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We study model and frame definability of various modal logics. Let ML(A+)denote the fragment of modal logic extended with the universal modality inwhich the universal modality occurs only positively. We show that a class ofKripke models is definable in ML(A+) if and only if the class is elementary andclosed under disjoint unions and surjective bisimulations. We also characterisethe definability of ML(A+) in the spirit of the well-known Goldblatt--Thomasontheorem. We show that an elementary class F of Kripke frames is definable inML(A+) if and only if F is closed under taking generated subframes and boundedmorphic images, and reflects ultrafilter extensions and finitely generatedsubframes. In addition we study frame definability relative to finitetransitive frames and give an analogous characterisation of ML(A+)-definabilityrelative to finite transitive frames. Finally, we initiate the study of modeland frame definability in team-based logics. We study (extended) modaldependence logic, (extended) modal inclusion logic, and modal team logic. Weestablish strict linear hierarchies with respect to model definability andframe definability, respectively. We show that, with respect to model and framedefinability, the before mentioned team-based logics, except modal dependencelogic, either coincide with ML(A+) or plain modal logic ML. Thus as a corollarywe obtain model theoretic characterisation of model and frame definability forthe team-based logics.

Characterising Modal Definability of Team-Based Logics via the Universal Modality