Representing finitely generated refinement monoids as graph monoids

TL;DR

This paper characterizes when finitely generated conical refinement monoids can be represented as graph monoids, linking their structure to properties of associated I-systems, with implications for algebraic and operator algebra contexts.

Contribution

It provides a precise characterization of finitely generated conical refinement monoids that are realizable as graph monoids based on their I-system structure.

Findings

Characterization of representability conditions for finitely generated conical refinement monoids as graph monoids.

Connection between the behavior of I-system maps at free primes and monoid representation.

Implications for non-stable K-theory and algebraic structures like Leavitt path algebras.

Abstract

Graph monoids arise naturally in the study of non-stable K-theory of graph C*-algebras and Leavitt path algebras. They play also an important role in the current approaches to the realization problem for von Neumann regular rings. In this paper, we characterize when a finitely generated conical refinement monoid can be represented as a graph monoid. The characterization is expressed in terms of the behavior of the structural maps of the associated $I$-system at the free primes of the monoid.