Identifying All Preorders on the Subdistribution Monad

The countable valuation monad, the countable distribution monad, and the countable subdistribution monad are often used in the coalgebraic treatment of discrete probabilistic transition systems. We identify preorders on them using a technique based on the preorder ⊤⊤-lifting and elementary facts about preorders on real intervals preserved by convex combinations. We show that there are exactly 15, 5, and 41 preorders on the countable valuation monad, the countable distribution monad, and the countable subdistribution monad respectively. We also give concrete definitions of these preorders. By applying Hesselink and Thijs's/ Hughes and Jacobs's construction to some preorder on the countable subdistribution monad, we obtain probabilistic bisimulation between Markov chains ignoring states with deadlocks.