An ultraproduct is the quotient set of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of the ultraproduct in which all factors are equal (analogous to an exponentiation). For example, ultrapowers can be used to construct new fieldss from given ones. A special case of this is the ultrapower of the real numbers, which satisfies the conditions for the hyperreal numbers.

This mathematical construction appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson.

Definition

a structure (all of the same signature), and an ultrafilter

of the Cartesian product declare them to be This binary relation is an equivalence relation

The is the quotient set

-equivalence class

is merely a filter is called a .

is a principal ultrafilter contains its kernel is not a principal ultrafilter), or equivalently, if every cofinite is consequently also usually infinite.

The ultraproduct acts as a filter product space where elements are equal if they are equal only at the filtered components (non-filtered components are ignored under the equivalence). One may define a finitely additive measure otherwise. Then two members of the Cartesian product are equivalent precisely if they are equal almost everywhere on the index set. The ultraproduct is the set of equivalence classes thus generated.

Finitary operations). Other relations is an ordered field then so is the ultraproduct.

Ultrapower

An ultrapower are equal. Explicitly, the

The

Examples

The hyperreal numbers are the ultraproduct of one copy of the real numbers defines an equivalence class representing a hyperreal number that is greater than any real number.

Analogously, one can define nonstandard integers, nonstandard complex numbers, etc., by taking the ultraproduct of copies of the corresponding structures.

belong to the same equivalence class.

In the theory of large cardinals-complete, then the ultraproduct will again be well-founded. (See measurable cardinal for the prototypical example.)

Łoś's theorem

Łoś's theorem, also called , is due to Jerzy Łoś (the surname is pronounced , approximately "wash", or ). It states that any first-order More precisely:

is an ultrafilter (and not just a filter) is used in the negation clause, and the axiom of choice is needed at the existential quantifier step. As an application, one obtains the transfer theorem for hyperreal fields.

Examples

. has an analog has the same property. That is, we can define a notion of the hyperrational numbers, which are a subset of the hyperreals, and they have the same first-order properties as the rationals.

Consider, however, the Archimedean property above.

Direct limits of ultrapowers (ultralimits)

In model theory and set theory, the direct limit of a sequence of ultrapowers is often considered. In model theory, this construction can be referred to as an ultralimit or limiting ultrapower.

form the direct limit of earlier stages. One may continue into the transfinite.

Ultraproduct monad

The ultrafilter monad is the codensity monad of the inclusion of the category of finite sets into the category of all sets.

Similarly, the of finitely-indexed families of sets of all indexed families of sets. So in this sense, ultraproducts are categorically inevitable. consists of a non-empty index set and an indexed family

See also

Notes

Proofs

References