On modal logics of model-theoretic relations

Given a class $mathcal C$ of models, a binary relation ${mathcal R}$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $mathcal C$ in $L$ where the modal operator is interpreted via $mathcal R$. We discuss how modal theories of $mathcal C$ and ${mathcal R}$ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside $L$. We calculate such theories for the submodel and the quotient relations. We prove a downward Lowenheim--Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.