Pairs of theories satisfying a Mordell-Lang condition

Alexi Block Gorman, Philipp Hieronymi, Elliot Kaplan

Research output: Contribution to journalArticlepeer-review


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 ⊆ R is R as an ordered K-vector space expanded by a predicate for 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.

Original languageEnglish (US)
Pages (from-to)131-160
Number of pages30
JournalFundamenta Mathematicae
Issue number2
StatePublished - 2020


  • Dense pairs
  • Geometric structures
  • H-structures
  • Lovely pairs
  • Mann groups
  • NIP
  • O-minimal structures
  • Open core
  • P-minimal structures
  • Pseudo real closed fields

ASJC Scopus subject areas

  • Algebra and Number Theory


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