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 $\mathbb {R}$ , so that F is not order isomorphic to $\mathbb {R}$ .

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).

References

Tall, David (March 1980), "Looking at graphs through infinitesimal microscopes, windows and telescopes" (PDF), Mathematical Gazette, 64 (427): 22–49, CiteSeerX 10.1.1.377.4224, doi:10.2307/3615886, JSTOR 3615886, S2CID 115821551

Bibliography

Dales, H. Garth; Woodin, W. Hugh (1996), Super-real fields, London Mathematical Society Monographs. New Series, vol. 14, The Clarendon Press Oxford University Press, ISBN 978-0-19-853991-9, MR 1420859
Gillman, L.; Jerison, M. (1960), Rings of Continuous Functions, Van Nostrand, ISBN 978-0442026912

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Tall, David (March 1980), "Looking at graphs through infinitesimal microscopes, windows and telescopes" (PDF), Mathematical Gazette, 64 (427): 22–49, CiteSeerX 10.1.1.377.4224, doi:10.2307/3615886, JSTOR 3615886, S2CID 115821551

Number systems
Sets of definable numbers	Natural numbers ( $\mathbb {N}$ ) Integers ( $\mathbb {Z}$ ) Rational numbers ( $\mathbb {Q}$ ) Constructible numbers Algebraic numbers ( $\mathbb {A}$ ) Closed-form numbers Periods Computable numbers Arithmetical numbers Set-theoretically definable numbers Gaussian integers
Composition algebras	Division algebras: Real numbers ( $\mathbb {R}$ ) Complex numbers ( $\mathbb {C}$ ) Quaternions ( $\mathbb {H}$ ) Octonions ( $\mathbb {O}$ )
Split types	Over $\mathbb {R}$ : Split-complex numbers Split-quaternions Split-octonions Over $\mathbb {C}$ : Bicomplex numbers Biquaternions Bioctonions
Other hypercomplex	Dual numbers Dual quaternions Dual-complex numbers Hyperbolic quaternions Sedenions ( $\mathbb {S}$ ) Split-biquaternions Multicomplex numbers Geometric algebra/Clifford algebra Algebra of physical space Spacetime algebra Plane-based geometric algebra
Other types	Cardinal numbers Extended natural numbers Irrational numbers Fuzzy numbers Hyperreal numbers Levi-Civita field Surreal numbers Transcendental numbers Ordinal numbers p-adic numbers (p-adic solenoids) Supernatural numbers Profinite integers Superreal numbers Normal numbers
Classification List