Lovely pairs of models: the non first order case

TL;DR

This paper extends the concept of lovely pairs from first-order theories to simple and thick compact abstract theories, establishing their properties and independence relations.

Contribution

It generalizes the theory of lovely pairs to non-first-order contexts, proving simplicity and characterizing independence in this broader setting.

Findings

Existence of a unique compact abstract theory T^P for every simple theory T

T^P is simple and its models are lovely pairs of T

Independence relations are characterized in the non-first-order setting

Abstract

We prove that for every simple theory $T$ (or even simple thick compact abstract theory) there is a (unique) compact abstract theory $T^\fP$ whose saturated models are the lovely pairs of $T$. Independence-theoretic results that were proved in \cite{ppv:pairs} when $T^\fP$ is a first order theory are proved for the general case: in particular $T^\fP$ is simple and we characterise independence.