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.

中文翻译：

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模型理论关系的模态逻辑

给定一类模型 $\mathcal C$、模型之间的二元关系 ${\mathcal R}$ 和模型理论语言 $L$，我们考虑 $\mathcal 理论的模态逻辑和模态代数$L$ 中的 C$，其中模态运算符通过 $\mathcal R$ 解释。我们讨论了 $\mathcal C$ 和 ${\mathcal R}$ 的模态理论如何依赖于模型理论语言、它们的 Kripke 完备性以及 $L$ 内模态的可表达性。我们为子模型和商关系计算这些理论。我们证明了一阶语言的向下Lowenheim-Skolem定理，用模态算子扩展模型之间的扩展关系。