Submission date: 15 December 2018

Abstract:

Let T be a simple theory and T^- is a reduct of T. For variables x, we call an ∅-invariant set Γ(x) of C with the property that for every formula φ^-(x,y) in L^-: for every a, φ^-(x,a) L^- -forks over &empty: iff Γ(x) ∧ φ^-(x,a) L-forks over ∅, a universal transducer. 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 S_y(T) refines the forking topology on S_y(T^-). 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. In the case where T^- is a theory with the wnfcp (the weak nfcp) and T is the theory of its lovely pairs we show Γ˜_x=(x=x) and give a more precise description of all its universal transducers in case T has the nfcp.

Mathematics Subject Classification: 03C45

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Full text arXiv 1812.06351: pdf, ps.