Pairs of theories satisfying a Mordell–Lang condition
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, at least in part, a couple concrete open questions raised about open cores and decidability. The first is: for which subfields $K \subseteq \mathbb R $ is $\mathbb R $ as an ordered $K$-vector space expanded by a predicate for $\mathbb Q $ decidable? The second is whether there is a subfield $K$ of a real closed field that is not real closed, yet every open set definable in the expansion of the real field by $K$ is semialgebraic.