Characterizing Frame Definability in Team Semantics via the Universal Modality

Let ML( u) denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We characterize the definability of ML( u) in the spirit of the well-known Goldblatt-Thomason-Theorem. We show that an elementary class F of Kripke frames is definable in ML( u) if and only if F is closed under taking generated subframes and bounded morphic images, and reflects ultrafilter extensions and finitely generated subframes. In addition, we initiate the study of modal frame definability in team-based logics. We show that, with respect to frame definability, the logics ML( u), modal logic with intuitionistic disjunction, and (extended) modal dependence logic all coincide. Thus we obtain Goldblatt-Thomason-style theorems for each of the logics listed above.