Finitely generated antisymmetric graph monoids.
type: Item
instance of: scholarly article
title: Finitely generated antisymmetric graph monoids.
zbMATH Open document ID: 1158.16006
author: Pere Ara
author: Francesc Perera
author: Friedrich Wehrung
published in: Journal of Algebra
publication date: 2008-10-22T00:00:00Z
review text: 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.
reviewed by: Kenneth R. Goodearl
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 19K14
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 20M05
zbMATH DE Number: 5355725
zbMATH Keywords: graph monoids
zbMATH Keywords: quivers
zbMATH Keywords: antisymmetric monoids
zbMATH Keywords: primely generated monoids
zbMATH Keywords: nonstable K-theory
zbMATH Keywords: Leavitt path algebras
zbMATH Keywords: Cuntz-Krieger \(C^*\)-algebras
zbMATH Keywords: finitely generated projective modules
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.06.013
OpenAlex ID: W2038967576
cites work: The Leavitt path algebra of a graph.
cites work: The regular algebra of a quiver.
cites work: Separative cancellation for projective modules over exchange rings
cites work: Nonstable \(K\)-theory for graph algebras.
cites work: Cancellation in primely generated refinement monoids.
cites work: Flat Semilattices
cites work: Q3848243
cites work: Simple \(C^*\)-algebras generated by isometries
cites work: Refinement monoids, Vaught monoids, and Boolean algebras
cites work: Q4865873
cites work: Semilattices of groups and inductive limits of Cuntz algebras
cites work: Representations of distributive semilattices in ideal lattices of various algebraic structures.
cites work: Bases of countable Boolean algebras
cites work: Q4681774
cites work: Non-measurability properties of interpolation vector spaces
cites work: The dimension monoid of a lattice
cites work: A uniform refinement property for congruence lattices
DOI: 10.1016/J.JALGEBRA.2008.06.013
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description: scientific article; zbMATH DE number 5355725