  The purpose of this paper is to provide a common framework for studying
various generalizations of Leavitt algebras and Leavitt path algebras. This
paper consists of two parts. In part I we define Cohn-Leavitt path algebras of
a new class of graphs with an additional structure called bi-separated graphs,
which generalize the constructions of Leavitt path algebras of various types of
graphs. We define and study the category \textbf{BSG} of bi-separated graphs
with appropriate morphisms so that the functor which associates a bi-separated
graph to its Cohn-Leavitt path algebra is continuous. We also characterize a
full subcategory of \textbf{BSG} whose objects are direct limits of finite
complete subobjects. We compute normal forms of these algebras and apply them
to study some algebraic theoretic properties in terms of bi-separated
graph-theoretic properties.
  In part II we specialize our attention to Cohn-Leavitt path algebras of a
special class of bi-separated graphs called B-hypergraphs. We investigate their
non-stable K-theory and show that the lattice of order-ideals of V-monoids of
these algebras is determined by bi-separated graph-theoretic data. Using this
information we study representations of Leavitt path algebras of regular
hypergraphs and also find a matrix criterion for Leavitt path algebras of
finite hypergraphs to have IBN property.
