# The Realization Problem for Finitely Generated Refinement Monoids

## TL;DR

This paper proves that every finitely generated conical refinement monoid can be realized as the monoid of finitely generated projective modules over a von Neumann regular ring, using adaptable separated graphs for representation.

## Contribution

It establishes a new realization theorem linking finitely generated conical refinement monoids to von Neumann regular rings via adaptable separated graphs.

## Key findings

- Every finitely generated conical refinement monoid is realizable as a monoid of projective modules over a von Neumann regular ring.
- Constructs a von Neumann regular algebra from an adaptable separated graph and field.
- Shows an isomorphism between the graph monoid and the algebra's projective module monoid.

## Abstract

We show that every finitely generated conical refinement monoid can be represented as the monoid $\mathcal V(R)$ of isomorphism classes of finitely generated projective modules over a von Neumann regular ring $R$. To this end, we use the representation of these monoids provided by adaptable separated graphs. Given an adaptable separated graph $(E, C)$ and a field $K$, we build a von Neumann regular $K$-algebra $Q_K (E, C)$ and show that there is a natural isomorphism between the separated graph monoid $M(E, C)$ and the monoid $\mathcal V(Q_K (E, C))$.