Counterexample of Goldblatt-Thomason theorem The Goldblatt-Thomason theorem states that a class of first-order definable Kripke frames is modally definable if it * is closed under disjoint unions, * is closed under generated subframes, * is closed under p-morphic images, * does not contain ultrafilter extensions of frames not in the class. I want to find an example of a FO definable class that is closed under the first three conditions, but that contains an ultrafilter extension of a frame not in the class. So far any frame class I can think of or find is either not closed under one of the first three conditions, or not FO definable.

I found one in Blackburn, Venema & de Rijke's _Modal Logic_ :

The class of frames defined by $\phi:=\forall x\exists y(xRy\land yRy)$ is closed under all of the first three conditions, but the ultrafilter extension of $\langle \mathbb N,<\rangle$ also is a model of $\phi$.