In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, the superreal numbers are a class of extensions of the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
, introduced by
H. Garth Dales and
W. Hugh Woodin
William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, ...
as a generalization of the
hyperreal numbers and primarily of interest in
non-standard analysis,
model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
, 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
David Orme Tall (born 15 May 1941) is Emeritus Professor in Mathematical Thinking at the University of Warwick. One of his early influential works is the joint paper with Vinner "Concept image and concept definition, Concept image and concept def ...
, which are
lexicographically ordered fractions of
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
over the reals.
Formal definition
Suppose ''X'' is a
Tychonoff space and C(''X'') is the
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
of continuous real-valued functions on ''X''. Suppose ''P'' is a
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
in C(''X''). Then the
factor algebra ''A'' = C(''X'')/''P'' is by definition an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
that is a real algebra and that can be seen to be
totally ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
. 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
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains number ...
being a very special case).
References
Bibliography
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{{DEFAULTSORT:Superreal Number
Field (mathematics)
Real closed field
Infinity