A note on bisimulation quantifiers and fixed points over transitive frames

We consider three basic questions regarding the extension of modal logic with a special kind of propositional quantifiers, known as bisimulation quantifiers, over arbitrary classes of frames: bisimulation invariance, uniform interpolation, and expressive power. In particular: we discuss the relation between bisimulation invariance of bisimulation quantifiers and the semantical notion of amalgamation of the class of frames; we consider a strong form of interpolation, uniform interpolation, and its relation with the closure under bisimulation quantifiers; we compare bisimulation quantifiers logic with the better known extension of modal logic with extremal fixed points.