On the forking topology of a reduct of a simple theory

Let $T$ be simple and $T^{-}$ a reduct of $T$. For variables $x$, we call an $\mathrm{\varnothing }$-invariant set $\mathrm{\Gamma }(x)$ of $\mathcal{C}$ with the property that for every formula ${\varphi }^{-}(x,y)\in L^{-}$: for every $a$, ${\varphi }^{-}(x,a)$ $L^{-}$-forks over $\mathrm{\varnothing }$ iff $\mathrm{\Gamma }(x)\wedge {\varphi }^{-}(x,a)$ $L$-forks over $\mathrm{\varnothing }$, a \em universal transducer\em. We show that there is a greatest universal transducer ${\tilde{\mathrm{\Gamma }}}_x$ (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 ${\tilde{\mathrm{\Gamma }}}_x$ 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 ${\tilde{\mathrm{\Gamma }}}_x=(x=x)$ and give a more precise description of all its universal transducers in case $T^{-}$ has the nfcp.