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The abstract model-theoretic concepts of compactness and Lowenheim–Skolem properties are investigated in the “softer” framework of pre-institutions [18]. Two compactness results are presented in this paper: a more informative reformulation of the compactness theorem for pre-institution transformations, and a theorem on natural equivalences with an abstract form of the first-order pre-institution. These results rely on notions of compact transformation, which are introduced as arrow-oriented generalizations of the classical, object-oriented notions of compactness. Furthermore, a notion of cardinal pre-institution is introduced, and a Lowenheim– Skolem preservation theorem for cardinal pre-institutions is presented.

Compactness and Löwenheim-Skolem properties in categories of pre-institutions