Duality and the Completeness of the Modal mu-Calculus

We consider the modal-calculus due to Kozen, which is a nitary modal logic with least and greatest xed points of monotone operators. We extend the existing duality between the category of modal algebras with homomorphisms and the category of descriptive modal frames with contractions to the case of having xed point operators. As a corollary, we obtain completeness results for two proof systems due to Kozen ((nitary and innnitary) with respect to certain classes of modal frames. The rules are sound in every model, not only for validity.