Pairs of Theories Satisfying a Mordell-Lang Condition

TL;DR

This paper introduces a unified framework for studying pairs of structures, encompassing many known classes and new examples, and uses it to resolve three open questions in the field.

Contribution

It develops a general setup for pairs of structures that includes various known and new classes, enabling the resolution of open problems.

Findings

Unified framework for pairs of structures

Includes new classes like real closed fields with pseudo real closed subfields

Answers three open questions in the study of pairs

Abstract

This paper proposes a new setup for studying pairs of structures. This new framework includes many of the previously studied classes of pairs, such as dense pairs of o-minimal structures, lovely pairs, fields with Mann groups, and $H$-structures, but also includes new ones, such as pairs consisting of a real closed field and a pseudo real closed subfield, and pairs of vector spaces with different fields of scalars. We use the larger generality of this framework to answer three concrete open questions raised in earlier work on this subject.