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 .

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.