  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.
