Finitely generated antisymmetric graph monoids

A graph monoid is a commutative monoid for which there is a particularly simple presentation, given in terms of a quiver. Such monoids are known to satisfy various nonstable K-theoretical representability properties for either von Neumann regular rings or C*-algebras. We give a characterization of graph monoids within finitely generated antisymmetric refinement monoids. This characterization is formulated in terms of the prime elements of the monoid, and it says that each free prime has at most one free lower cover. We also characterize antisymmetric graph monoids of finite quivers. In particular, the monoid Z∞ = {0, 1, 2, . . .}∪{∞} is a graph monoid, but it is not the graph monoid of any finite quiver.