Generic derivations on o-minimal structures
A derivation on a field K is a map d from K to K such that d(x + y) = d(x) + d(y) and d(x y) = x d(y) + d(x) y.
Given an o-minimal structure M in a language L, we introduce the notion L-derivation, i.e derivation compatible with L. For example, if M is the field of reals with exponentiation, then we further require that the derivation d satisfies d(exp x) = exp(x) d(x).
We show that if T is the theory of M, and Td is the theory of L-derivations on M, then (assuming T has elimination of quantifiers) Td has a model completion Tdg, the theory of "generic" L-derivations. We study the properties of Tdg: for instance, T is the open core of Tdg. The case when T is the theory of real closed ring has already been studied by Singer, Tressl, and others: in that case, every derivation is already an L-derivation. Similar notions and results can be done in the settings of p-adics, etc.