On the forking topology of a reduct of a simple theory

Abstract

Let $T$ be simple and $T^-$ a reduct of $T$. For variables $x$, we call an $\emptyset$-invariant set $\Gamma(x)$ of ${{\cal C}}$ with the property that for every formula $\phi^-(x,y)\in L^-$: for every $a$, $\phi^-(x,a)$ $L^-$-forks over $\emptyset$ iff $\Gamma(x)\wedge \phi^-(x,a)$ $L$-forks over $\emptyset$, a \em universal transducer\em. 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 $S_y(T^-)$. 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. 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 $\tilde\Gamma_x=(x=x)$ and give a more precise description of all its universal transducers in case $T^-$ has the nfcp.