The regular algebra of a poset

Abstract

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