Goldblatt-Thomason-style Theorems for Graded Modal Language.
Katsuhiko Sano, Minghui Ma · 2010
We prove two main Goldblatt-Thomason-style Theorems for graded modal language in Kripke semantics: full Goldblatt-Thomason Theorem for elementary classes and relative Goldblatt-Thomason Theorem within the class of finite transitive frames. Two different semantic views on GML allow us to prove these results: neighborhood semantics and graph semantics. By neighborhood semantic view, we can define a natural generalization of Jankov-Fine formula for GML and establish relative Goldblatt-Thomason Theorem. By extracting graph semantics from Fine’s completeness proof of GML (1972), we introduce a new notion of graded ultrafilter images and establish full Goldblatt-Thomason Theorem. Therefore we revive Fine’s old idea in the new context of Goldblatt-Thomason-style characterization.