Lovely pairs for independence relations

TL;DR

This paper introduces a generalized concept of lovely pairs for an independence relation that encompasses both simple and geometric theories, extending key theorems to this broader framework.

Contribution

It proposes a unified notion of lovely pairs for independence relations, bridging the gap between simple and geometric theories with extended theorems.

Findings

Unified framework for lovely pairs across theories

Extension of main theorems to the generalized setting

Broader applicability of independence relation concepts

Abstract

In the literature there are two different notions of lovely pairs of a theory T, according to whether T is simple or geometric. We introduce a notion of lovely pairs for an independence relation, which generalizes both the simple and the geometric case, and show how the main theorems for those two cases extend to our general notion.