An NIP structure which does not interpret an infinite group but whose Shelah expansion interprets an infinite field

A P-adic structure which does not interpret an infinite field but whose Shelah completion does

“…Proof. We may suppose that O ⊆ M m and that for every O-definable X ⊆ O n there is an In [20] we described a (2 ℵ0 ) + -saturated NIP structure H such that H does not interpret an infinite group but H Sh interprets (R, +, ×). So H trace defines (R, +, ×).…”

“…In [21,20] we described NIP structures H such that H does not interpret an infinite field but the Shelah completion of H does. Here we describe a more natural example, modulo a reasonable conjecture.…”

“…This is why we start with a proper elementary extension of Q p . [21,20] are weakly o-minimal, hence dp-rank one. It is easy to see that the dp-rank of B is two, but E is dp-rank one.…”