The Goldblatt-Thomason theorem characterizes elementary frame classes that are modally definable to be exactly those that are closed under p-morphic images, generated subframes and disjoint unions, and that in addition reflect ultrafilter extensions. We give variations on this theorem by restricting the frame classes (to finite or image-finite frames) and by generalizing the modal language (with the path quantifier and/or counting modalities).

The second part of this work (Chapter 5) generalizes the concept of definability, which is given in terms of validity of formulas. The validity on the level of frames corresponds to formulas of monadic second order logic, that is, quantification over sets of the universum. We introduce a new concept of validity that allows, from the perspective of second order logic, quantification over (binary) relations.