nLab Goldblatt-Thomason theorem

Goldblatt-Thomason theorem is a theorem in modal logic which characterizes the elementary classes which are also modally definable classes. It says that if $K$ is an elementary class of Kripke frames, then $K$ is modally definable iff $K$ is closed under taking bounded morphic images, generated subframes, disjoint unions, and reflects ultrafilter extensions.

Idea

The theorem is based on the duality between the category of Boolean algebras and Sets

$$BA \stackrel{\overset{\Pi}{\leftarrow}}{\underset{\Sigma}{\to}} Set^{op}$$

where $\Pi$ is the power set functor and $\Sigma$ assigns to a Boolean algebra the set of its ultrafilters. $\Sigma$ is left-adjoint to $\Pi$, but in general they do not form a dual equivalence of categories. This only holds in the finite case, where it is enough for $\Sigma$ to assign to a Boolean algebra the set of its atoms.

The idea of the proof is to translate between Kripke frames and Modal algebras, and obtain the above result by an application of Birkhoff's HSP theorem.

References

• Brief summary of Goldblatt-Thomason theorem, pdf

• The Goldblatt-Thomason Theorem for Coalgebras, pdf

category: logic