Abstract
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 \(\emptyset \)-invariant set \(\Gamma (x)\) in \({\mathcal {C}}\) a universal transducer if for every formula \(\phi ^-(x,y)\in L^-\) and every a,
We show that there is a greatest universal transducer \(\tilde{\Gamma }_x\) (for any x) and it is type-definable. In particular, the forking topology on \(S_y(T)\) refines the forking topology on \(S_y(T^-)\) for all y. Moreover, we describe the set of universal transducers in terms of certain topology on the Stone space and show that \(\tilde{\Gamma }_x\) 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 \(\tilde{\Gamma }_x=(x=x)\) and give a more precise description of the set of universal transducers for the special case where \(T^-\) has the nfcp.
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Shami, Z. On the forking topology of a reduct of a simple theory. Arch. Math. Logic 59, 313–324 (2020). https://doi.org/10.1007/s00153-019-00691-w
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DOI: https://doi.org/10.1007/s00153-019-00691-w