Let L be a language of first-order predicate logic, let I be an indexing set, and for each i element I, let A_i be a structure of the language L. Let u be an ultrafilter in the power set Boolean algebra P(I). Then the ultraproduct of the family (A_i)_(i element I) is the structure A that is given by the following: 1. For each fundamental constant c of the language L, the value of c^(A) is the equivalence class of the tuple (c^(A_i))_(i element I), modulo the ultrafilter u. 2. For each n-ary fundamental relation R of the language L, the value of R^(A) is given as follows: The tuple ([x_1]_u, ..., [x_n]_u) is in R^(A) if and only if the set {i element I|(x_1(i), ..., x_n(i))} is a member of the ultrafilter u.

ultrafilter | ultrapower