The regular algebra of a poset

TL;DR

This paper constructs a von Neumann regular algebra from a finite poset, linking algebraic structures to poset relations, and extends the realization problem for such rings.

Contribution

It introduces a functorial method to associate a von Neumann regular algebra to any finite poset, expanding the class of monoids with known realization solutions.

Findings

The monoid of projective modules corresponds to the poset with specific relations.

The construction provides a positive solution to the realization problem for a broader class of regular rings.

The algebraic structure reflects the order relations of the underlying 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.