Pseudofinite counting functions mod n

An Euler characteristic χ is definable if for every definable function f : X → Y and every r ∈ R, the set {y ∈ Y : χ(f−1(y)) = r} is definable. If R = Z/nZ and M is a finite structure, there is a (unique) Euler characteristic χ : Def(M) → Z/nZ assigning every set its size mod n. This χ is always strong and ∅definable. If M is an ultraproduct of finite structures, then there is a canonical strong Euler characteristic χ : Def(M) → Z/nZ coming from the ultraproduct. Specifically, if M is an ultraproduct ∏ i∈IMi/U , and X is a definable set in M of the form φ(M ; a), and a is the class of some tuple 〈ai〉i∈I ∈ ∏ i∈IMi, then take χ(X) to be the ultralimit of |φ(Mi; ai)| mod n. This limit Euler characteristic will always be strong, but need not be definable.