  Dimer models have been used in string theory to construct path algebras with
relations that are 3-dimensional Calabi-Yau Algebras. These constructions
result in algebras that share some specific properties: they are finitely
generated modules over their centers and their representation spaces are toric
varieties. In order to describe these algebras we introduce the notion of a
toric order and show that all toric orders which are 3-dimensional Calabi-Yau
algebras can be constructed from dimer models on a torus.
  Toric orders are examples of a much broader class of algebras: positively
graded cancellation algebras. For these algebras the CY-3 condition implies the
existence of a weighted quiver polyhedron, which is an extension of dimer
models obtained by replacing the torus with any two-dimensional compact
orientable orbifold.
