I am away from base so cannot check, but I seem to remember that it is exactly a relational version of Birkhoff’s theorem. There is a survey of Algebraic Polymodal Logic by Goldblatt that I have used and I remember that it has a discussion of the theorem. Note there is a paper linking the theorem with coalgebras. It starts:
”The Goldblatt-Thomason theorem [11] states that a class of Kripke frames closed under ultrafilter extensions is modally definable if and only if it reflects ultrafilter extensions and is closed under generated subframes, homomorphic images and disjoint unions….”
]]>I’m not really understanding the theorem, but it sounds similar to the Birkhoff characterization of equational varieties.
]]>Thanks.
I have added a link to this from modal logic.
]]>