What does ultraproduct mean?
Definitions for ultraproduct
ul·tra·prod·uct
This dictionary definitions page includes all the possible meanings, example usage and translations of the word ultraproduct.
Wiktionary
ultraproductnoun
A quotient of the direct product of a family of structures
ChatGPT
ultraproduct
An ultraproduct is a mathematical construction that forms a new object by combining a collection of objects from a given category along with a selected ultrafilter on the index set of the collection. Ultraproducts are commonly used in model theory and set theory to study properties of structures and to construct new structures with certain desired properties.
Wikidata
Ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient 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 non-standard analysis, which was pioneered by Abraham Robinson.
Numerology
Chaldean Numerology
The numerical value of ultraproduct in Chaldean Numerology is: 5
Pythagorean Numerology
The numerical value of ultraproduct in Pythagorean Numerology is: 7
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"ultraproduct." Definitions.net. STANDS4 LLC, 2024. Web. 26 Apr. 2024. <https://www.definitions.net/definition/ultraproduct>.
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