Algebraic characterisation of pseudo-elementary and second-order classes

TL;DR

This paper offers algebraic characterisations for classes definable in second-order logic and pseudo-elementary classes, resolving open problems and providing structural classifications.

Contribution

It provides the first purely algebraic characterisations of pseudo-elementary classes and their subclasses, advancing the understanding of second-order definability.

Findings

Algebraic characterisation of PC_{ ext{Δ}} classes via closure properties

Structural classification of second-order equivalent structures

Algebraic characterisations of classes definable by second-order formulas

Abstract

In this paper we provide purely model-theoretic (algebraic) characterisations for classes definable in second-order logic and for pseudo-elementary classes (including PC and PC_{\Delta} classes). Classical results of this flavour include Keisler-Shelah type theorems (characterising first-order definability by closure under ultraproducts and ultraroots) and Birkhoff's HSP theorem; a key starting point for this paper is S\'agi's work, which provides an algebraic description of classes definable by existential second-order sentences. Here we resolve several open problems from the literature. Our main results are the following. We solve the long-standing problem of giving a purely algebraic characterisation of pseudo-elementary classes: we characterise PC_{\Delta} classes by intrinsic closure properties. We also give a characterisation for the basic pseudo-elementary classes (PC). We provide a structural classification of second-order equivalent structures, and we obtain purely algebraic characterisations of the classes definable by second-order formulas as well as those definable by finitely many second-order sentences.