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...

real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes...

is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, as typically construed, states that given...

mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Such fields will contain infinitesimal and...

numbers. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any Dedekind-complete ordered field is isomorphic...

mathematics Nested set collection Order polytope Ordered field – Algebraic object with an ordered structure Ordered group – Group with a compatible partial orderPages...

an ordered field, with the usual ordering ≥. The Artin–Schreier theorem states that a field can be ordered if and only if it is a formally real field, which...

an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of...

{Q} } ⁠ is an ordered field that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique...

involved. More generally, this applies for an ordered field. For more information, see § Ordered fields. The property for the additive inverse states...

rationals and reals in fact form ordered fields.) The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order...

well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible...

complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field 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 ordered field such that, in this...

they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that...

generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders. There are...

In statistics, the ordered logit model (also ordered logistic regression or proportional odds model) is an ordinal regression model—that is, a regression...

mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose...

include both hyperreal cardinal and ordinal numbers, which is the largest ordered field. Vladimir Arnold wrote in 1990: Nowadays, when teaching analysis, it...

states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. The theorem was conjectured...

nonstandard reals (usually denoted as *R), denote an ordered field that is a proper extension of the ordered field of real numbers R and satisfies the transfer...

fields that are widely used in number theory. They have absolute values which are very different from those of complex numbers. For any ordered field...

equipped with a (not necessarily unique) ordering that makes it an ordered field. The definition given above is not a first-order definition, as it requires...

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,...

an ordered field satisfying some version of the completeness axiom. Different versions of this axiom are all equivalent in the sense that any ordered field...

Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb Ordered field – Algebraic object with an ordered structure...

of scalars in an ordered field is considered. A subset C {\displaystyle C} of a vector space V {\displaystyle V} over an ordered field F {\displaystyle...

In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for...

In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite...

orders, numerous special kinds of ordered sets have been defined, some of which have grown into mathematical fields of their own. In addition, order theory...