On the forking topology of a reduct of a simple theory

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 ∅ -invariant set Γ (x) in C a universal transducer if for every formula ϕ^-(x, y) ∈ L^- and every a, ϕ-(x,a)L--forksover∅iffΓ(x)∧ϕ-(x,a)L-forksover∅.We show that there is a greatest universal transducer Γ ~ _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 Γ ~ _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 Γ ~ _x= (x= x) and give a more precise description of the set of universal transducers for the special case where T^- has the nfcp.