
Proof. If , it means that for some , we have but . Thus, and .
I also claim no two irreflexive points can be adjacent.
Proof. Suppose that are irreflexive points w.r.t. formula and , respectively and suppose that they are adjacent in the finite list of filtration. Take any number . Clearly, . But and are adjacent in the filtration, then we must have for all such . This means that for all , particularly , we have implies . But, we have .
Properties of non-simple cluster
Now, we prove that for any reflexive point, it belongs to a non-simple cluster.
Proof. If there is no , then all the points belong to the same cluster. If there exist , then for reflexive points , means for all , implies . This means, must not be the greatest number satisfying formula for any . Because there is only finite number of singletons (due to the finiteness of , we can choose rational number such that all the numbers in are all reflexive (not a singleton). Now pick any . Clearly . We show now. Because it means for all , implies , and because is not a singleton, we then just need to check implies . Suppose but . This means that there’s a singleton for which is a contradiction. Thus belongs to a cluster which also contains at least the whole .
Because every point in filtration is either reflexive or irreflexive, these properties show that the result of filtration is a finite list of clusters, may be interspersed by some irreflexive singletons, no two of which are adjacent.
This is approximately what I wrote in my homework. I have not got any feedback yet at the time I am writing this. I hope it got almost full marks (like 8 or more).