Refinement monoids and adaptable separated graphs

TL;DR

This paper introduces adaptable separated graphs and characterizes their monoids, demonstrating their role in representing finitely generated conical refinement monoids and advancing the understanding of the Realization Problem for von Neumann regular rings.

Contribution

It defines adaptable separated graphs, characterizes their monoids, and proves that any finitely generated conical refinement monoid can be represented by such graphs.

Findings

Monoids of adaptable separated graphs are primely generated conical refinement monoids.

Explicit determination of associated I-systems for these monoids.

Representation of all finitely generated conical refinement monoids by adaptable separated graphs.

Abstract

We define a subclass of separated graphs, the class of adaptable separated graphs, and study their associated monoids. We show that these monoids are primely generated conical refinement monoids, and we explicitly determine their associated I-systems. We also show that any finitely generated conical refinement monoid can be represented as the monoid of an adaptable separated graph. These results provide the first step toward an affirmative answer to the Realization Problem for von Neumann regular rings, in the finitely generated case.