  A system (P_a: a in A) of probability measures on a common state space S
indexed by another index set A can be ``realized'' by a system (X_a: a in A) of
S-valued random variables on some probability space in such a way that each X_a
is distributed as P_a. Assuming that A and S are both partially ordered, we may
ask when the system (P_a: a in A) can be realized by a system (X_a: a in A)
with the monotonicity property that X_a <= X_b almost surely whenever a <= b.
When such a realization is possible, we call the system (P_a: a in A)
``realizably monotone.'' Such a system necessarily is stochastically monotone,
that is, satisfies P_a <= P_b in stochastic ordering whenever a <= b. In
general, stochastic monotonicity is not sufficient for realizable monotonicity.
However, for some particular choices of partial orderings in a finite state
setting, these two notions of monotonicity are equivalent. We develop an
inverse probability transform for a certain broad class of posets S, and use it
to explicitly construct a system (X_a: a in A) realizing the monotonicity of a
stochastically monotone system when the two notions of monotonicity are
equivalent.
