Ultrapower

A structure obtained from a given structure M by taking the Cartesian product M^I of copies of M indexed by a set I and then factoring by an ultrafilter U on I: elements are equivalence classes of functions I → M that agree on a set in U. Ultrapowers and more general ultraproducts are central tools for producing elementary extensions and nonstandard models.

An ultrapower of a structure M by an ultrafilter U on an index set I is the ultraproduct of the constant family (M)_{i∈I} modulo U; elements are equivalence classes of sequences in M, and Łoś's theorem makes the ultrapower an elementary extension of M under nonprincipal ultrafilters.