The regular algebra of a poset

Let K be a fixed field. We attach to each finite poset ℙ avon Neumann regular K-algebra Qk (ℙ) in a functorial way. We show that the monoid of isomorphism classes of finitely generated projective Qk (ℙ)-modules is the abelian monoid generated by ℙ with the only relations given by p = p + q whenever q< p in ℙ. This extends the class of monoids for which there is a positive solution to the realization problem for von Neumann regular rings. © 2009 American Mathematical Society.