We explore and relate two notions of monotonicity, stochastic and realizable, for a system of probability measures on a common finite partially ordered set (poset) S when the measures are indexed by another poset A. We give counterexamples to show that the two notions are not always equivalent, but for various large classes of S we also present conditions on the poset A that are necessary and sufficient for equivalence. When A = S, the condition that the cover graph of S have no cycles is necessary and sufficient for equivalence. This case arises in comparing applicability of the perfect sampling algorithms of Propp and Wilson and the first author of the present paper.

40 pages. See also http://www.mts.jhu.edu/~fill/ and http://www.mts.jhu.edu/~machida/ . Accepted (subject to revision); will appear in either Annals of Probability or Annals of Applied Probability