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# Ultraproduct

Let be a language of first-order predicate logic, let be an indexing set, and for each , let be a structure of the language . Let be an ultrafilter in the power set Boolean algebra . Then the ultraproduct of the family is the structure that is given by the following:

1. For each fundamental constant of the language , the value of is the equivalence class of the tuple , modulo the ultrafilter .

2. For each -ary fundamental relation of the language , the value of is given as follows: The tuple is in if and only if the set is a member of the ultrafilter .

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

The ultraproduct of the family is typically denoted .

Ultrafilter, Ultrapower

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

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## 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. http://www.thoralf.uwaterloo.ca/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.

Ultraproduct

## Cite this as:

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