Abstract

The construction of the Leavitt path algebra associated to a directed graph E is extended to incorporate a family C consisting of partitions of the sets of edges emanating from the vertices of E. The new algebras, L_K(E, C), are analyzed in terms of their homology, ideal theory, and K-theory. These algebras are proved to be hereditary, and it is shown that any conical abelian monoid occurs as the monoid 𝒱(L_K(E, C)) of isomorphism classes of finitely generated projective modules over one of these algebras. The lattice of trace ideals of L_K(E, C) is determined by graph-theoretic data, namely as a lattice of certain pairs consisting of a subset of E⁰ and a subset of C. Necessary conditions for 𝒱(L_K(E, C)) to be a refinement monoid are developed, together with a construction that embeds (E, C) in a separated graph (E₊, C⁺) such that 𝒱(L_K(E₊, C⁺)) has refinement.