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 .