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Preservation Theorems Through the Lens of Topology

Aliaume Lopez

TL;DR

This paper develops a unifying topological framework for preservation theorems by introducing logically presented pre-spectral spaces (lpps) and diagram bases, which connect semantic preservation to syntactic fragments via topological openness. A generic preservation theorem shows that preservation holds exactly when the definable opens form compact sets in a topology generated by FO-definable sets, enabling construction of new preservation results by composing spaces through morphisms, subspaces, finite sums, and finite products. The authors relate the framework to Noetherian and spectral spaces, prove stability under closures and FO-interpretations, and show that Rossman’s finite-homomorphism preservation result extends to classes with the finite model property via projective limits. They further demonstrate that all pre-spectral spaces arise as limits of Noetherian spaces, providing a comprehensive, modular approach to classical and novel preservation theorems with applications to finite structure classes and beyond.

Abstract

In this paper, we introduce a family of topological spaces that captures the existence of preservation theorems. The structure of those spaces allows us to study the relativisation of preservation theorems under suitable definitions of surjective morphisms, subclasses, sums, products, topological closures, and projective limits. Throughout the paper, we also integrate already known results into this new framework and show how it captures th essence of their proofs.

Preservation Theorems Through the Lens of Topology

TL;DR

This paper develops a unifying topological framework for preservation theorems by introducing logically presented pre-spectral spaces (lpps) and diagram bases, which connect semantic preservation to syntactic fragments via topological openness. A generic preservation theorem shows that preservation holds exactly when the definable opens form compact sets in a topology generated by FO-definable sets, enabling construction of new preservation results by composing spaces through morphisms, subspaces, finite sums, and finite products. The authors relate the framework to Noetherian and spectral spaces, prove stability under closures and FO-interpretations, and show that Rossman’s finite-homomorphism preservation result extends to classes with the finite model property via projective limits. They further demonstrate that all pre-spectral spaces arise as limits of Noetherian spaces, providing a comprehensive, modular approach to classical and novel preservation theorems with applications to finite structure classes and beyond.

Abstract

In this paper, we introduce a family of topological spaces that captures the existence of preservation theorems. The structure of those spaces allows us to study the relativisation of preservation theorems under suitable definitions of surjective morphisms, subclasses, sums, products, topological closures, and projective limits. Throughout the paper, we also integrate already known results into this new framework and show how it captures th essence of their proofs.

Paper Structure

This paper contains 45 sections, 23 theorems, 7 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preservation Theorems
  3. Classical Preservation Theorems
  4. Notations.
  5. Abstract Preservation.
  6. Monotone Preservation.
  7. The Łoś-Tarski Theorem in Topological Terms
  8. 'Syntactic' Base.
  9. Compactness.
  10. Pre-spectral Spaces and Diagram Bases
  11. Pre-spectral Spaces
  12. Logical presentations.
  13. Diagram Bases
  14. A Generic Preservation Theorem
  15. Downward Closures.
  16. ...and 30 more sections

Key Result

Theorem 3.4

Let $\uptau$ be a topology on $X$, $\mathcal{L}$ a bounded sublattice of $\wp(X)$, and $\mathcal{L}'$ a sublattice of $\mathcal{L}$. The following are equivalent:

Figures (3)

  • Figure 1: Different induced subspace constructions.
  • Figure 2: Saturation of maximal cylinder open subsets in the case of a binary product $X \times Y$.
  • Figure 3: The commutative diagram of a projective system.

Theorems & Definitions (69)

  • Claim 2.1
  • proof
  • Claim 2.2
  • proof
  • Remark 2.3: Compact sets in $\uptau_{\leq}$
  • proof : 2.2.3. Proof of the Łoś-Tarski Theorem
  • Definition 3.1: Pre-spectral space
  • Definition 3.2: Logically presented pre-spectral space
  • Definition 3.3: Diagram base
  • Theorem 3.4: Generic preservation
  • ...and 59 more