Scott-Karp analysis without sentences

TL;DR

This paper refines Scott and Karp's analysis by removing the need for infinitary formulas, creating a more abstract framework that extends to topological group actions and introduces game-theoretic equivalences.

Contribution

It offers a more abstract, infinitary-free version of Scott-Karp analysis and extends the framework to Hjorth's global similarity in topological group actions, including new game-theoretic characterizations.

Findings

Hierarchies still provide desired equivalences in classical settings

Framework extends to Hjorth's global similarity in topological groups

Introduces novel game-theoretic equivalents for similarity relations

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 Fra\"iss\'e, novel game theoretic equivalents for Hjorth's global similarity relations and some natural variants.