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.
Formal definition.
Suppose "X" is a Tychonoff space, also called a T3 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 formula_1, so that "F" is not order isomorphic to formula_1.
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).