Core Concepts
Preservation theorems in first-order logic can be characterized by the topological properties of logically presented pre-spectral spaces.
Abstract
The paper introduces a topological framework for investigating preservation theorems in first-order logic. It defines the notion of logically presented pre-spectral spaces, which capture the key ingredients behind the classical proofs of preservation theorems like the Łoś-Tarski Theorem.
The main contributions are:
Defining logically presented pre-spectral spaces and diagram bases, and showing that a preservation theorem holds if and only if the space under consideration is logically presented pre-spectral (Theorem 3.4).
Establishing the stability of logically presented pre-spectral spaces under various topological constructions, such as under subclasses, finite sums and products, and limits (Sections 5-7).
Illustrating the framework by re-casting known preservation results, like Rossman's Theorem on homomorphism preservation in the finite, in the new topological setting.
The topological viewpoint allows preservation theorems to be obtained as byproducts of topological constructions, rather than requiring complex model-theoretic arguments from scratch for each new result.
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