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.