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 KK is an elementary class of Kripke frames, then KK is modally definable iff KK 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

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

where Π\Pi is the power set functor and Σ\Sigma assigns to a Boolean algebra the set of its ultrafilters. Σ\Sigma is left-adjoint to Π\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 Σ\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

Last revised on March 22, 2017 at 11:13:36. See the history of this page for a list of all contributions to it.