Finitely generated antisymmetric graph monoids.
Some of the monoids arising in the nonstable K-theory of certain rings and \(C^*\)-algebras are characterized in terms of elementary properties. For any ring (or \(C^*\)-algebra) \(R\), the monoid \(V(R)\) consists of the isomorphism classes of finitely generated projective \(R\)-modules, with addition induced from direct sum. \textit{P. Ara, M. A. Moreno}, and \textit{E. Pardo} determined these monoids when \(R\) is either the Leavitt path algebra \(L_K(E)\) or the Cuntz-Krieger graph \(C^*\)-algebra \(C^*(E)\) associated to a row finite quiver (directed graph) \(E\) [Algebr. Represent. Theory 10, No. 2, 157-178 (2007; Zbl 1123.16006)]. More specifically, they defined a monoid \(M(E)\), called the graph monoid associated to \(E\), presented by generators and relations determined by \(E\), and proved that \(V(L_K(E))\cong V(C^*(E))\cong M(E)\). Further, \textit{P. Ara} and \textit{M. Brustenga} constructed a von Neumann regular ring of fractions \(Q_K(E)\) for \(L_K(E)\) and proved that \(V(Q_K(E))\cong M(E)\) [J. Algebra 309, No. 1, 207-235 (2007; Zbl 1119.16012)]. The present paper provides a characterization of graph monoids within the class of finitely generated antisymmetric refinement monoids, formulated in terms of elementary properties of prime elements. Antisymmetric graph monoids of finite quivers are also characterized. A surprising consequence is that there exist finitely generated graph monoids which are not graph monoids of any finite quivers.
scientific article; zbMATH DE number 5355725
scientific article