Algebras associated to acyclic directed graphs

We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings.