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Scott-Karp analysis without sentences

Andreas Brunner, Charles Morgan, Darllan Concieção Pinto

TL;DR

This paper develops a formula-free version of the local strand of Scott–Karp analysis, providing a back-and-forth framework that equates local invariants with global similarities without recourse to infinitary sentences. It introduces hierarchies of invariant-function data $F^{\alpha}_{M,\bar a}$, $H^{\alpha}_{M,\bar a}$, and $G^{\alpha}_{M,N,\bar a}$, proving they determine $\sim_{\alpha}$ and recover Karp-style equivalences. The approach also extends to Hjorth’s broader setting of topological group actions, yielding new game-theoretic equivalents for Hjorth’s global similarity relations. Equivalences are connected to infinitary sentences via a translation and to Karp's theorem, enabling reading of global structure from local invariants. The framework provides a unifying, language-agnostic toolkit for comparing structures and their tuples, with potential applications in descriptive set theory and model theory.

Abstract

Scott and Karp gave an analysis which provides a level-by-level equivalence between global similarity between two structures and local commonality in terms of sharing particular invariants. Scott and Karp's local invariants were certain infinitary formulae. We give a more abstract version of the local side of Scott-Karp analysis which avoids the use of infinitary languages. We show the resulting hierarchies are still provide desired equivalences in the classical setting. Moreover, the abstract nature of our analysis, as we show, makes it suitable to provide local level-by-level equivalents to Hjorth's much more general version of global similarity in the context of topological group actions on topological groups. We, furthermore, provide, analogously to the classical work of Ehrenfeucht and Fraïssé, novel game theoretic equivalents for Hjorth's global similarity relations and some natural variants.

Scott-Karp analysis without sentences

TL;DR

This paper develops a formula-free version of the local strand of Scott–Karp analysis, providing a back-and-forth framework that equates local invariants with global similarities without recourse to infinitary sentences. It introduces hierarchies of invariant-function data , , and , proving they determine and recover Karp-style equivalences. The approach also extends to Hjorth’s broader setting of topological group actions, yielding new game-theoretic equivalents for Hjorth’s global similarity relations. Equivalences are connected to infinitary sentences via a translation and to Karp's theorem, enabling reading of global structure from local invariants. The framework provides a unifying, language-agnostic toolkit for comparing structures and their tuples, with potential applications in descriptive set theory and model theory.

Abstract

Scott and Karp gave an analysis which provides a level-by-level equivalence between global similarity between two structures and local commonality in terms of sharing particular invariants. Scott and Karp's local invariants were certain infinitary formulae. We give a more abstract version of the local side of Scott-Karp analysis which avoids the use of infinitary languages. We show the resulting hierarchies are still provide desired equivalences in the classical setting. Moreover, the abstract nature of our analysis, as we show, makes it suitable to provide local level-by-level equivalents to Hjorth's much more general version of global similarity in the context of topological group actions on topological groups. We, furthermore, provide, analogously to the classical work of Ehrenfeucht and Fraïssé, novel game theoretic equivalents for Hjorth's global similarity relations and some natural variants.
Paper Structure (8 sections, 23 theorems, 33 equations)

This paper contains 8 sections, 23 theorems, 33 equations.

Table of Contents

  1. Preliminaries
  2. Equivalence relations for tuples from two structures
  3. Scott-Karp analysis without formulae
  4. Relationship with the original Scott-Karp analysis
  5. Degrees and ranks
  6. Karp's theorem
  7. Hjorth analysis and games
  8. Hjorth analysis, a Karp-style theorem

Key Result

Lemma 3.4

Let $M$ be an $L$-structure, $\bar{a}\in {^{<\lambda}|M|}$, and $\alpha\in\mathop{\mathrm{On}}$. Let $K$ be an $L$-structure, $\bar{s}\in {^{\mathop{\mathrm{lh}}(\bar{a})}|K|}$ and $\bar{t}\in {^{<\mu}N}$. Let $\beta<\alpha$. Then Moreover, if further $N$ is an $L$-structure and $\bar{b}\in {^{\mathop{\mathrm{lh}}(\bar{a})}|N|}$

Theorems & Definitions (68)

  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1
  • ...and 58 more