Cohn-Leavitt path algebras of bi-separated graphs

The purpose of this article is to provide a common framework for studying various generalizations of Leavitt path algebras. We first define Cohn-Leavitt path algebras of graphs with an additional structure called bi-separated graphs. We then define and study the category BSG of bi-separated graphs with appropriate morphisms so that the functor which associates bi-separated graphs to their Cohn-Leavitt path algebras is continuous. Next, we define two sub-categories of BSG, compute basis for the algebras corresponding to one of those subcategories and study some algebraic properties in terms of bi-separated graph-theoretic properties. Finally, we compute nu-monoid for some particular cases.