Ultraproduct

An ultraproduct is a structure obtained by identifying elements of a direct product of structures that agree on a set of indices belonging to an ultrafilter. To define it, let be a formal language for first-order logic, let be an index set, and for each , let be a structure for with underlying set . Let be an ultrafilter in the power set Boolean algebra . On the Cartesian product , define the equivalence relation iff . The universe of the ultraproduct is the set of equivalence classes for this relation. Its constants, relations, and operations are interpreted as follows:

1. For each constant of , the value of is the equivalence class of the family .

2. For each -ary relation of , the n-tuple is in iff the set ${i in I|(x_1(i),...,x_n(i)) in R^((A_i))}$ is a member of the ultrafilter .

3. For each -ary operation of , and for each n-tuple , the value of is .

The ultraproduct of the family is typically denoted .

See also

Ultrafilter, Ultrapower

This entry contributed by Matt Insall (author's link)

Explore with Wolfram|Alpha

More things to try:

References

Bell, J. L. and Slomson, A. B. Models and Ultraproducts: an Introduction. Amsterdam, Netherlands: North-Holland, 1971.Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981. https://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html.Enderton, H. B. A Mathematical Introduction to Logic. New York: Academic Press, 1972.Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985.

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W., with contributions by Matt Insall. "Ultraproduct." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Ultraproduct.html

Subject classifications