Ultraproduct - Definition

Definition

The general method for getting ultraproducts uses an index set I, a structure M_i for each element i of I (all of the same signature), and an ultrafilter U on I. The usual choice is for I to be infinite and U to contain all cofinite subsets of I. Otherwise the ultrafilter is principal, and the ultraproduct is isomorphic to one of the factors.

Algebraic operations on the Cartesian product

are defined in the usual way (for example, for a binary function +, (a + b) _i = a_i + b_i ), and an equivalence relation is defined by a ~ b if and only if

and the ultraproduct is the quotient set with respect to ~. The ultraproduct is therefore sometimes denoted by

One may define a finitely additive measure m on the index set I by saying m(A) = 1 if A ∈ U and = 0 otherwise. Then two members of the Cartesian product are equivalent precisely if they are equal almost everywhere on the index set. The ultraproduct is the set of equivalence classes thus generated.

Other relations can be extended the same way:

where denotes the equivalence class of a with respect to ~.

In particular, if every M_i is an ordered field, then so is the ultraproduct.

An ultrapower is an ultraproduct for which all the factors M_i are equal:

More generally, the construction above can be carried out whenever U is a filter on I; the resulting model is then called a reduced product.