The regular algebra of a poset

Abstract

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