TY - JOUR

T1 - On the forking topology of a reduct of a simple theory

AU - Shami, Ziv

N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - Let T be a simple L-theory and let T- be a reduct of T to a sublanguage L- of L. For variables x, we call an ∅ -invariant set Γ (x) in C a universal transducer if for every formula ϕ-(x, y) ∈ L- and every a, ϕ-(x,a)L--forksover∅iffΓ(x)∧ϕ-(x,a)L-forksover∅.We show that there is a greatest universal transducer Γ ~ x (for any x) and it is type-definable. In particular, the forking topology on Sy(T) refines the forking topology on Sy(T-) for all y. Moreover, we describe the set of universal transducers in terms of certain topology on the Stone space and show that Γ ~ x is the unique universal transducer that is L--type-definable with parameters. If T- is a theory with the wnfcp (the weak nfcp) and T is the theory of its lovely pairs of models we show that Γ ~ x= (x= x) and give a more precise description of the set of universal transducers for the special case where T- has the nfcp.

AB - Let T be a simple L-theory and let T- be a reduct of T to a sublanguage L- of L. For variables x, we call an ∅ -invariant set Γ (x) in C a universal transducer if for every formula ϕ-(x, y) ∈ L- and every a, ϕ-(x,a)L--forksover∅iffΓ(x)∧ϕ-(x,a)L-forksover∅.We show that there is a greatest universal transducer Γ ~ x (for any x) and it is type-definable. In particular, the forking topology on Sy(T) refines the forking topology on Sy(T-) for all y. Moreover, we describe the set of universal transducers in terms of certain topology on the Stone space and show that Γ ~ x is the unique universal transducer that is L--type-definable with parameters. If T- is a theory with the wnfcp (the weak nfcp) and T is the theory of its lovely pairs of models we show that Γ ~ x= (x= x) and give a more precise description of the set of universal transducers for the special case where T- has the nfcp.

KW - Forking topology

KW - Reduct

KW - Universal transducer

UR - http://www.scopus.com/inward/record.url?scp=85072021772&partnerID=8YFLogxK

U2 - 10.1007/s00153-019-00691-w

DO - 10.1007/s00153-019-00691-w

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AN - SCOPUS:85072021772

SN - 0933-5846

VL - 59

SP - 313

EP - 324

JO - Archive for Mathematical Logic

JF - Archive for Mathematical Logic

IS - 3-4

ER -