  Discrete partially ordered sets can be turned into distance spaces in several
ways. The distance functions may or may not satisfy the triangle inequality,
and restriction of the distance to finite chains may or may not coincide with
the natural, difference-of-height distance measured in a chain. For
semilattices, a semimodularity condition ensures the good behavior of the
distances considered. This condition is trivially satisfied by trees, and in
lattices it coincides with the usual semimodularity property. For a large class
of distance functions the triangle inequality is equivalent to semimodularity.
