Construction of Diffusion Algebras

In cond-mat/0103603 Diffusion algebras have been introduced in the context of one-dimensional stochastic processes with exclusion in statistical mechanics. While this reference is focused on the needs of the physicist reader and thus states results without proofs and focuses on the discussion of lower-dimensional examples, it is the purpose of this paper to present a construction formalism for Diffusion algebras and to use the latter to prove the results in that reference.