Superreal number
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.[1]
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 [math]\displaystyle{ \R }[/math], so that F is not order isomorphic to [math]\displaystyle{ \R }[/math].
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]
References
- ↑ Tall, David (March 1980), "Looking at graphs through infinitesimal microscopes, windows and telescopes", Mathematical Gazette 64 (427): 22–49, doi:10.2307/3615886, http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot1980a-microscopes-etc.pdf
Bibliography
- Dales, H. Garth; Woodin, W. Hugh (1996), Super-real fields, London Mathematical Society Monographs. New Series, 14, The Clarendon Press Oxford University Press, ISBN 978-0-19-853991-9, http://www.oup.com/us/catalog/general/subject/Mathematics/PureMathematics/?view=usa&ci=9780198539919
- Gillman, L.; Jerison, M. (1960), Rings of Continuous Functions, Van Nostrand, ISBN 978-0442026912
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