  The class of refinement monoids (abelian monoids satisfying the Riesz
refinement property) is subdivided into those which are tame, defined as being
an inductive limit of finitely generated refinement monoids, and those which
are wild, i.e., not tame. It is shown that tame refinement monoids enjoy many
positive properties, including separative cancellation ($2x=2y=x+y \implies
x=y$) and multiplicative cancellation with respect to the algebraic ordering
($mx\le my \implies x\le y$). In contrast, examples are constructed to exhibit
refinement monoids which enjoy all the mentioned good properties but are
nonetheless wild.
