A Correlation Inequality for Markov Processes in Partially Ordered State Spaces

Let $E$ be a finite partially ordered set and $M_p$ the set of probability measures in $E$ giving a positive correlation to each pair of increasing functions on $E$. Given a Markov process with state space $E$ whose transition operator (on functions) maps increasing functions into increasing functions, let $U_t$ be the transition operator on measures. In order that $U_tM_p \subset M_p$ for each $t \geqq 0$, it is necessary and sufficient that every jump of the sample paths is up or down.