  In earlier work, we had introduced the Kantorovich probability monad on
complete metric spaces, extending a construction due to van Breugel. Here we
extend the Kantorovich monad further to a certain class of ordered metric
spaces, by endowing the spaces of probability measures with the usual
stochastic order. It can be considered a metric analogue of the probabilistic
powerdomain.
  The spaces we consider, which we call L-ordered, are spaces where the order
satisfies a mild compatibility condition with the metric itself, rather than
merely with the underlying topology. As we show, this is related to the theory
of Lawvere metric spaces, in which the partial order structure is induced by
the zero distances.
  We show that the algebras of the ordered Kantorovich monad are the closed
convex subsets of Banach spaces equipped with a closed positive cone, with
algebra morphisms given by the short and monotone affine maps. Considering the
category of L-ordered metric spaces as a locally posetal 2-category, the lax
and oplax algebra morphisms are exactly the concave and convex short maps,
respectively.
  In the unordered case, we had identified the Wasserstein space as the colimit
of the spaces of empirical distributions of finite sequences. We prove that
this extends to the ordered setting as well by showing that the stochastic
order arises by completing the order between the finite sequences, generalizing
a recent result of Lawson. The proof holds on any metric space equipped with a
closed partial order.
