On the forking topology of a reduct of a simple theory

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, $$\begin{aligned} \phi ^-(x,a) L^-\text{-forks \text{ over } \emptyset \text{ iff \Gamma (x)\wedge \phi ^-(x,a) L\text{-forks \text{ over } \emptyset . \end{aligned}$$ 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.