Leavitt path algebras of separated graphs

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 $\text{mon}{L}_K(E,C)$ 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 $E^0$ and a subset of $C$. Necessary conditions for $\text{mon}{L}_K(E,C)$ to be a refinement monoid are developed, together with a construction that embeds $(E,C)$ in a separated graph $(E_{+},C^{+})$ such that $\text{mon}{L}_K(E_{+},C^{+})$ has refinement.