Given a family of structures of the same signature in the sense of model theory (or more specially, universal algebra), one can (assuming the ultrafilter principle, a weak form of the axiom of choice) use ultrafilters to form a certain congruence on the direct product and construct a quotient object, a new structure of the same signature, called an ultraproduct. An ultraproduct of some number of copies of the same structure may be called an ultrapower.

The hyperreal numbers (wikipedia) and nonstandard integers in nonstandard analysis are obtained as countable ultrapowers with help of free ultrafilters on \mathbb{N}. Such ultrafilters contain all cofinite? subsets of integers, but not only them. See wikipedia:ultraproduct.

From Michael Barr’s Models of Sketches

Unlike limits and colimits, an ultraproduct is not defined by any universal mapping property. Of course, if the category has limits and (filtered) colimits, then it has ultraproducts constructed as colimits of products … But usually the category of models of a coherent theory (such as the theory of fields) lacks products and hence does not have categorical ultraproducts.

  • M. Makkai, Ultraproducts and categorical logic, Lectures Notes in Math. 1130, Springer 1985, pp. 222–309

Revised on May 31, 2012 18:08:08 by Zoran Škoda (