Modules for Leavitt path algebras via extended algebraic branching systems

For a graph $E$, we introduce the notion of an extended $E$-algebraic branching system, generalising the notion of an $E$-algebraic branching system introduced by Gonçalves and Royer. We classify the extended $E$-algebraic branching systems and show that they induce modules for the corresponding Leavitt path algebra $L(E)$. Among these modules we find a class of nonsimple modules whose endomorphism rings are fields.