Notes on modal definability (Q1119622)

The paper contains a series of results in Kripke model theory for propositional modal logics related mainly to the notion of an ultrafilter extension [\textit{J. van Benthem}, ``Canonical modal logics and ultrafilter extensions'', J. Symb. Logic 44, 1-8 (1979; Zbl 0405.03011)]. It is proved that a subframe logic (i.e. the logic preserved under subframes, not necessarily generated) is elementary iff it is preserved under ultrafilter extensions. A consequence of this: A subframe logic is canonical iff it is elementary and complete. Another new result is: On the finite transitive frames, a class of frames is modally definable iff it is closed under generated subframes, disjoint sums and p-morphic images. Also a modal formula is built which is non-elementary in the class of all finite frames.