  We describe the structure of the monoid of natural-valued monotone functions
on an arbitrary poset. For this monoid we provide a presentation, a
characterization of prime elements, and a description of its convex hull. We
also study the associated monoid ring, proving that it is normal, and thus
Cohen-Macaulay. We determine its Cohen-Macaulay type, characterize the
Gorenstein property, and provide a Gr\"obner basis of the defining ideal. Then
we apply these results to the monoid of quasi-arithmetic multiplicities on a
uniform matroid. Finally we state some conjectures on the number of
irreducibles for the monoid of multiplicities on an arbitrary matroid.
