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. In this article we show that the answers to these questions that are valid for the class of all frames do not generalize to arbitrary classes, but they do generalize if we restrict to classes of (finite) transitive or (finite) transitive and reflexive frames.