superreal number
{{short description|Class of extensions of the real numbers}}
In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theory, and the study of Banach algebras. The field of superreals is itself a subfield of the surreal numbers.
Dales and Woodin's superreals are distinct from the super-real numbers of David O. Tall, which are lexicographically ordered fractions of formal power series over the reals.{{citation |first=David |last=Tall |title=Looking at graphs through infinitesimal microscopes, windows and telescopes |journal=Mathematical Gazette |volume=64 |issue=427 |pages=22–49 |date=March 1980 |jstor=3615886 |url=http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot1980a-microscopes-etc.pdf |doi=10.2307/3615886|citeseerx=10.1.1.377.4224 |s2cid=115821551 }}
Formal definition
Suppose X is a Tychonoff space and C(X) is the algebra of continuous real-valued functions on X. Suppose P is a prime ideal in C(X). Then the factor algebra A = C(X)/P is by definition an integral domain that is a real algebra and that can be seen to be totally ordered. The field of fractions F of A is a superreal field if F strictly contains the real numbers , so that F is not order isomorphic to .
If the prime ideal P is a maximal ideal, then F is a field of hyperreal numbers (Robinson's hyperreals being a very special case).{{citation needed|date=December 2020}}
References
== Bibliography ==
- {{Citation | last1=Dales | first1=H. Garth | last2=Woodin | first2=W. Hugh | title=Super-real fields | url=http://www.oup.com/us/catalog/general/subject/Mathematics/PureMathematics/?view=usa&ci=9780198539919 | publisher=The Clarendon Press Oxford University Press | series=London Mathematical Society Monographs. New Series | isbn=978-0-19-853991-9 |mr=1420859 | year=1996 | volume=14}}
- {{citation |first1=L. |last1=Gillman |first2=M. |last2=Jerison |title=Rings of Continuous Functions |publisher=Van Nostrand |year=1960 |isbn=978-0442026912 }}
{{Number systems}}
{{DEFAULTSORT:Superreal Number}}