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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.〔David Tall, "Looking at graphs through infinitesimal microscopes, windows and telescopes," Mathematical Gazette, 64 22– 49, reprint at http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot1980a-microscopes-etc.pdf〕
==Formal Definition==
Suppose X is a Tychonoff space, also called a T3.5 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 which is a real algebra and which 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).
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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