  We formalize and analyze the notions of stochastic monotonicity and
realizable mono-tonicity 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.
