The Alternation Hierarchy of the μ-calculus over Weakly Transitive Frames

It is known that the mu-calculus collapses to its alternation-free fragment over transitive frames and to modal logic over equivalence relations. We adapt a proof by D'Agostino and Lenzi to show that the mu-calculus collapses to its alternation-free fragment over weakly transitive frames. As a consequence, we show that the mu-calculus with derivative topological semantics collapses to its alternation-free fragment. We also study the collapse over frames of S4.2, S4.3, S4.3.2, S4.4 and KD45, logics important for Epistemic Logic. At last, we use the mu-calculus to define degrees of ignorance on Epistemic Logic and study the implications of mu-calculus's collapse over the logics above.