In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard orderings.
Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Every Dedekind-complete ordered field is isomorphic to the reals. Squares are necessarily non-negative in an ordered field. This implies that the complex numbers cannot be ordered since the square of the imaginary unit i is −1 (which is negative in any ordered field). Finite fields cannot be ordered.
Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn. This grew eventually into the Artin–Schreier theory of ordered fields and formally real fields.
an orderedfield is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields...
real numbers form the unique (up to an isomorphism) Dedekind-complete orderedfield. Other common definitions of real numbers include equivalence classes...
numbers. Every orderedfield contains an ordered subfield that is isomorphic to the rational numbers. Any Dedekind-complete orderedfield is isomorphic...
is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typically construed, states that given two...
mathematics Nested set collection Order polytope Orderedfield – Algebraic object with an ordered structure Ordered group – Group with a compatible partial orderPages...
rationals and reals in fact form orderedfields.) The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order...
Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb Orderedfield – Algebraic object with an ordered structure...
{Q} } is an orderedfield that has no subfield other than itself, and is the smallest orderedfield, in the sense that every orderedfield contains a unique...
complete orderedfield that does not contain any smaller complete orderedfield. Such a definition does not prove that such a complete orderedfield exists...
first-order language of fields is true in F if and only if it is true in the reals. There is a total order on F making it an orderedfield such that, in this...
generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders. There are...
they form an orderedfield. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal orderedfield in the sense that...
equipped with a (not necessarily unique) ordering that makes it an orderedfield. The definition given above is not a first-order definition, as it requires...
nonstandard reals (usually denoted as *R), denote an orderedfield that is a proper extension of the orderedfield of real numbers R and satisfies the transfer...
In statistics, the ordered logit model (also ordered logistic regression or proportional odds model) is an ordinal regression model—that is, a regression...
include both hyperreal cardinal and ordinal numbers, which is the largest orderedfield. Vladimir Arnold wrote in 1990: Nowadays, when teaching analysis, it...
mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose...
article, only the case of scalars in an orderedfield is considered. A subset C of a vector space V over an orderedfield F is a cone (or sometimes called a...
In mathematics, a Euclidean field is an orderedfield K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for...
totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered set,...
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