ultraproduct
Sign in to saveAlso known as ultrapower
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.
~12 min read
Encyclopedic overview
10 sectionsContents
- Definition
- Ultrapower
- Examples
- Łoś's theorem
- Examples
- Direct limits of ultrapowers (ultralimits)
- Ultraproduct monad
- See also
- Notes
- References
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.
Excerpted from Wikipedia’s “ultraproduct” article, available under the CC BY-SA 4.0 licence.