# Cohn-Leavitt path algebras of bi-separated graphs

## TL;DR

This paper introduces Cohn-Leavitt path algebras for bi-separated graphs, generalizing Leavitt path algebras, and explores their algebraic properties and K-theory, especially for hypergraph cases.

## Contribution

It defines a new class of algebras for bi-separated graphs, studies their categorical properties, and analyzes their K-theory and IBN property for hypergraph cases.

## Key findings

- Normal forms of the algebras are computed.
- The lattice of order-ideals is characterized by graph data.
- A matrix criterion for IBN property is established.

## Abstract

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.