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Graph monoids arise naturally in the study of non-stable K-theory of graphC*-algebras and Leavitt path algebras. They play also an important role in thecurrent approaches to the realization problem for von Neumann regular rings. Inthis paper, we characterize when a finitely generated conical refinement monoidcan be represented as a graph monoid. The characterization is expressed interms of the behavior of the structural maps of the associated $I$-system atthe free primes of the monoid.

Representing finitely generated refinement monoids as graph monoids