Finitely generated antisymmetric graph monoids

Pere Ara (1) , Francesc Perera (1) , Friedrich Wehrung (2)

1 UAB - Departament de Matemàtiques [Barcelona]
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

A graph monoid is a commutative monoid for which there is a particularly simple presentation, given in terms of a row-finite quiver. Such monoids are known to satisfy various nonstable K-theoretical representability properties for either von Neumann regular rings or C*-algebras. We give a characterization of graph monoids within finitely generated antisymmetric refinement monoids. This characterization is formulated in terms of the so-called prime elements of the monoid, and it says that each free prime has at most one free lower cover. We also characterize antisymmetric graph monoids of finite quivers. In particular, the monoid Z^\infty={0,1,2,...} union {\infty} is a graph monoid, but it is not the graph monoid of any finite quiver.

Mots clés: Monoid; commutative; quiver; graph monoid; separative; refinement; primely generated; primitive; antisymmetric; prime element; free element; regular refinement; von Neumann regular; C*-algebra

Domaines: Mathématiques générales [math.GM]; Anneaux et algèbres [math.RA]

Pere Ara, Francesc Perera, Friedrich Wehrung. Finitely generated antisymmetric graph monoids. Journal of Algebra, 2008, 320 (5), pp.1963--1982. ⟨10.1016/j.jalgebra.2008.06.013⟩. ⟨hal-00156906v2⟩