Stable definability and generic relations

An amalgamation base p in a simple theory is stably definable if its canonical base is interdefinable with the set of canonical parameters for the double-struck P sign-definitions of p as double-struck P sign ranges through all stable formulae. A necessary condition for stably definability is given and used to produce an example of a supersimple theory with stable forking having types that are not stably definable. This answers negatively a question posed in [8]. A criterion for and example of a stably definable amalgamation base whose restriction to the canonical base is not axiomatised by stable formulae are also given. The examples involve generic relations over non CM-trivial stable theories.