  For Markov chains with a partially ordered finite state space we show strong
stationary duality under the condition of M\"obius monotonicity of the chain.
We show relations of M\"obius monotonicity to other definitions of monotone
chains. We give examples of dual chains in this context which have transitions
only upwards. We illustrate general theory by an analysis of nonsymmetric
random walks on the cube with an application to networks of queues.
