Realizable monotonicity for continuous-time Markov processes

We formalize and analyze the notions of stochastic monotonicity and realizable monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which stochastic monotonicity and realizable monotonicity coincide in continuous-time but not in discrete-time.