  We work out some relations between duality and intertwining in the context of
discrete Markov chains, fixing up the background of previous relations first
established for birth and death chains and their Siegmund duals. In view of the
results, the monotone properties resulting from the Siegmund dual of birth and
death chains are revisited in some detail, with emphasis on the non neutral
Moran model. We also introduce an ultrametric type dual extending the Siegmund
kernel. Finally we discuss the sharp dual, following closely the Diaconis-Fill
study.
