On the μ-calculus over transitive and finite transitive frames

We prove that the modal μ-calculus collapses to first order logic over the class of finite transitive frames. The proof is obtained by using some byproducts of a new proof of the collapse of the μ-calculus to the alternation free fragment over the class of transitive frames.Moreover, we prove that the modal μ-calculus is Büchi and co-Büchi definable over the class of all models where, in a strongly connected component, vertexes are distinguishable by means of the propositions they satisfy.