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. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this.

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.

The general method for getting ultraproducts uses an index set ${\displaystyle I,}$ a structure ${\displaystyle M_{i}}$ (assumed to be non-empty in this article) for each element ${\displaystyle i\in I}$ (all of the same signature), and an ultrafilter ${\displaystyle {\mathcal {U}}}$ on ${\displaystyle I.}$

For any two elements ${\displaystyle a_{\bullet }=\left(a_{i}\right)_{i\in I}}$ and ${\displaystyle b_{\bullet }=\left(b_{i}\right)_{i\in I}}$ of the Cartesian product ${\textstyle {\textstyle \prod \limits _{i\in I}}M_{i},}$ declare them to be ${\displaystyle {\mathcal {U}}}$-equivalent, written ${\displaystyle a_{\bullet }\sim b_{\bullet }}$ or ${\displaystyle a_{\bullet }=_{\mathcal {U}}b_{\bullet },}$ if and only if the set of indices ${\displaystyle \left\{i\in I:a_{i}=b_{i}\right\}}$ on which they agree is an element of ${\displaystyle {\mathcal {U}};}$ in symbols, $${\displaystyle a_{\bullet }\sim b_{\bullet }\;\iff \;\left\{i\in I:a_{i}=b_{i}\right\}\in {\mathcal {U}},}$$ which compares components only relative to the ultrafilter ${\displaystyle {\mathcal {U}}.}$ This binary relation ${\displaystyle \,\sim \,}$ is an equivalence relation on the Cartesian product ${\displaystyle {\textstyle \prod \limits _{i\in I}}M_{i}.}$

The ultraproduct of ${\displaystyle M_{\bullet }=\left(M_{i}\right)_{i\in I}}$ modulo ${\displaystyle {\mathcal {U}}}$ is the quotient set of ${\displaystyle {\textstyle \prod \limits _{i\in I}}M_{i}}$ with respect to ${\displaystyle \sim }$ and is therefore sometimes denoted by ${\displaystyle {\textstyle \prod \limits _{i\in I}}M_{i}\,/\,{\mathcal {U}}}$ or ${\displaystyle {\textstyle \prod }_{\mathcal {U}}\,M_{\bullet }.}$

Explicitly, if the ${\displaystyle {\mathcal {U}}}$-equivalence class of an element ${\displaystyle a\in {\textstyle \prod \limits _{i\in I}}M_{i}}$ is denoted by $${\displaystyle a_{\mathcal {U}}:={\big \{}x\in {\textstyle \prod \limits _{i\in I}}M_{i}\;:\;x\sim a{\big \}}}$$ then the ultraproduct is the set of all ${\displaystyle {\mathcal {U}}}$-equivalence classes $${\displaystyle {\prod }_{\mathcal {U}}\,M_{\bullet }\;=\;\prod _{i\in I}M_{i}\,/\,{\mathcal {U}}\;:=\;\left\{a_{\mathcal {U}}\;:\;a\in {\textstyle \prod \limits _{i\in I}}M_{i}\right\}.}$$

Although ${\displaystyle {\mathcal {U}}}$ was assumed to be an ultrafilter, the construction above can be carried out more generally whenever ${\displaystyle {\mathcal {U}}}$ is merely a filter on ${\displaystyle I,}$ in which case the resulting quotient set ${\displaystyle {\textstyle \prod \limits _{i\in I}}M_{i}/\,{\mathcal {U}}}$ is called a reduced product.