Let L be a language of first-order predicate logic, let I be an indexing set, and for each i in I, let A_i be a structure of the language L. Let u be an ultrafilter in the power set Boolean algebra P(I). Then the ultraproduct of the family (A_i)_(i in I) is the structure A that is given by the following:

1. For each fundamental constant c of the language L, the value of c^((A)) is the equivalence class of the tuple (c^((A_i)))_(i in I), modulo the ultrafilter u.

2. For each n-ary fundamental relation R of the language L, the value of R^((A)) is given as follows: The tuple ([x_1]_u,...,[x_n]_u) is in R^((A)) if and only if the set {i in I|(x_1(i),...,x_n(i))} is a member of the ultrafilter u.

3. For each n-ary fundamental operation f of the language L, and for each n-tuple ([x_1]_u,...,[x_n]_u), the value of f^((A))([x_1]_u,...,[x_n]_u) is [f^((A))(x_1,...,x_n)]_u.

The ultraproduct A of the family (A_i)_(i in I) is typically denoted (A_i)_(i in I)/u.

See also

Ultrafilter, Ultrapower

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

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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., 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.

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Insall, Matt. "Ultraproduct." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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