Characterizing Relative 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 relative definability of ML(U^+) relative to finite transitive frames in the spirit of the well-known Goldblatt-Thomason theorem. We show that a class F of Kripke frames is definable in ML(U^+) relative to finite transitive frames if and only if F is closed under taking generated subframes and bounded morphic images. In addition, we study modal definability in team-based logics. We study (extended) modal dependence logic, (extended) modal inclusion logic, and modal team logic. With respect to global model definability we obtain a trichotomy and with respect to frame definability a dichotomy. As a corollary we obtain relative Goldblatt--Thomason -style theorems for each of the logics listed above.