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 setBoolean 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 .