TY - JOUR
T1 - Pairs of theories satisfying a Mordell-Lang condition
AU - Gorman, Alexi Block
AU - Hieronymi, Philipp
AU - Kaplan, Elliot
N1 - Funding Information:
The first author was partially supported by a DOE GAANN fellowship. The second author was partially supported by NSF grant DMS-1654725.
PY - 2020
Y1 - 2020
N2 - 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.
AB - 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.
KW - Dense pairs
KW - Geometric structures
KW - H-structures
KW - Lovely pairs
KW - Mann groups
KW - NIP
KW - O-minimal structures
KW - Open core
KW - P-minimal structures
KW - Pseudo real closed fields
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U2 - 10.4064/fm857-1-2020
DO - 10.4064/fm857-1-2020
M3 - Article
SN - 0016-2736
VL - 251
SP - 131
EP - 160
JO - Fundamenta Mathematicae
JF - Fundamenta Mathematicae
IS - 2
ER -