We study the construction of preorders on
-monads by the semantic ⊤ ⊤-lifting. We show the universal property of this construction, and characterise the class of preorders on a monad as a limit of a
-chain. We apply these theoretical results to identifying preorders on some concrete monads, including the powerset monad, maybe monad, and their composite monad. We also relate the construction of preorders and coalgebraic formulation of simulations.