The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. 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 this construction in which all factors are equal.
For example, ultrapowers can be used to construct new fields from given ones. A special case of this is the ultrapower of the real numbers, which satisfies the conditions for the hyperreal numbers.
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
The general method for getting ultraproducts uses an index set <math>I,</math> a structure <math>M_i</math> (assumed to be non-empty in this article) for each element <math>i \in I</math> (all of the same signature), and an ultrafilter <math>\mathcal{U}</math> on <math>I.</math>
For any two elements <math>a_\bull = \left(a_i\right)_{i \in I}</math> and <math>b_\bull = \left(b_i\right)_{i \in I}</math> of the Cartesian product
<math display=inline>{\textstyle\prod\limits_{i \in I M_i,</math>
declare them to be , written <math>a_\bull \sim b_\bull</math> or <math>a_\bull =_{\mathcal{U b_\bull,</math> if and only if the set of indices <math>\left\{i \in I : a_i = b_i\right\}</math> on which they agree is an element of <math>\mathcal{U};</math> in symbols,
<math display=block>a_\bull \sim b_\bull \; \iff \; \left\{i \in I : a_i = b_i\right\} \in \mathcal{U},</math>
which compares components only relative to the ultrafilter <math>\mathcal{U}.</math>
This binary relation <math>\, \sim \,</math> is an equivalence relation
Similarly, the is the codensity monad of the inclusion of the category <math>\mathbf{FinFam}</math> of finitely-indexed families of sets into the category <math>\mathbf{Fam}</math> of all indexed families of sets. So in this sense, ultraproducts are categorically inevitable.