Branching systems and representations of Cohn-Leavitt path algebras of separated graphs

We construct for each separated graph (E;C) a family of branching systems over a set X and show how each branching system induces a representation of the Cohn-Leavitt path algebra associated to (E;C) as homomorphisms over the module of functions in X. We also prove that the abelianized Cohn-Leavitt path algebra of a separated graph with no loops can be written as an amalgamated free product of abelianized Cohn-Leavitt algebras that can be faithfully represented via branching systems.