Orderability of Belief States in a POMDP?

Question

Consider a POMDP with integer states $1,2,\ldots,N$, where $N$ is finite. We thus have a complete order over the states.

It seems reasonable to think that belief states for this POMDP may be orderable in some partial order sense.

Does this orderability translate into any structure of the optimal policy? Anyone have any relevant literature?