Ultraproduct

An ultraproduct is a mathematical construction, which is used in abstract algebra to construct new fields from given ones. Certainly the most important case is the construction of the hyperreal numbers by taking the ultraproduct of countably infinitely many copies of the field of real numbers.

The general construction uses an index set I, a field F_i for each element i of I, and an ultrafilter U on I (the usual choice is for I to be infinite and U to contain all cofinite subsets of I).

Algebraic operations on the cartesian product are defined in the usual way (such that ), and an equivalence relation is defined by a ~ b if and only if the set is in U, and the ultraproduct is the quotient set with regard to ~.

Other relations can be extended the same way: if and only if is in U. In particular, if every F_i is an ordered field, then so is the ultraproduct.

Examples

The hyperreal numbers are the ultraproduct of one copy of the real numbers for every natural number, with regard to an ultrafilter containing all cofinite sets of natural numbers. Their order is the extension of the order of the real numbers.

Analogously, you could define nonstandard complex numbers by taking the ultraproduct of copies of the field of complex numbers.