Towards a Typed Omega Algebra

We propose axioms for 1-free omega algebra, typed 1-free omega algebra and typed omega algebra. They are based on Kozen’s axioms for 1-free and typed Kleene algebra and Cohen’s axioms for the omega operation. In contrast to Kleene algebra, several laws of omega algebra turn into independent axioms in the typed or 1-free variants. We set up a matrix algebra over typed 1-free omega algebras by lifting the underlying structure. The algebra includes non-square matrices and care has to be taken to preserve type-correctness. The matrices can represent programs in total and general correctness. We apply the typed construction to derive the omega operation on two such representations, for which the untyped construction does not work. We embed typed 1-free omega algebra into 1-free omega algebra, and this into omega algebra. Some of our embeddings, however, do not preserve the greatest element. We obtain that the validity of a universal formula using only +, ·, , ω and 0 carries over from omega algebra to its typed variant. This corresponds to Kozen’s result for typed Kleene algebra.