On the µ-calculus over transitive and finite transitive frames

This paper regards the problem of understanding what happens when we restrict the semantics of modal μ-calculus to special classes of frames. In this paper 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 which strongly connected component have a bounded cardinality, modulo bisimulation.